review article qcd thermodynamics on the...

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2013 Article ID 452978 26 pageshttpdxdoiorg1011552013452978

Review ArticleQCD Thermodynamics on the Lattice

Sayantan Sharma

Fakultat fur Physik Universitat Bielefeld 33615 Bielefeld Germany

Correspondence should be addressed to Sayantan Sharma sayantanphysikuni-bielefeldde

Received 11 April 2013 Accepted 3 June 2013

Academic Editor Jan E Alam

Copyright copy 2013 Sayantan Sharma This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A remarkable progress has been made in the understanding of the hot and dense QCDmatter using lattice gauge theoryThe issueswhich are very well understood as well as those which require both conceptual and algorithmic advances are highlightedThe recentlattice results on QCD thermodynamics which are important in the context of the heavy ion experiments are reviewed Instancesof greater synergy between the lattice theory and the experiments in the recent years are discussed where lattice results could bedirectly used as benchmarks for experiments and results from the experiments would be a crucial input for lattice computations

1 Introduction

A large part of the visible matter in our universe is made outof protons and neutrons collectively called hadrons Hadronsaremade out ofmore fundamental particles called quarks andgluons The quantum theory for these particles is quantumchromodynamics (QCD) QCD is a strongly interacting the-ory and the strength of interaction becomes vanishinglysmall only at asymptotically high energies Due to this reasonthe quarks and gluons are not visible directly in ourworld andremain confinedwithin the hadrons Lattice gauge theory hasemerged as the most successful nonperturbative tool to studyQCD with very precise lattice results available for hadronmasses and decay constants which are in excellent agreementwith the experimental values [1]

It is expected that at high enough temperatures thatexisted in the early universe the hadrons would melt intoa quark gluon plasma (QGP) phase Signatures of such aphase have been seen during the last decade in the heavyion collision experiments at the relativistic heavy ion col-lider (RHIC) in Brookhaven National Laboratory This isparticularly exciting for the lattice theory community whichhas been predicting such a phase transition since a longtime [2] The formation of the QGP phase occurs at tem-peratures near ΛQCD where QCD is strongly interactingwhich means lattice is the most reliable tool to understandthe properties of the hot QCD medium Over the past threedecades the lattice community has contributed significantly

to the understanding of the physics of heavy ion experimentsand strongly interacting matter under extreme conditions ingeneral Lattice computations are entering into the precisionregimewhere lattice data can be directly used for interpretingthe experimental results and set benchmarks for the heavyion experiments at RHIC and at the ALICE facility in CERNIt is now generally believed that the hot and dense mattercreated due to the collision of two heavy nuclei at RHICand ALICE equilibrates within 1 fmc of the initial impactThe equilibrated QGPmedium then expands and cools downin the process ultimately forming hadrons at the chemicalfreezeout The evolution of the fireball from its equilibra-tion till the chemical freezeout is described by relativistichydrodynamics [3] The QCD Equation of State (EoS) isan input for the hydrodynamic equations and lattice canprovide a nonperturbative estimate of this quantity from firstprinciplesThe lattice data for the speed of sound in the QCDmedium is also an important input for the hydrodynamicstudy once bulk viscosity is considered

In this paper I have selected the most recent results fromlattice QCD thermodynamics that are relevant for the heavyion phenomenology I have tried to review the necessarybackground but not attempted to provide a comprehensiveaccount of the development of the subject throughout theseyears I have divided this paper into two major sectionsThe first section deals with QCD at finite temperature andzero baryon density where lattice methods are very robustI have given a basic introduction to the lattice techniques

2 Advances in High Energy Physics

and how the continuum limit is taken which is essential torelate the lattice data with the real world experiments I havediscussed the current understanding we have of the nature ofQCD phase transition as a function of quark masses inferredfrom lattice studies Subsequently the different aspects of thehot QCD medium for physical quark masses are discussedthe EoS the nature and the temperature of transition andthe behaviour of various thermodynamic observables in thedifferent phases In the study of thermodynamics the contri-bution of the lighter 119906 119889 and 119904 quarks is usually consideredThe effect of heavier charm quarks onQCD thermodynamicsis discussed in this section in view of their relevance forthe heavy ion experiments at LHC where hydrodynamicevolution is expected to set in already at temperatures about500MeV and also for the physics of early universe Therelevance of chiral symmetry for theQCDphase diagram andthe effects of chiral anomaly are discussed in detailThe chiralanomaly is believed to have an important role in shaping thephase diagram and several lattice studies in the recent yearsare trying to understand its effect It is a difficult problemand I have tried to compile the recent results and review thegeneral understanding within the community about how toimprove upon them

The second section is about lattice QCD at finite densitywhere there is an inherent short coming of the lattice algo-rithms due to the so-called sign problem A brief overviewof the different methods used and those being developed bythe lattice practitioners to circumvent this problem is givenIt is an active field of research with a lot of understandingof the origin and the severity of this problem gained inrecent years which is motivating the search for its possiblecure In the regime where the density of baryons is not toolarge which is being probed by the experiments at RHIClattice techniques have been used successfully to producesome interesting results One such important proposal inthe recent time is the first principles determination of thechemical freezeout curve using experimental data on theelectric charge fluctuations This and the lattice results onthe fluctuations of different quantum numbers in the hotmedium and the EoS at finite baryon density are discussedin detail An important feature of the QCD phase diagramis the possible presence of a critical end-point for thechiral first order transition Since critical end-point searchis one of the main objectives at RHIC I have reviewedthe current lattice results on this topic The presence ofthe critical end-point is still not conclusively proven fromlattice studies It is a very challenging problem and I mentionthe further work in progress to address this problem effec-tively Fermions with exact chiral symmetry on the latticeare important in this context I have discussed the recentsuccessful development to construct fermion operators thathave exact chiral symmetry even at finite density whichwould be relevant for future studies on the critical end-pointThe signatures of the critical end-point could be detectedin the experiments if the critical region is not separatedfrom the freezeout curve It is thus crucial to estimate thecurvature of the critical line fromfirst principles and I devotean entire subsection to discuss the lattice results on thistopic

I apologize for my inability to include all the pioneeringworks that have firmly established this subject and also toreview the extensive set of interesting contemporary worksFor a comprehensive review of the current activity in latticethermodynamics at finite temperature and density I refer tothe excellent review talks of the lattice conference 2012 [4 5]

2 QCD at Finite Temperature on the Lattice

The starting point of any thermodynamic study is the parti-tion functionThe QCD partition function for119873119891 flavours ofquarks in the canonical ensemble is given as

ZQCD (119879 119881) = intD119880120583 (119909)

119873119891

prod

119891=1

det119863119891119890minus119878119866 (1)

where 119863119891 is the fermion operator for each flavour ofquark 119891 119880120583 is the gauge link defined as 119880120583(119909) =

exp(minus119894119892 int119909119860120583(119909

1015840)119889119909

1015840) in terms of gauge fields 119860120583 which

are adjoint representation of the SU(3) color group and 119892

is the strength of the gauge coupling 119878119866 is the gluon actionin Euclidean space of finite temporal extent of size denotedby the inverse of the temperature of the system 119879 LatticeQCD involves discretizing the spacetime into a lattice with aspacing denoted by 119886The volume of the lattice is given as119881 =

11987331198863 where119873 are the number of lattice sites along the spatial

directions and the temperature being 119879 = 1(119873120591119886) where119873120591 are the number of sites along the temporal direction Thelattice is usually denoted as 1198733

times 119873120591 The gluon action andthe fermion determinant are discretized on the lattice Thesimplest gluon action known asWilson plaquette action is ofthe form

119878119866 =6

1198922sum

119909120583]120583lt]

(1 minus1

3Tr Re119880120583] (119909))

119880120583] (119909) = 119880120583 (119909)119880] (119909 + 120583)119880dagger120583 (119909 + ]) 119880dagger

] (119909)

(2)

where 119880120583](119909) is called a plaquette The naive discretizationof the continuum Dirac equation on the lattice results in thefermion operator of the form

119863119891 (119909 119910)

= sum

119909119910

[

4

sum

120583=1

1

2120574120583 (119880120583 (119909) 120575119910119909+120583 minus 119880

dagger120583 (119910) 120575119910119909minus120583) + 119886119898119891120575119909119910]

(3)

where in each of the expressions the site index 119909 = 1 minus 1198733times

119873120591 The discretization of the gluon and fermion operatorsare not unique and there are several choices which givethe correct continuum limit Usually discretized operatorswith small finite 119886 corrections are preferred Reducing 119886-dependent corrections by adding suitable ldquoirrelevantrdquo termsin the Renormalization Group (RG) sense is known asthe improvement of the operator Another issue relatedto the discretization of the fermion operator is called theldquofermion doubling problemrdquo It arises because the naive

Advances in High Energy Physics 3

discretization of the continuum fermion operator introducesextra unphysical fermion species called the doublers Theexistence of the doublers can be traced back to a No-Gotheorem [6] on the lattice which states that fermion actionswhich are ultralocal have exact chiral symmetry and have thecorrect continuum limit cannot be free from the doublersDoublers are problematic since in the continuum limit wewould get a theory with 16 fermion species and QCD with16 flavours which is very close to the upper bound of thenumber of flavours beyond which the asymptotic freedomis lost It is thus important to ensure that the discretefermion operator should be free of the doublers In orderto do so the chiral symmetry is explicitly broken on thelattice like for the case of Wilson fermions [7] or only aremnant of it is preserved for the staggered fermions [8]The staggered fermion discretization retains the doublingproblem in a milder form In the continuum limit thestaggered fermion determinant would give contribution offour degenerate fermion species or tastes However on a finitelattice there is a considerable mixing among the tastes so asimple fourth root of the determinant would not yield thecontribution of a single fermion flavour This is called therooting problem The severity of rooting problem can beminimized by choosing either the stout-smeared staggeredquarks [9] or the highly Improved staggered suarks (HISQ)[10] Other improved versions of staggered fermions usedfor QCD thermodynamics are the p4 and asqtad fermions[11ndash14] Only the overlap [15 16] and the domain wallfermions [17] have exact chiral symmetry on the lattice atthe expense of breaking the ultralocality condition of theNielsen-Ninomiya No-go theorem As a result overlap anddomain wall fermions are much more expensive to simulatecompared to the staggered and the Wilson fermions ForQCD thermodynamics the staggered and to some extent theWilson fermions are favourites with very high precision dataavailable with improved versions of staggered quarks [18 19]With the advent of faster computing resources and smarteralgorithms even large scale simulations with chiral fermionsare becoming a reality [20ndash23]

With the choice of a suitable gauge and the fermionopera-tors on the lattice different physical observables aremeasuredon statistically independent configurations generated usingsuitable Monte-Carlo algorithms To make connection withthe continuum physics one needs to take the 119886 rarr 0 limit ofthe observables measured on the lattice The gauge couplingis related to the lattice spacing through the beta-function andthe continuum limit in turn implies 119892 rarr 0 In the space ofcoupling constants and the fermion masses the continuumlimit is a second order fixed point and the approach to thefixed point should be done along the correct RG trajectoryor the lines of constant physics The line of constant physicsis defined by setting the mass of hadrons on the lattice to thecontinuum values at each value of the coupling constantThenumber of such relations required depends on the numberof fermion flavours To relate the lattice hadron masses totheir experimental values one has to define a scale to expressthe lattice spacing 119886 in terms of some physical units Thereare two often used methods in QCD to set the scale usingthe quantities 1199031 and the kaon decay constant 119891119870 The 1199031

scale is defined from the quark-antiquark potential 119881119902119902(119903)

as

(1199032120597119881119902119902 (119903)

120597119903)

119903=1199031

= 10 (4)

On the lattice one measures 119881119902119902(119903) and 1199031 is extracted fromit using a suitable fit ansatz for the potential To quantifythe value of 1199031 in physical units one uses either the piondecay constant or the splitting of energy levels of bottommesons to set the lattice spacing [24] Advantage of thisscale is that it is not sensitive to fermion discretizationeffects and to the choice of quark masses that defines theline of constant physics However the accurate determi-nation of the potential requires very good statistics Onecan also set the scale by choosing the 119891119870 measured onthe lattice to its physical value The 119891119870 is known withvery high accuracy from the experiments Once the line ofconstant physics is set one has to take care of the finitesize and lattice spacing effects such that the continuumextrapolation is correctly performed To minimize suchcorrections the correlation length which is given by theinverse of the mass of the lowest excitation of the systemshould be much larger that the lattice spacing but sufficientlysmaller than the spatial size Also for thermodynamicsit is crucial to minimize finite volume corrections whichis ensured for the choice 120577 ge 3 where 120577 = 119873119873120591

To characterize different phases one needs to define asuitable order parameterwhich depends on the symmetries ofthe theory In the limit of infinitely heavy quarkmasses QCDis just a pure gauge theory with an exact order parameter theexpectation value of the Polyakov loop is given as

119871 (x) = 1

3Tr 119875

119873120591

prod

1199094=1

1198804 (x 1199094) 119875 997904rArr path ordering (5)

The phase transition from a phase of confined colour degreesof freedom to the deconfined regime of free gluons isof first order and is established very firmly from latticestudies [25] The corresponding transition temperature is119879119888 (pure gauge) = 276(2)MeV [26] using string tensionradic120590 value to be 425MeV to set the scale If the quarksare massless the QCD partition function with 119873119891 quarkflavours has an exact SU(119873119891) otimes SU(119873119891) chiral symmetryAt some temperature there is a phase transition from achiral symmetry broken phase to the symmetry restoredphase characterized by the order parameter called the chiralcondensate

⟨120595119891120595119891⟩ = lim119898119891rarr0

lim119881rarrinfin

119879

119881

120597 lnZQCD

120597119898119891

119891 = 1 119873119891

(6)

The phase transition in the chiral limit for 119873119891 = 3 isexpected to be of first order and there are several latticeresults supporting this [27ndash31] For119873119891 = 2 the lattice resultsare contradictory with some claiming a first order transition[32 33] whereas recent results showing that the second ordertransition is also a possibility [34] The current status of


119873119891 = 2 QCD phase transition in the chiral limit would bediscussed again in a later subsection For any finite value ofquark masses however there is no unique order parameterand no sharp phase transition is expected but only a gradualcrossover

Based on effective field theories with same symmetriesas QCD using universality arguments and renormalizationgroup inspired techniques a schematic diagram of differentphases of QCD as a function of quark mass is summarizedin the famous ldquoColumbia plotrdquo [35] The first order regionsin the quenched and the chiral limits are separated from thecrossover region by second order lines belonging to the 119885(2)universality class These boundaries are schematic thoughand it is important to estimate the precise location of thephysical point in this diagram Lattice studies over the yearshave helped to redraw the boundaries more quantitatively Alatest version of the ldquoColumbia plotrdquo is shown in Figure 1With the high precision lattice data with physical lightand strange quark masses it is now known that the QCDtransition in our world is a crossover [36ndash38] The boundaryof the first order region in the upper right hand corner ofFigure 1 is fairly well known [39]The extent of the first orderregion in the bottom left hand is now believed to be smalland much far away from the physical point [40 41] Howeverthe extent of the 119885(2) line in the left hand corner is still notwell established it can either continue along the 119898119906119889 = 0

axis to the 119898119904 rarr infin corner or end at a tricritical point Abetter understanding of this issue is currently underway Thekey to the resolution of this issue is to understand the effectsof chiral anomaly through rigorous lattice computationsSince the light 119906 119889-quark masses are much smaller thanΛQCD the QCD action has an approximate SU(2) times SU(2) times119880119861(1) symmetry with an additional classical119880119860(1) symmetrybroken explicitly by quantum effects This is known as the119880119860(1) anomaly [42ndash44] At zero temperature the magnitudeof this anomaly is related to the instanton-density If themagnitude of this anomaly is temperature independent thephase transition along the 119898119906119889 = 0 axes has to be of secondorder belonging to the 119874(4) universality class [45] Thiswould mean that the 119885(2) line has to end at a tricritical pointcharacterized by the strange quark mass119898tric

119904 The differencebetween the physical and tricriticalmass for the strange quarkis not yet known with a good precision

In the following subsections the lattice results for theQCD EoS for physical quark masses are discussed which isan input for the hydrodynamics of the QGP medium Thecurrent results on the pseudocritical temperature the entropydensity and the speed of sound are also shown All the resultsare for 2 + 1 flavour QCD that is two light degenerate 119906

and 119889 quarks and a heavier strange quark mass The effect ofthe heavy charm quarks on the thermodynamic quantities isalso highlighted At the end of this section I touch upon the119873119891 = 2QCD near the chiral limit and the effects of the119880119860(1)

anomaly for QCD thermodynamics

21 Equation of State The Equation of State (EoS) is therelation between the pressure and energy density of a systemin thermal equilibrium For estimating the QCD EoS themost frequently used method by the lattice practitioners is

Crossover

Physical pointms

Nf = 1

Nf = 3

Pure gauge

Firstorder

Firstorder

Second order

infin

infin

Nf = 2

mc = (ms270 ms270)

O(4)Z(2)

Z(2) secondorder line


mud

mtrics

Figure 1 The present status of the Columbia plot

the integral method [46] In this method one first computesthe trace anomaly 119868(119879) which is the trace of the energy-momentum tensor This is equal to the quantity 120598 minus 3119901 where120598 is the energy density of the system and 119901 is the pressureMoreover it is related to the pressure of the system throughthe following relation

119868 (119879) = 1198795 120597

120597119879

119901

1198794 (7)

So if 119868(119879) is known the pressure can be computed byintegrating 119868(119879) over a range of temperature with the lowervalue of temperature chosen such that the correspondingvalue of pressure is vanishingly small The trace anomaly isrelated to the chiral condensate and the gluon action as

119868 (119879)

1198794= minus119873

4120591 (119886

119889120573

119889119886(⟨119878119866⟩ minus ⟨119878119866⟩0)

+ sum

119891

119886

119889 (119898119891119886)

119889119886(⟨120595119891120595119891⟩ minus ⟨120595119891120595119891⟩0

))

120573 =6

1198922

(8)

where the subscript zero denotes the vacuum expectationvalues of the corresponding quantities The subtraction isnecessary to remove the zero temperature ultraviolet diver-gences and the vacuum expectation values are usually com-puted on a lattice with number of sites (119873120591)0 in the temporaldirection equal to the corresponding spatial number of sites119873The subtraction is an unavoidable expense of this methodA new idea of deriving thermodynamic observables fromcumulants of momentum distribution has emerged wherethe vacuum subtraction is not required [47] and it would beinteresting to check the application of this method in QCDAlso one needs to know the functional dependence of theinverse of QCD coupling constant 120573 and the quark masseswith the lattice spacing 119886 along the line of constant physics


1

2

3

4

5

6

150 200 250 300 350

(120576minus3p

)T4

T (MeV)

Stout cont

r1 scale

N120591 = 6

N120591 = 8

N120591 = 10

N120591 = 12

HISQtreeN120591 = 4

(a)

05

1

15

2

25

3

35

4

300 400 500 600 700 800

(120576minus3p

)T4

T (MeV)

N120591 = 6

N120591 = 8

N120591 = 6N120591 = 8

N120591 = 10 Stout cont

HISQtreeN120591 = 4 p4 N120591 = 4

(b)

Figure 2The results for the trace anomaly using the HISQ action for low (a) and high (b) temperatures for lattice sizes with temporal extent119873120591 and spatial size 4119873120591 from [48] Also in (b) the HISQ results are compared to the results using p4 fermions which has an improvedbehaviour at high temperatures and to the continuum perturbation theory results at 1-loop (solid line) and 2-loop (dashed line) for the traceanomaly The stout data are the continuum estimates from the119873120591 = 6 8 10 data in [19]

5

4

3

2

1

0

(120576minus3p

)T4

100 150 200 250 300 350 400 450 500 550T (MeV)

Continuum363 times 6323 times 8

323 times 10

323 times 12HRG model

(a)

4

3

2

1

2

1

I(T)T4

0 200 400 600 800 1000T (MeV)

Cont estParametrization

0 50 100 150

HRG

(b)

Figure 3 The latest data with the stout smeared fermions (a) from [50] In (b) the fit to the trace anomaly data from the continuumextrapolation of the 119873120591 = 6 8 results from [19] The results are in perfect agreement with the Hadron resonance gas model calculationsfor 119879 lt 140MeV

On the lattice 119868(119879) is known only for a finite number oftemperature valuesThe pressure computed by the numericalintegration of the 119868(119879) data has errors both due to statisticalfluctuations and systematic uncertainties involved in thenumerical interpolation of the data

The results for the trace anomaly are available for differentlattice discretizations of the fermions For staggered quarks

there are two sets of results one from theHotQCD collabora-tion usingHISQdiscretization [48 49] and the other from theBudapest-Wuppertal collaboration using stout smeared stag-gered quarks [19 50] These results are compiled in Figures 2and 3 For the HISQ results the bare lattice parameters arefixed by setting the lowest strange pseudoscalar meson massto its physical value at about 686MeV and 119898120587 = 160MeV


which defines the line of constant physics The kaon decayconstant 119891119870 = 1561MeV or alternatively the 1199031 = 03106 fmfrom the static quark potential is used to set the scale Thecorresponding parameters for the stout smeared quarks are119898120587 = 135MeV119898119870 = 498MeV and the kaon decay constantFrom Figure 2 it is evident that there is a good agreementbetween the two sets of results for 119879 lt 180MeV and alsofor high enough temperatures 119879 gt 350MeV The stoutcontinuum results in the figure were obtained extrapolationwith the 119873120591 = 6 8 10 data from [19] In the intermediatetemperature range there is some discrepancy specially thepeaks of the interaction measure do not coincide for thesetwo different discretization schemes which may be due tofinite lattice spacing effects However the HISQ 119873120591 = 12

data is inching closer to the stout results in this regime Therecent continuum stout results obtained from continuumextrapolation of the new 119873120591 = 12 data in addition to theolder data are consistent with theHISQ results with the peakposition shifting to 200MeV (Figure 3(a)) There is also agood agreement of the HISQ and stout data with the traceanomaly obtained from the Hadron Resonance Gas (HRG)model for119879 lt 140MeVandwith the resummed perturbationtheory results at high temperatures Using the 119873120591 = 6 8

data which is available upto temperatures of 1000MeV acontinuum extrapolation of the stout data was performed theresult ofwhich is shown in Figure 3(b) For this entire range oftemperature there is a useful parameterization characterizingthe trace anomaly [19] with the following parametric form

119868 (119879)

1198794= 119890

minusℎ1119905minusℎ21199052

sdot (ℎ0 +1198910 [tanh (1198911119905 + 1198912) + 1]

1 + 1198921119905 + 11989221199052

)

119905 =119879

200MeV

(9)

where the best fit parameters are

ℎ0 = 01396 ℎ1 = minus018 ℎ2 = 0035

1198910 = 276 1198911 = 679 1198912 = minus529

1198921 = minus047 1198922 = 104

(10)

This parametric form could be a useful input for the hydrody-namical simulations which usually uses the lattice EoS beforehadronization and that from the HRG after the freezeout ofhadrons

There are lattice results for the EoS using alternativefermion discretizations the Wilson fermions The WHOT-QCD collaboration has results for 2 + 1 flavours of improvedWilson fermions [51] with the physical value of strange quarkmass but a large pion mass equal to 063119898120588 The tmfTcollaboration has results for 2 flavours of maximally twistedWilson fermions [52] with119898120587 gt 400MeV Both these resultsare compiled in Figure 4 These are in rough qualitativeagreementwith the staggered fermion data specially the peakfor theWHOT-QCD data occurring at 200MeV is consistentwith the HISQ and stout results A more quantitative agree-ment at this stage is difficult since the pion masses for theWilson fermions are much larger than the physical value

22 The Pseudocritical Temperature We recall that the QCDtransition from a phase of color singlet states to a phase ofcolored quantum states is an analytic crossover for physicalquark masses This is fairly well established by now from lat-tice studies using two different approaches One approach isto monitor the behaviour of the thermodynamic observablesin the transition region for physical values of quark masseswhile the other is to map out the chiral critical line as afunction of light quark mass [53] The absence of a sharpphase transition implies that there is no unique transitiontemperature but only different pseudocritical temperaturescorresponding to different observables There is no orderparameter but the observables like the renormalizedPolyakovloop 119871119877 has a point of inflexion across the crossover regionAnother observable relevant in the crossover regime is therenormalized chiral condensate which has been defined[54] in the following manner to take into account themultiplicative renormalization as well as additive ones due toa finite bare quark mass

Δ 119897119904 (119879) =

⟨120595120595⟩119897119879 minus (119898119897119898119904) ⟨120595120595⟩119904119879

⟨120595120595⟩1198970 minus (119898119897119898119904) ⟨120595120595⟩1199040

119897 = 119906 119889 (11)

The normalized chiral susceptibility 120594119877 for the light quarksdefined as

120594119877 =1

1198811198793119898

2119897

1205972

1205971198982119897

(lnZ (119879) minus lnZ (0)) (12)

is a good observable as well Both 119871119877 and Δ 119897119904(119879) have apoint of inflexion at the pseudocritical temperature and 120594119877

has a smooth peak From the continuum extrapolated dataof the stout-smeared staggered fermions the pseudocriticaltemperatures corresponding to these observables for physicalquark masses are

119879119888 =

170 (4) (3) for 119871119877

157 (3) (3) Δ 119897119904

147 (2) (3) 120594119877

(13)

The data for 119871119877 and Δ 119897119904 with the HISQ discretization isshown in Figure 5 These are for lattices of size119873120591 times (4119873120591)

3The HISQ data are in good agreement with the continuumextrapolated stout-smeared staggered results from [55] Thefact that the rise of 119871119877 is more gradual than the corre-sponding rise of Δ 119897119904 signals that the crossover is morelikely influenced by the chiral symmetry restoration Previousscaling studies of the renormalized chiral condensate withthe p4-staggered quarks showed that the physical light quarksalready approximate the 119874(4) critical behaviour of the chiralquarks [34] Using the119874(4) scaling of the renormalized chiralcondensate the 119879119888 obtained for HISQ quarks through chiraland continuum extrapolation is 154 plusmn 9MeVThis value is inexcellent agreement with the stout result implying that thecontinuum extrapolation done with the staggered fermionsis quite robust

23 Comparing Results for Different Fermion DiscretizationsThe results for the EoS and the pseudocritical temperature


0

5

10

15

20

120576T4

700500 600300 400100 200T (MeV)

3pT4

(120576 minus 3p)T4

(a)

900700500300100

10

8

6

4

2

0

minus2

T (MeV)

Interpolation

N120591 = 4N120591 = 6N120591 = 8

N120591 = 10

N120591 = 12

(120598minus3p

)T4

(b)

Figure 4The results for the pressure energy density and the trace anomaly with clover-improvedWilson fermions on a 323 times8 lattice from[51] (a) and the trace anomaly data with the twisted mass Wilson fermions from [52] (b)

1

08

06

04

02

0

Δls

fK scale

T (MeV)120 140 160 180 200

AsqtadN120591 = 8

N120591 = 12

HISQtreeN120591 = 6

N120591 = 8

N120591 = 12

N120591 = 8 ml = 0037ms

Stout cont

(a)

fK scale

T (MeV)120 140 160 180 200

04

035

03

025

02

015

01

005

0

Lre

n(T

)

HISQtreeN120591 = 6

N120591 = 8

N120591 = 12

AsqtadN120591 = 8

N120591 = 12

Stout cont

(b)

Figure 5The results for the subtracted chiral condensate (a) and the renormalized Polyakov loop (b) from the HotQCD collaboration from[49] These data are compared with the continuum results using stout smeared fermions from [55]

discussed so far have been obtained using different improvedversions of the staggered quarks For these fermion speciesthe so called ldquorootingrdquo problemmay alter the continuum limitdue to breaking of the119880119860(1) anomaly [56] though some otherwork refutes this claim [57] It is important to check the effectsof the rooting procedure on the continuum extrapolationof finite temperature observables The Budapest-Wuppertal

collaboration has recently compared the continuum extrap-olated results for different observables using the Wilson andstaggered fermions [58] as the former discretization does notsuffer from the rooting problem The scale for the Wilsonfermions was determined using 119898Ω = 1672MeV and theline of constant physics was set using 119898120587119898Ω sim 03 and119898119870119898Ω sim 036 For the staggered quarks the line of constant


125 150 175 200 225 250 275(MeV)

0005

0

minus0005

minus001

minus0015

minus002

minus003

minus0025

minus0035008 01 012 014 016

TmΩ

Staggered continuumWilson continuum

mR

R120595Rm

1205874

120595

(a)

150 175 200 225 250 275(MeV)

008 01 012 014 016TmΩ


2

15

1

05

0

LR

(b)

Figure 6 The continuum extrapolated renormalized chiral condensate (a) and the Polakov loop (b) are compared for Wilson and stout-smeared staggered fermions from [58]

physics was set such that the ratios 119898120587119898Ω and 119898119870119898Ω

are within 3 of the corresponding values for the Wilsonfermions This means that the pions are quite heavy with119898120587 sim 540MeV for both these discretizationsThe continuumextrapolated results for 119871119877 and the renormalized chiral con-densate are shown in Figure 6The continuumresults for boththese quantities are in good agreement for the whole rangeof temperature implying that these two different fermiondiscretizations indeed have the correct continuum limit Inall these computations an improved Wilson operator wasused in which the dominant O(119886) correction terms due toexplicit breaking of chiral symmetry by these fermions werecancelled It ensured that in both the studies the approachto the continuum limit was chosen to be the same Howeverat this large value of quark masses the rooting problem maybe mild enough to show any adverse effects and it would bedesirable to perform a similar comparison at physical valueof the quark masses

Since the effects of chiral symmetry persist in the crosso-ver region it is important to compare the existing results for119879119888 with those using fermions with exact chiral symmetry onthe lattice For the Wilson and the staggered action even formassless quarks the full SU(2) otimes SU(2) chiral symmetry isrealized only in the continuum limit For chiral fermions onthe lattice like the overlap or the domain wall fermions thechiral and the continuum limits are disentangled allowingus to understand the remnant effects of chiral symmetry inthe crossover region even on a finite lattice However latticeQCD with overlap fermions is computationally prohibitive[59] and currently better algorithms are being developedto simulate them with comparatively lesser effort [60] Thedomain wall fermions have exact chiral symmetry only whenthe extent of the fifth dimension1198735 of the five dimensionallattice on which these fermions are defined is infiniteFor smooth gauge fields the chiral symmetry violation on

a finite lattice is suppressed as an exponential of 1198735 but thesuppression could be much slower as 11198735 for rough gaugeconfigurations in the crossover region Better algorithms havebeen employed to ensure exponential suppression even forrough gauge fields [61]Themost recent results for the overlapfermions from the Budapest-Wuppertal collaboration [21]and the domain wall fermions from the HotQCD collabo-ration [61] are shown in Figure 7 The renormalized chiralcondensate for the overlap fermions is qualitatively consistentwith the continuum staggered fermion results even for smallvolumes and large pion masses of about 350MeV aroundthe crossover region The lattice cut-off effects seem to bequite small for 119873120591 = 8 The renormalized chiral condensateand the Δ 119897119904 for the domain wall fermions are shown inFigure 7 The lattice size is 16

3times 8 with the number of

lattice sites along the fifth dimension taken to be 32 for119879 gt 160MeV and 48 otherwise and the pion mass is about200MeVThe lattice volume is comparatively small thereforethese results do not show a sharp rise in the crossoverregionWith larger volumes the rise in these thermodynamicquantities is expected to be much steeper The value of 119879119888

estimated from the peak of the chiral susceptibility that is thederivative of the chiral condensate is between 160ndash170MeVwhich is consistentwith the continuumresults from theHISQfermions

24TheThermodynamical Observables Thermodynamic ob-servables characterize the different phases across a phasetransition From the behaviour of these observables one caninfer about the degrees of freedom of the different phasesand the nature of the interactions among the constituents Itwas already known from an important lattice study that thepressure in high temperature phase of QCD showed a strongdependence on the number of quark flavours [62] signalingdeconfinement of the quark and gluon degrees of freedom


001

0

minus001

minus002

minus003

minus004

minus005

minus006

minus007

minus008

120 140 160 180 200 220 240 260T (MeV)

01 012 014 016 018 02 022 024

6 times 123

8 times 163

Staggered

Tw0

mR120595120595Rm

1205874

(a)

0002

00015

0001

00005

0140 150 160 170 180 190 200

T (MeV)

12059511205951T3

Δ lsT3

(b)

Figure 7 The renormalized chiral condensate for the overlap quarks is compared to the continuum extrapolated results using the stoutsmeared staggered quarks in (a) from [21] In (b) the behaviour of different chiral condensates defined using the domain wall fermions isshown in the critical region from [61]

Recent results for the pressure entropy density and the speedof sound for QCD using the stout-smeared staggered quarksare compiled in Figure 8Though in our world there is no realphase transition the entropy density increases rapidly withtemperature again signaling the liberation of a large numberof colour degrees of freedom The entropy density for QCDis almost 20 off from the value of a free gas of quarks andgluons even at temperatures about 1000MeV The deviationof the pressure of QGP computed at similar temperaturesfrom its free theory value is even more close to about 25 ofits value Another observable that characterizes the differentphases is the speed of sound 119888119904 If QGP at high temperatureswas qualitatively close to a strongly interacting conformaltheory then the speed of sound would be exactly 1radic3However the deviation from conformality is quite significanteven at temperatures about 119879 = 500MeV which hints thatthe AdS-CFT inspired study of the QGP medium should bedone withmore careThe values of entropy density computedwith different discretizations of staggered fermions like theasqtad or the p4 fermions [63] show about 10 deviationfrom the free theory value at very high temperatures Thedeparture from AdS-CFT values is even more severe usingthese fermions The stout results are about 10 lower thanthe corresponding asqtad and p4 results This deviation isattributed to the fact that the latter discretizations havesmaller cut-off effects at higher temperatures and would bemore closer to the continuum results The stout continuumvalues shown in the figure were obtained by averaging the119873120591 = 8 10 data A proper continuum extrapolation ofthe results for both the fermion discretizations is necessaryfor resolving the difference and for use of these values forthe real world calculations However the lattice results withat least 10 off from the free theory values even at very

high temperatures implies that the QGP phase is stronglyinteracting more like a liquid rather than a gas of quarksand gluons confirming the similar prediction from theRHIC experiments For 119879 lt 119879119888 the results for all theseobservables are in agreement with Hadron resonance gasmodel predictions

25 Effects of Charm Quarks on the EoS The effects of charmquarks to the pressure in the QGP phase were estimatedsometime ago using next-to leading order perturbationtheory [64] It was observed that the contribution of charmquarks becomes significant for temperatures 119879 gt 2119879119888 Pre-liminary data from the LHC already indicates that the charmquarks would thermalize quickly as the lighter quarks Itwould then affect the EoS and thus the hydrodynamicalevolution of the fireball formed at LHC energies Latticestudies are important to quantify the contribution of charmto the EoS in the QGP phase The first lattice studies weredone by the RBC [65] as well as the MILC collaboration [66]with quenched charm quarks that is by neglecting quantumfluctuations due to the charm quarks The quenched charmresults for the EoS differ from the 2 + 1 flavour resultsalready at 12119879119888 Recent results from the Budapest-Wuppertalcollaboration with dynamical charm quarks [50] howevershow that the effects of charm quarks show up only around300MeV more in agreement with the perturbative estimates(Figure 9) Both the approaches highlight the fact that theeffects of charm quark should be considered for the EoS usedas an input for the hydrodynamical evolution of the fireball atLHC energies which may set in at 119879 sim 500MeV It would bealso important for the EoS of the standard model importantfor the cosmological evolution in the early universe [67 68]


200 400 600 800 1000T (MeV)

20

15

10

5

15

10

5

SB

100 150 200 250

s(T)T3

N120591 = 6N120591 = 8N120591 = 10

(a)

200 400 600 800 1000T (MeV)

5

4

3

2

1

SB

100 150 200 250

25215105

p(T

)T4

N120591 = 6N120591 = 8N120591 = 10

(b)

c2 s(T

)

200 400 600 800 1000T (MeV)

035

03

025

02

015

01

035030250201501

SB

100 150 200 250 300

N120591 = 6N120591 = 8N120591 = 10

(c)

Figure 8 The entropy density pressure and the speed of sound for the stout-smeared fermions as a function of temperature from [19]

26 The 2 Flavour QCD Transition and the Fate of the 119880119860(1)

Anomaly The chiral phase transition for119873119891 = 2QCD is stillnotwell understood from lattice studies aswas emphasized atthe beginning of this sectionThough the lattice results for 2+1 flavours with different fermion discretizations are in goodagreement the corresponding ones for the two light flavourcase are still inconclusive Two major approaches have beenundertaken in the recent years to understand the order ofthis transition One of them is to check the scaling propertiesof the order parameter If the phase transition is indeed asecond order one then the order parameter would show119874(4)

scaling in the transition region The second approach is tounderstand the effects of the 119880119860(1) anomaly near the phasetransition If the quantum fluctuations responsible for this119880119860(1) anomaly decrease significantly with temperature itwould result in the degeneracy of the masses of mesons ofcertain quantum numbers and a characteristic behaviour ofthe density of low lying eigenmodes of the fermion operatorI discuss themajor lattice results using both these approaches

in the following paragraphs Most of these approaches arehinting that the two flavour chiral phase transition may bea second order one

261 Scaling Analysis in the Critical Region The order param-eter that characterizes the chiral phase transition is the chiralcondensate A suitable dimensionless definition of the chiralcondensate used in the lattice study by the BNL-Bielefeldcollaboration [34] is

119872119887 = 119898119904

⟨120595120595⟩

1198794 (14)

The additive ultraviolet divergences are not explicitly sub-tracted from the condensate and hence it is the bare valuedenoted by subscript 119887 This additive divergence would beincluded in the regular part and in the transition regionwould be much smaller in magnitude than the singular part


20

15

10

5

0

2 + 1 + 1 flavors2 + 1 flavors

120576T4

IT4

pT4

150 200 250 300 350 400T (MeV)

(a)

200 300 400 500 600 700 800 900 1000T (MeV)

6

5

4

3

2

1

0

PT

4

Nf = 2 + 1 EOS Nf = 2 + 1 + 1 N120591 = 8

Nf = 2 + 1 + 1 N120591 = 6 Nf = 2 + 1 + 1 N120591 = 10

(b)

Figure 9 In (a) the effects of quenched charm quark to the pressure energy density and trace anomaly are shown as a function oftemperature from [66] The lattice size is 243 times 6 In (b) the effects of dynamical charm quarks to the pressure are shown as a functionof temperature from [50]

of 119872119887 In the vicinity of the transition region the orderparameter can be written as

119872119887 (119879119867) = ℎ1120575

119891119866 (119905

ℎ1120573120575) + 119891reg (119879119867) (15)

where 119891119866 is the universal scaling function known fromanalysis of the 119874(119873) spin models [69ndash71] with 120573 and 120575 beingthe corresponding critical exponents The quantities ℎ and 119905

are dimensionless parameters that determine the deviationsfrom the critical point and are defined as

119905 =1

1199050

119879 minus 1198791198880

1198791198880

ℎ =119867

ℎ0

119867 =119898119897

119898119904

(16)

with 1198791198880 being the transition temperature in the chiralregime that is for ℎ rarr 0 and ℎ0 and 1199050 are nonuniversalconstants One of the choices of the regular part of the orderparameter used in the lattice study is

119891reg = 119867(1198860 + 1198861

119879 minus 1198791198880

1198791198880

+ 1198862(119879 minus 1198791198880

1198791198880

)

2

) (17)

where one assumes that the regular part is an analyticfunction of the relevant parameters around the transitionpoint The BNL-Bielefeld collaboration used an improvedvariety of the staggered quarks called the p4 quarks tocompute the order parameter defined in (14) and 120594119898 itsderivative with respect to 119898119897 for different values of the lightquark masses 119898119897 The strange quark mass was fixed at itsphysical value These quantities were fitted to the functionalform given in (15) and its derivative respectively The scalinganalysis was done for a fixed lattice of size1198733

times4 so the orderparameter and its derivatives are expected to have an 119874(2)

scaling in the chiral regime since the fermion discretization

only retains a remnant of the continuum 119874(4) symmetrygroup From the plots for the order parameter in Figure 10(a)it is evident that for 119898119897119898119904 = 180 the phase transitionis indeed a second order one with 119874(2) critical exponentsthough 119874(4) scaling cannot be ruled out completely withthe current precision available In the scaling regime thevariable119872119887ℎ

1120575 should be a universal function of 119905ℎ1120573120575 InFigure 10(b) the scaled chiral condensate is seen to be almostuniversal for 119898119897119898119904 lt 120 which provides a hint that evenfor the physical quark masses there is a remnant effect of thechiral symmetry The crossover transition for 2 + 1 flavourQCD should be sensitive to the effects of chiral symmetry andtherefore also to the effects of the 119880119860(1) anomaly

262TheEffects of 119880119860(1)Anomaly TheQCDpartition func-tion breaks 119880119860(1) symmetry explicitly However its effectvaries with temperature since we know that at asymptoticallyhigh temperatures we approach the ideal Fermi gas limitwhere this symmetry is restored It is important to under-stand the temperature dependence of 119880119860(1) breaking nearthe chiral phase transition If 119880119860(1) breaking is significantlyreduced from that at zero temperature one would then claimthat the symmetry is effectively restoredThis would result inthe degeneracy of the mass of the isospin triplet pseudoscalar(pion) and scalar (delta) mesons The order parameter forsuch an effective restoration is the quantity defined as

120594120587 minus 120594120575 = int1198894119909 [⟨120595 (119909) 12059121205745120595 (119909) 120595 (0) 12059121205745120595 (0)⟩

minus ⟨120595 (119909) 1205912120595 (119909) 120595 (0) 1205912120595 (0)⟩]

(18)

and the order parameter for the restoration of the chiralsymmetry is the chiral condensate These quantities are also


000

050

100

150

200

250

094 096 098 100 102 104 106 108TTc

Mb

120

110

140

180

Chiral limit

mlms

(a)

000

050

100

150

200

All masses

th1120573120575

Mbh

1120575

O(2)

15

25

110

120

140

180

minus5 minus4 minus3 minus2 minus1 0 1 2 3 4 5

mlms

(b)

Figure 10 The interpolated data for 119872119887 for different light quark masses are compared with the corresponding plot for an 119874(4) spin modelin the continuum denoted by the solid blue line (a) In (b) the scaling plots for the chiral condensate for QCD are shown to match with theuniversal function with 119874(2) symmetry for119898119897119898119904 lt 120 Both the plots are for p4 staggered quarks from [34]

related to the fundamental theory through the density ofeigenvalues 120588(120582) of the Dirac operator as

⟨120595120595⟩ = int119889120582120588 (120582119898)2119898

1198982 + 1205822

120594120587 minus 120594120575 = int119889120582120588 (120582119898)4119898

2

(1198982 + 1205822)2

(19)

Different scenarios that could lead to different functionalbehaviour of 120588(120582) were discussed in detail in [61] I summa-rize the arguments below

(i) From dilute instanton gas approximation 120588(120582119898) =

11988801198982120575(120582) rArr ⟨120595120595⟩ sim 119898 and 120594120587 minus 120594120575 sim 2

(ii) Analyticity of 120588(120582119898) as a function of 120582 and 119898 whenchiral symmetry is restored To the leading order120588(120582119898) = 119888119898119898 + 119888120582120582 + O(1198982

1205822)

If 120588(120582119898) sim 120582 rArr ⟨120595120595⟩ sim minus2119898 ln119898 120594120587 minus 120594120575 sim 2If 120588(120582119898) sim 119898 rArr ⟨120595120595⟩ sim 120587119898 120594120587 minus 120594120575 sim 120587

In fact to understand the effect of anomaly it is desirable touse fermions with exact chiral symmetry on the lattice Theoverlap and the domain wall fermions are such candidatesfor which the chiral anomaly can be defined Indeed theoverlap fermions satisfy an exact index theorem on the lattice[72] A recent study of the eigenvalue spectrum with thedomain wall fermions from the HotQCD collaboration [73]seems to favour 120588(120582119898) = 1198880119898

2120575(120582) + 1198881120582 for the density

of eigenvalues This would imply that in the chiral limit the119880119860(1) anomaly would still survive when the chiral symmetryis restored This is also consistent with the behaviour of120594120587 minus 120594120575 as a function of temperature shown in Figure 11(a)

At crossover temperature around 160MeV the 120594120587 minus 120594120575 is farfrom zero implying that the effects of the anomaly may belarge in the crossover region

A recent theoretical study [74] with the overlap fermionsshows that in the chiral symmetry restored phase where⟨120595120595⟩ = 0 the eigenvalue density in the chiral limit shouldbehave as

lim119898rarr0

⟨120588 (120582119898)⟩ = lim119898rarr0

⟨120588 (119898)⟩1205823

3+ O (120582

4) (20)

which would imply that 120594120587minus120594120575 rarr 0 as119898 rarr 0 Moreover itis argued that if an operator is invariant under some symme-try transformation then its expectation value becoming zerowould not necessarily imply that the symmetry is restoredwhereas the converse is true [74] This would mean that theobservable 120594120587 minus 120594120575 may not be a good candidate to study the119880119860(1) restoration Rather the equality of the correlators ofthe pion and delta meson could be a more robust observableto indicate the restoration of the 119880119860(1) symmetry Recentresults from the JLQCD collaboration with 2 flavours ofoverlap fermions seem to indicate that the 119880119860(1) may berestored near the chiral symmetry restoration temperaturemaking it a first order transition [75 76] Two of their mainresults are compiled in Figure 12The correlators of the scalarmesons become degenerate at about 196MeV and at the sametemperature a gap opens up in the small eigenvalue regionof the eigenvalue spectrum 119879 = 196MeV is slightly abovethe transition temperature which is nearly about 177MeV For119879 = 177MeV there is no degeneracy between the scalarand the pseudoscalar correlators and the density of zeromodes is finite implying that the chiral symmetry is brokenwhich means that the 119880119860(1) changes rapidly near the phasetransition However the lattice size is 163 times 8 which is small


350

300

250

200

150

100

50

0

T (MeV)140 150 160 170 180 190 200

120594disc T2

1205945disc T2 120594top T

2(ml + mres )2

(120594120587 minus 120594120575)T2

(a)

0025

002

0015

001

0005

00

120588(120582)

002 004 006 008 01120582

163 times 8

Min(120582100)ml

ms

(b)

Figure 11 The susceptibilities for different meson quantum states constructed with the domain wall fermions are shown as a function oftemperature in (a) from [61] The eigenvalue distribution with domain wall fermions shown in (b) from [73] has a peak in the near zeromode distribution at 177MeV The lattice size is 163 times 8 times 1198735 where1198735 = 32 for 119879 ge 160MeV and1198735 = 48 otherwise

T ⋍ 170MeV

T ⋍ 210MeV

1

05

0

0 100 200 300 400 500

120588(120582)

(MeV

3)

120588(120582)

(MeV

3)

120588(120582)

(MeV

3)

1

05

0

1

05

0

120582 (MeV)

T ≃ 180sim190MeV

120573 = 218 am = 005

120573 = 218 am = 001

120573 = 225 am = 001120573 = 220 am = 001120573 = 220 am = 0025120573 = 220 am = 005

120573 = 240 am = 001

120573 = 230 am = 001

120573 = 230 am = 0025

120573 = 230 am = 005

(a)

PS connected (120587)PS all (120578998400)

S connected (120575)S all (120590)

0 2 4 6 8 10 12 14 16

times10minus7

35

3

25

2

15

1

Distance

= 001120573 = 225 (Tsim192) ma

(b)

Figure 12 In (a) the quark mass dependence of eigenvalue distribution for the overlap quarks is compared at different temperatures from[75 76] In (b) the degeneracy of the scalar and pseudoscalar mesons for overlap quarks are shown at a temperature of 192MeV which isslightly higher than the corresponding pseudocritical temperature from [75 76]

enough to introduce significant finite volume and cut-offeffects in the present results

With the chiral fermions the fate of 119880119860(1) in the crosso-ver region is still undetermined and more work needs to bedone for conclusive understanding of this issue WithWilsonand staggered quarks the anomaly is recovered only in thecontinuum limit For fine enough lattice spacings one can

however check the behaviour of the low lying eigenmodesand the meson masses for different quantum numbers tounderstand the effects of the remnant 119880119860(1) anomaly usingthese fermions From the eigenvalue distribution of HISQoperator shown in Figure 13(a) [77] it is evident that theeffect of 119880119860(1) still persists at 119879 = 330MeV The long tailin the low lying eigenmodes is not a finite volume artifact


120588(120582)

120582a

0 004 008 012 016

323 times 8483 times 8

T = 3301MeVmlms = 120

10eminus02

10eminus03

10eminus04

10eminus05

10eminus06

(a)

14

12

1

08

06

04

02

M(2120587

T)

085 09 095 1 105 11 115 12TTc

P

S

V

A

(b)

Figure 13 The density of eigenvalues at 119879 = 3301MeV for HISQ discretization showing a long tail even with large volumes from [77](a) In (b) the screening masses for scalar pseudo-scalar vector and axial vector mesons using Wilson fermions are shown as a function oftemperature from [78]

since it persists even for very large volumes However thedata is quite noisy and more statistics are required formaking a final conclusion The screening masses for themesons of different quantum numbers were obtained fromlattice studies with improved Wilson fermions (Figure 13(b)[78]) In the transition region the scalar and pseudoscalarmesons are not degenerate and an agreement seen only fortemperatures above 12119879119888 However the input quark massesare quite large compared to the physical values andmore datais needed to take a final call At present the effects of quantumanomalies are not yet understood from lattice studies

3 Lattice QCD at Finite Density

QCD with a finite number of baryons is relevant for thephysics of neutron stars and supernovae It is the theoreticalsetup for the heavy ion physics phenomena occurring atlow center of mass energy radic119904 of the colliding nuclei Someof these low radic119904 collisions are being investigated at theRHIC and to be probed further with the start of the heavyion experiments at FAIR GSI and NICA Dubna In factan interesting feature of the QCD phase diagram is thecritical end-point related to chiral symmetry restorationTheexistence of the critical point has important consequences onthe QCD phase diagram and it is the aim of the extensivebeam energy scan (BES) program at the RHIC to search forit

To explain these experimental results from first prin-ciples we need to extend the lattice QCD formulation toinclude the information of finite baryon density One of themethods is to work in a grand canonical ensemble In such anensemble the partition function is given by

ZQCD (119879 120583) = Tr (119890HQCDminus120583119873) = intD119880120583

119873119891

prod

119891=1

det119863119891 (120583) 119890minus119878119866

(21)

where the chemical potential 120583 is the Lagrange multipliercorresponding to the conserved number density 119873 thatcommutes with the QCD Hamiltonian 119867QCD 119873 can be thebaryon number or the net electric charge The 120583 enters intothe lattice fermion action as exp(plusmn120583119886) factors multiplyingthe forward and backward temporal links respectively [7980] referred to as the Hasenfratz-Karsch method The naivefermion operator at finite120583 on the latticewould be of the form

119863119891(120583)119909119910 = [

3

sum

119894=1

1

2120574119894 (119880119894 (119909) 120575119910119909+119894 minus 119880

dagger119894 (119910) 120575119910119909minus119894)

+1

21205744 (119890

1205831198861198804 (119909) 120575119910119909+4 minus 119890

minus120583119886119880

dagger120583 (119910) 120575119910119909minus4)

+ 119886119898119891120575119909119910]

(22)

This is not a unique way of introducing 120583 and it could bealso done in several different ways [81] The lattice fermiondeterminant at finite 120583 like in the continuum is no longerpositive definite since

det119863dagger119891 (120583) = det119863119891 (minus120583) 997904rArr det119863119891 (120583) =

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816e119894120579(23)

and the interpretation of intD119880 det119863119891(120583)119890minus119878119866 as a probability

weight in the standard Monte Carlo simulations is no longerwell defined This is known as the ldquosign problemrdquo One mayconsider only the real part of the fermion determinant forMonte Carlo algorithms and generate configurations by theso-called phase quenching Once the partition function isknown in the phase quenched limit one can then use thereweighting techniques to generate the partition function ofthe full theory at different values of 120583 The expectation value


of the phase of the determinant needed for reweighting atsome finite 120583 is given as

⟨e119894120579⟩ =

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816e119894120579119890minus119878119866

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816119890minus119878119866

= 119890minus119881Δ119865119879

(24)

where Δ119865 is the difference between the free energy densitiesof the full and the phase quenched QCD For two degeneratequark flavours the phase quenched theory is equivalent toa theory with a finite isospin chemical potential [82] andΔ119865 is the difference of free energies of QCD with finitebaryon (quark) chemical potential and that at an isospinchemical potential These two theories are qualitatively quitedifferent and the sign problem results in a very smalloverlap between these two theories For isospin QCD thecharged pions are the lightest excitations and these canundergo a Bose-Einstein condensation for 120583 gt 1198981205872 Thedifference between the respective free energies in this regimeis quite large leading to a severe sign problem This is analgorithmic problem that can arise for any theory which haschiral symmetry breaking A better understanding of thesign problem has been achieved in the recent years with aknowledge of the regions in the phase diagram with severesign problem and thosewhere it is controllable [83ndash85]Thereare several methods followed to circumvent this problem onthe lattice some of which are listed below

(i) reweighting of the 120583 = 0 partition function [86ndash89](ii) Taylor series expansion [90ndash92](iii) canonical ensemble method [93ndash96](iv) imaginary chemical potential approach [97ndash100](v) complex Langevin algorithm [101ndash105](vi) worm algorithms [106 107]

The Taylor series method has been widely used in the latticeQCD studies in the recent years which has led to interestingresults relevant for the experiments One such proposalis the determination of the line of chemical freezeout forthe hadrons in the phase diagram at small baryon densityfrom first principles lattice study It was first proposed thatcumulants of baryon number fluctuations could be used fordetermining the freezeout parameters [108] on the latticeLast year another interesting suggestion was made [109]where the experimental data on cumulants of electric chargefluctuations could be used as an input to compute thefreezeout curve using lattice dataThis and some other resultsare discussed in the subsequent subsections Most of theresults are obtained with improved versions of staggeredfermions It has been known that the rooting problem maybe more severe at finite density [110] It is thus important toexplore other fermion formulations as well for lattice studiesWilson fermions have been used but it is important to usechiral fermions especially for the study of the critical point Ioutline in the next subsection the theoretical efforts in therecent years that have led to the development of fermionoperators at finite density with exact chiral symmetry on thelattice which can be used for future lattice studies on thecritical point

31 Chiral Fermions at Finite Density The contribution of the119880119860(1) anomaly is believed to affect the order of the chiralphase transition at zero density and hence is crucial for thepresence or absence of the critical point If the anomaly isnot represented correctly at finite density it may affect thelocation of the critical point in the phase diagram if it existsOverlap fermions have exact chiral symmetry on the latticein the sense that the overlap action is invariant under suitablechiral transformations known as the Luscher transformations[111] It can be further shown that the fermion measure in thepath integral is not invariant under Luscher transformationsand its change gives the chiral anomaly The index theoremrelating the anomaly to the difference between the fermionzero modes can be proved for them [72] Thus the overlapfermions have the properties analogous to the fermions inthe continuum QCD In the continuum it is known that theanomaly is not affected in presence of a finite baryon chemicalpotential It would be desirable to preserve this continuumproperty with the overlap fermions as well such that thephysical properties important for the existence of the criticalpoint are faithfully presented on a finite lattice Defining anoverlap fermion action at finite chemical potential is non-trivial as the conserved currents have to be defined with care[112] The first attempt to define an overlap fermion operatorat finite density [113] was done in the last decade and anindex theorem at finite 120583was also derived for them Howeverthese overlap fermions did not have exact chiral symmetryon a finite lattice [114] Moreover the index theorem forthem was 120583-dependent unlike in the continuum Recentlyoverlap fermion at finite density has been defined from thefirst principles [115] which has exact chiral symmetry on thelattice [116] and preserves the120583-independent anomaly as wellA suitable domain wall fermion action has been also definedat finite density [116] which was shown to reproduce theoverlap action in the appropriate limit It would be importantto check the application of these overlap and domain wallfermion operators at finite 120583 for future large scale QCDsimulations

32 Correlations and Fluctuations on the Lattice The studiesof fluctuations of the conserved charges are important tounderstand the nature of the degrees of freedom in a ther-malized medium and the interactions among them [117 118]The diagonal susceptibility of order 119899 defined as

120594119883119899 =

119879

119881

120597119899 lnZ120597120583

119899119883

119883 equiv 119861 119878 119876 (25)

measures the fluctuations of the conserved quantum num-ber 119883 In a heavy-ion experiment the relevant conservednumbers are the baryon number 119861 and electric charge 119876The strangeness 119878 is zero at the initial time of collision ofheavy nuclei but strange quark excitations are producedat a later time in the QGP and is also believed to be agood quantum number These fluctuations can be computedexactly on the lattice at 120583 = 0 from the quark numbersusceptibilities [119] Continuum extrapolated results for thesecond order susceptibilities of baryon number strangeness


and electric charge exist for both HISQ [120] and stoutsmeared staggered quarks [121] The fluctuations of baryonnumber are very well explained by the hadron resonancegas model for 119879 lt 160MeV However the fluctuations ofthe strangeness are usually larger than the HRG values byabout 20 in the freezeout region characterized by 160 le

119879 le 170MeV The electric charge fluctuations on theother hand are smaller than the corresponding HRG valuesby 10 in the same region The ratio of 120594

1198762 120594

1198612 (120583 =

0) ≃ 029ndash035 in the freezeout region A first principledetermination of this ratio is crucial as it would allow usto relate the net baryon number fluctuations with the netproton number fluctuations which is an observable in theheavy ion experiments [120] At high temperatures thesefluctuations slowly approach the corresponding free theoryvalue with the continuum extrapolated data for the baryonnumber susceptibility showing about 20 deviation from thefree theory value even at 2119879119888 [120] The data are in goodagreement with resummed perturbation theory estimates atthese temperatures [122 123] indicating that the QGP isstill fairly strongly interacting even at temperatures around2119879119888

To relate to the results of the heavy ion experimentsat a lower collision energy radic119904 one has to compute thefluctuations on the lattice at a finite value of 120583 The mostwidely used lattice method to compute the susceptibilities ata finite value of quark chemical potential 120583 is through theTaylor expansion of the corresponding quantity at 120583 = 0 forexample

1205941198612 (120583)

1198792=1205941198612 (0)

1198792+

1205832

211987921205941198614 (0) +

1205834

411987941205941198616 (0) 119879

2+ sdot sdot sdot

(26)

The light and strange quark susceptibilities have been com-puted at finite but small densities from Taylor expan-sion using asqtad staggered quarks [66] and the ratiosof baryon number susceptibilities using the unimprovedstaggered fermions [108] in the region of interest for theRHIC experiments All these ratios agree well with theestimates from the HRG model [108] the results for whichare compiled in Figure 16(b) The ratios of susceptibilitiesserve as a good observable for comparing the lattice and theexperimental data since these are free from the unknownquantities like the volume of the fireball during freezeout[124]

The higher order susceptibilities 120594119899 for 119899 gt 4 areimportant even in the 120583 = 0 regime In the chiral limit it isexpected that the fourth order baryon number susceptibilitywould have a cusp and the sixth order would diverge with119874(4) scaling at the critical temperature Even for physicalquark masses 120594119861

6 for QCD would show oscillations nearthe pseudocritical temperature and 120594

1198618 would have negative

values in the same region [125] quite contrary to the HRGpredictions Thus the signatures of critical behaviour couldbe understood by the careful study of these quantities alreadyat 120583 sim 0 which is probed by the experiments at LHC[125]

Other important quantities of relevance are the off-diag-onal susceptibilities These defined as

120594119861119878119876119894119895119896 =

119879

119881

120597119894+119895+119896 lnZ

120597120583119894119861120597120583

119895

119878120597120583119896119876

(27)

are a measure of the correlations between different quantumnumbers and hence good observables to estimate the effectsof interactions in the different phases of the QCD mediumIt has been suggested that the quantity 119862119861119878 = minus3120594

11986111987811 120594

1198782

is a good observable to characterize the deconfinement inthermal QCD [126] If the strangeness is carried by quarklike excitations the value of 119862119861119878 would be identity andwould be much smaller than unity in the phase where onlythe baryons and mesons carry the strangeness quantumnumber Recent results from the HotQCD collaborationusing HISQ action [120] show that 119862119861119878 approaches unityvery quickly at around 200MeV implying that almost nostrange hadrons survive in the QGP phase above 119879119888 Thisis compiled in Figure 14(a) The HotQCD data is consistentwith the corresponding continuum extrapolated data withthe stout smeared fermions [121] Also 119862119861119878 is not sensitiveto the sea strange quark masses for 119879 gt 119879119888 since thefirst partially quenched results [127] for this quantity areconsistent with the full QCD results The other importantobservable is the baryon-electric charge correlation In theconfined phase electric charge in the baryon sector is mainlycarried by protons and antiprotons therefore the correlationwould rise exponentially with temperature if this phasecould be described as a noninteracting gas consisting ofthese particles At high temperatures however quark-likeexcitations would be important and their masses being muchsmaller than the temperature this correlation would fall tozero From the behaviour of the continuum extrapolatedHISQ data for 120594119861119876

11 compiled in Figure 14(b) it is evidentthat near the pseudocritical temperature there is a changein the fundamental properties of the degrees of freedomof the medium with quark-like excitations dominating at15119879119888

33 The Freezeout Curve from Lattice To relate the resultsfrom heavy ion experiments with the lattice data it is crucialto map the center of mass energy of the colliding nucleiin the heavy ion collisions radic119904 to the corresponding pointin the 119879 minus 120583119861 plane of the QCD phase diagram This iscalled the freezeout curve Phenomenologically the freezeoutcurve is obtained from a particular parameterization of theHRG model obtained through fitting the experimental dataon hadron abundances [128] At chemical freezeout thechemical composition of the baryons gets frozen meaningthat the inelastic collisions between these species become lessprobable under further cooling of the system However thesystematic uncertainties in determining the hadron yieldsare not taken into account in the phenomenological deter-mination of the freezeout curve Recent work by the BNL-Bielefeld collaboration shows how lattice techniques canprovide first principle determination of the freezeout curvethrough suitable experimental observables As emphasized in


SB

HRG

1

08

06

04

02

0

minus3120594

BS

11120594

S 2

120 140 160 180 200 220 240T (MeV)

N120591 = 12

N120591 = 8N120591 = 6

fK scale

(a)

HRG

005

004

003

002

001

0

120594BQ

11T

2

fK scale

120 140 160 180 200 220 240T (MeV)

N120591 = 12 N120591 = 6N120591 = 8Cont extrap

(b)

Figure 14 The HISQ data for 119862119861119878 (a) and 12059411986111987611 119879

2 (b) as a function of temperature from [120]

the last subsection the ratios of susceptibilities are believedto be good observables for comparing the lattice and theexperimental data Two such observables proposed in [109]are

11987711988312 equiv

119872119883

1205902119883

=120583119861

119879(119877

119883112 +

1205832119861

1198792119877119883312 + O (120583

4119861))

11987711988331 equiv

1198781198831205903119883

119872119883

= 119877119883131 +

1205832119861

1198792119877119883331 + O (120583

4119861)

(28)

where 119872119883 120590119883 119878119883 denotes the mean variance and theskewness in dimensionless units for the conserved quantumnumber 119883 These observables are chosen because these areodd and even functions of 120583119861 allowing us to independentlydetermine 119879 and 120583119861 from these two quantities The quantumnumber 119883 can either be chosen to be the net electric charge119876 or the net baryon number 119861 In the experiments onecan only measure the proton number fluctuations and itis not clear whether the proton number fluctuations couldbe a proxy for the net baryon fluctuation [129] It was thussuggested that the ratios of net charge fluctuations would bea better observable to compare with the experiments Oncethe 119877119876

31 is known from experiments one can determine thefreezeout temperature 119879119891 from it by comparing with thecontinuum extrapolated lattice data Analogously one canobtain the 120583119861 at freezeout from comparison of the 119877119876

12 dataIn Figure 15(a) the results for 119877119876

31 are shown as a functionof temperature It is evident that the first order correctionto the value of the ratio is within 10 of the leading ordervalue for 120583119861119879 lt 13 and in the freezeout region that is119879 gt 140MeV From the leading order results of 119877119876

31 one canestimate the freezeout temperature For radic119904 in the range of39ndash200GeV currently probed in the beam energy scan (BES)

experiment at RHIC the freezeout temperature from theHRG parameterization of the hadron multiplicities is about165MeV At this temperature the ratio 119877

11987631 calculated from

the HRGmodel is quite larger than the lattice estimate whichwould mean that the freezeout temperature estimated fromlattice data would differ from the model results by atleast 5Similarly if 119877119876

12 is known from the experiments 120583119861 can beaccurately estimated and is expected to be different from thecurrent HRG estimates This is not very surprising becausethe freezeout of the fluctuations happens due to diffusiveprocesses and is due to a different mechanism from thefreezeout of hadrons due to decreasing probability of inelasticcollisions Another question that was addressed in this workwas how relevant are the other parameters like 120583119878 and 120583119876

for the phase diagram and the freezeout curve It was seenthat 120583119878 and 120583119876 are significantly smaller than 120583119861 and theratios of these quantities have a very small 120583119861 dependencein the entire temperature range of 140ndash170MeV relevant forthe freezeout studies It signifies that the relevant axes for thephase diagram are indeed119879 and120583119861 and these two parametersare sufficient for characterizing the freezeout curve

34 Physics Near the Critical Point It is known from modelswith the same symmetries as QCD that the chiral phasetransition at 119879 = 0 and finite 120583 is a first order one At zerodensity and high enough temperatures QCD undergoes acrossover from the hadron to the QGP phase By continuityit is expected that the first order line should end at a criticalend-point in the phase diagram [130ndash132]The determinationof its existence from first principles lattice computation hasbeen quite challenging and the currently available latticeresults are summarized in Figure 16(a)These are all obtainedusing staggered fermionsThe first lattice study on the criticalpoint was done using reweighting technique Configurations


RQ 31

HRG

30

25

20

15

10

05

00

Free

T (MeV)140 150 160 170 180 190 200 210 220 230 240

120583BT = 1

120583BT = 0

N120591 = 6

N120591 = 8

(a)

0035

0030

0025

0020HRG

HRG

T = 160MeV

T = 160MeV

T = 170MeV

T = 170MeV

T = 150MeV

T = 150MeV

030

025

020

015

120583S120583

B

0 20 40 60 80 100 120 140 160 180 200120583B (MeV)

minus120583Q120583

B(b)

Figure 15 In (a) the leading term for 11987711987631 shown in the yellow band is compared to its NLO term denoted by the blue band in the continuum

limit In (b) the ratios of 120583119876 and 120583119878 with respect to 120583119861 are compared with the HRGmodel predictions at different temperatures Both figuresare from [109]

07

08

09

1

11

0 1 2 3 4 5

Freezeout curve

120583BT

TT

c

30 GeV20 GeV

18 GeV (CERN)10 GeV

(a)

HRG

Power law

10

1

01

001

0001

m1

1 10 100 1000 10000radicSNN (GeV)

(b)

Figure 16 The estimates of the critical point from lattice studies are shown in (a) from [137] The magenta solid circle box and star denotethe 119873120591 = 4 6 8 data respectively for 2 flavours of staggered quarks [135ndash137] and the open circles denote 119873120591 = 4 data for 2 + 1 flavoursobtained with reweighting techniques [87 133] In (b) the ratio of the third and the second order baryon number susceptibility is plotted asa function ofradic119904 relevant for the RHIC and LHC experiments and compared with the HRG model data from [108]

generated at the critical value of the gauge coupling for 120583119861 = 0

were used to determine the partition function at differentvalues of 119879 and 120583119861 using two-parameter reweighting [87] Byobserving the finite volume behaviour of the Lee-Yang zeroesof the partition function it was predicted that for 2+1 flavourQCD there is a critical end-point at 119879119864 = 160(4)MeV and120583119861 = 725(35)MeV In this study the light quark was fourtimes heavier than its physical value Reducing the light quarkmass shifted the critical end-point to 120583119861 = 360(40)MeV

with 119879119864 = 162(2) remaining the same [133] However thisresult is for a rather small lattice of size 163times4 and is expectedto change in the continuum limit and with larger volumesReweighting becomes more expensive with increasing vol-ume of the lattice so going to a larger lattice seems difficultwith this method

The other results for the critical point were obtainedusing the Taylor series method In this method the baryonnumber susceptibility at finite density is expanded in powers


of 120583119861119879 as a Taylor series as shown in (26) for each value oftemperature The baryon number susceptibility is expectedto diverge at the critical end-point [134] so the radius ofconvergence of the series would give the location of thecritical end-point [92] However on a finite lattice there areno divergences but the different estimates of the radius ofconvergence given as

119903119899 (119899 = odd) = radic120594119861119899+1

1198792120594119861119899+3

119903119899 (119899 = even) = [1205941198612

119879119899120594119861119899+2

]

1119899

(29)

should all be positive and equal within errors at the criticalend-point Currently the state of the art on the lattice isestimates of baryon number susceptibilities upto 120594

1198618 This

gives five different independent estimates of the radius ofconvergence upto 1199036 which were shown to be consistentwithin errors for 119873120591 = 4 6 8 at 119879119864 = 094(1)119879119888 [135ndash137]The radius of convergence after finite volume correction is120583119861119879119864 = 17(1) [137] which means 120583119861 = 246(15)MeVat the critical end-point if we choose 119879119888 = 154MeV Theinput pion mass for this computation is about 15 timesthe physical value and could affect the final coordinatesof the end-point Moreover the different estimates for theradius of convergence 119903119899 in (29) agrees with each other forasymptotically large values of 119899 and one might need to checkthe consistency of the results with the radii of convergenceestimates beyond 1199036 Hints of the critical end-point were alsoobtained [138] using a different fermion discretization and adifferent methodology as well Working with the canonicalensemble of improved Wilson fermions the presence of acritical point was reported at 119879119864 = 0925(5)119879119888 and 120583119861119879119888 =

260(8) This is a very preliminary study though with a smalllattice volume and a very heavy pion mass of about 700MeV

Though there is growing evidence in support for theexistence of the critical end-point the systematics for all theselattice studies are still not under control It would be desirableto follow a different strategy to determine its existenceThe alternate method suggested [139] was to determinethe curvature of the surface of second order chiral phasetransitions as a function of the baryon chemical potential 120583119861If the chiral critical surface bends towards larger values of119898119906119889with increasing baryon chemical potential and for a fixedvalue of the strange quark mass it would pass through thephysical point ensuring the existence of a critical end-pointHowever if the curvature is of opposite sign the chiral criticalend-point would not exist For lattice size of 83times4 the criticalvalue of the light quarks was estimated upto O(1205834

119861) [140]

119898119888 (120583119861)

119898119888 (0)= 1 minus 39 (8) (

120583119861

3120587119879)

2

minus sdot sdot sdot (30)

with the strange quark mass fixed at its physical value Theleading value of the curvature has the same sign even for afiner lattice of extent119873120591 = 6 [141]These studies show that theregion of first order transition shrinks for small values of 120583119861which wouldmean that the critical point may not exist in thisregime of120583119861 However for larger values of120583119861 the higher order

terms could be important and may bend the chiral criticalline towards the physical values of quark masses The finitecut-off effects are still sizeable and it is currently prematureto make any definite predictions in the continuum limit withthis method

It is equally important to understand the possible exper-imental signatures of the critical point The search of thecritical end-point is one of the important aims for theextensive BES program at RHIC In a heavy ion experimentonemeasures the number of charged hadrons at the chemicalfreezeout and its cumulants During the expansion of thefireball the hot and dense QCDmediumwould pass throughthe critical region and cool down eventually forming hadronsIf the freezeout and the critical regions are far separated thesystem would have no memory of the critical fluctuationsand the baryon number susceptibility measured from theexperiments could be consistent with the predictions fromthermal HRG models which has no critical behaviour Ifthe freezeout region is within the critical region the criticalfluctuations would be larger than the thermal fluctuations Itis thus important to estimate the chiral critical line for QCDfrom first principles The curvature of the chiral critical linehas been estimated by the BNL-Bielefeld collaboration [142]by extending the scaling analysis of the dimensionless chiralcondensate 119872119887 outlined in Section 261 for finite values ofbaryon chemical potential using Taylor series expansionThecorresponding scaling variables at finite 120583119861 are

119905 =1

1199050

(119879 minus 1198791198880

1198791198880

+ 120581119861

120583119861

3119879) ℎ =

119898119897

ℎ0119898119904

(31)

The quantity119872119887 can be expanded as a Taylor series in 1205831198613119879as

119872119887 (120583) = 119872119887 (0) +120594119898119861

2119879(120583119861

3119879)

2

+ O(120583119861

3119879)

4

(32)

where 120594119898119861 is the mixed susceptibility defined as 120594119898119861 =

(1198792119898119904)120597

2119872119887120597(1205831198613119879)

2 computed at 120583119861 = 0 In the criticalregion it would show a scaling behaviour of the form

120594119898119861

119879=2120581119861119879

1199050119898119904

ℎminus(1minus120573)120573120575

1198911015840119866 (

119905

ℎ1120573120575) (33)

The universality of the scaled 120594119898119861 data is clearly visible inFigure 17(b) both for p4 staggered quarks on 119873120591 = 4 latticewithmass ratios of light and strange quarks varying from 120

to 180 and with HISQ discretization on a 323 times8 lattice withthe mass ratio fixed at 120 The fit of the complete latticedata set to the scaling relation for 120594119898119861 gave the value of 120581119861 =

000656(66) At non-vanishing 120583119861 the phase transition pointis located at 119905 = 0 which implies that the critical temperatureat finite density can be parameterized as

119879119888 (120583119861)

119879119888 (0)= 1 minus 120581119861(

120583119861

3119879)

2

+ O(120583119861

3119879)

4

997904rArr 119879119888 (120583119861) ≃ 154 (1 minus 00066(120583119861

3119879)

2

) MeV

(34)


150

100

50

Tem

pera

ture

(MeV

)

200 400Baryonic chemical potential (MeV)

Freezeout

T120594119904c (120583)

T120595120595c (120583)

RHIC radicSNNsim130GeVSPS radicSNNsim17GeV

SPS radicSNNsim9GeVAGS radicSNNsim5GeV

(a)

007

006

005

004

003

002

001

0minus2 minus1 0 1 2 3

z

minus2120581qf998400G(z) 120

140180

minusm

st0h(1minus120573)120573120575120594mqT

2

N120591 = 8 mlms = 120

N120591 = 4 mlms = 110

(b)

Figure 17 In (a) the width of the pseudocritical region for chiral condensate is shown as a blue curve and that for strange quark susceptibilityis shown as a red curve from [143] In (b) the scaling of the mixed susceptibility is shown for different light quark masses and at the physicalvalue of strange quark mass from [142]

This estimate of the curvature is about three times larger thanthe corresponding prediction from the hadron resonance gasmodel It would be interesting to compare the curvature ofthe freezeout line computed on the lattice with that of thecritical line once the experimental data for the electric chargecumulants are available

Another complimentary study about the fate of thecritical region at finite density was done by the Budapest-Wuppertal group [143] It was suggested that if the criticalregion shrinks with increasing 120583119861 it would imply that oneslowly converges to the critical end-point The width of thecritical region was measured from two different observablesthe renormalized chiral condensate and the strange quarknumber susceptibility Stout smeared staggered quarks wereemployed and the continuum limit was taken with the119873120591 =

6 8 10dataThe results are summarized in Figure 17(a) Fromthe plots it seems that the width of the crossover regiondoes not change from its 120583119861 = 0 value significantly for120583119861 lt 500MeVwhich implies either that the critical end-pointdoes not exist at all or is present at a higher value of 120583119861 Thecorresponding curvature measured for the light quark chiralcondensate is 00066(20) which is consistent with the resultfrom the BNL-Bielefeld collaboration The results indicatethat the chiral pseudocritical line and the phenomenologicalfreezeout curve would separate apart at larger values of 120583119861and would be further away at the critical end-point

It was noted that the higher order fluctuations are morestrongly dependent on the correlation length of the system[144] and would survive even if the chiral and freezeout linesare far apart It has been proposed [125] that the signature ofthe critical point can be detected bymonitoring the behaviourof the sixth and higher order fluctuations of the electriccharge along the freezeout curve

35The EoS at Finite Density TheEoS at finite density wouldbe the important input for understanding the hydrodynami-cal evolution of the fireball formed at low values of the colli-sional energy at theRHICand the future experiments at FAIRandNICA It is believed that there is no generation of entropyonce the fireball thermalizes [145] In that case as pointedout in [146] it is important to determine the EoS alonglines of constant entropy per net baryon number 119878119899119861 torelate the lattice results with the experiments The isentropedetermined by a fixed value of 119878119899119861 that characterizes theevolution of the fireball is 119878119899119861 ≃ 300 for RHIC experimentsat radic119904 = 200GeV For the future experiments at FAIR theisentropes would be labelled by 119878119899119861 = 30 nearly as same asthe early SPS experiments at CERN where 119878119899119861 sim 45 Fortwo flavour QCD with p4 staggered quarks and with pionmass heavier than its physical value it was already observedthat the ratio of pressure and energy density showed littlevariation as a function of 119878119899119861 The pressure and the energydensity at finite 120583 are usually computed on the lattice as aTaylor series about its value at zero baryon density as

119875 (120583119897 120583119904)

1198794=119875 (0)

1198794+sum

119894119895

120594119894119895 (119879)

1198794minus119894minus119895(120583119897

119879)

119894

(120583119904

119879)

119895

(35)

The formula is valid for two degenerate light quark flavoursand a heavier strange quarkThe coefficients 120594119894119895 are the quarknumber susceptibilities at 120583 = 0 and are non-zero for 119894 + 119895 =

even The corresponding expression for the trace anomaly isgiven as

119868 (120583119897 120583119904)

1198794= minus

1198733120591

1198733

119889 lnZ119889 ln 119886

=119868 (0)

1198794+sum

119894119895

119887119894119895 (119879) (120583119897

119879)

119894

(120583119904

119879)

119895

(36)


8

6

4

2

0

IT4

100 200 300 400 500 600T (MeV)

SnB = 30

SnB = infinSnB = 45

SnB = 300

Filled Nt = 6

Empty Nt = 4

(a)

6

4

2

5 10 15

pT

4

120598T4

SNL = 30 latticeSNL = 300 lattice

120583 = 0 latticeIdeal limit

(b)Figure 18The EoS for different isentropes using asqtad quarks is shown in (a) from [66] In (b) the data for the energy density and pressureis compared for different isentropes using stout-smeared staggered quarks from [147]

The 120594119894119895s can be also obtained from the coefficients 119887119894119895 byintegrating the latter along the line of constant physics For 2+1 flavours of improved asqtad staggered quarks with physicalstrange quark mass and119898119897 = 11989811990410 the interaction measurewas computed upto O(1205836

) for two different lattice spacings(Figure 18(a)) The interaction measure did not change sig-nificantly from the earlier results with heavier quarks andshowed very little sensitivity to the cut-off effects along theisentropes [66] However it was observed that the light andthe strange quark number susceptibilities change significantlyfrom the zero temperature values along the isentropes Nopeaks were found in the quark number susceptibilities atisentropes 119878119899119887 = 300 which led to the conclusion that thecritical point may not be observed at the RHIC [66] TheEoS and the thermodynamic quantities were computed forphysical values of quark masses by the Budapest-Wuppertalcollaboration [147] In this study they set the values of thelight quark chemical potentials such that 120583119897 = 1205831198613 and thestrange quark susceptibility is 120583119904 = minus2120583119897120594

11990611990411120594

1199042 to mimic the

experimental conditions where the net strangeness is zeroThe pressure and the energy density was computed uptoO(1205832

) The ingredients that went into the computations were(a) the near continuumvalues of the interactionmeasure datafrom the119873120591 = 10 lattice and (b) the spline interpolated valuesof 120594119904

2 12059411990611990411 for the range 125 lt 119879 lt 400MeV obtained using

the continuum extrapolated data for 1205941199042 120594

11990611990411 It was observed

as evident from Figure 18(b) that the finite density effectsalong the RHIC isentropes are negligible consistent with theearlier work However for isentropes given by 119878119899119861 = 30 thefinite density effects become more important The effect oftruncation at O(1205832

) was also estimated on a reasonably large119873120591 = 8 lattice It was observed that

119901 up to O ((120583119861119879)4)

119901 up to O ((120583119861119879)2)

le

11 for120583119861

119879le 2

135 for120583119861

119879le 3

(37)

implying that the fourth and higher order terms need to bedetermined for even modest values of 120583119861 in the Taylor seriesmethod An independent study about the truncation effectsof the Taylor series was performed in [148]The derivatives ofpressure were computed for two flavour QCD with staggeredquarks at imaginary chemical potential These derivativesare related to the successive terms of the Taylor coefficientsof pressure evaluated at 120583 = 0 By fitting the imaginary120583 data with a polynomial ansatz these Taylor coefficientswere obtained and compared with the exact values It wasobserved that for 119879119888 le 119879 le 104119879119888 atleast the 8th orderTaylor coefficient is necessary for a good fit This highlightsthe necessity to evaluate higher order susceptibilities beyondthe currently measured eighth order in the studies of EoS orthe critical end-point New ideas to extend the Taylor series tohigher order susceptibilities are evolving [148 149] and theseshould be explored in full QCD simulations

4 Summary

As emphasized in the introduction I have tried to compiletogether some of the important instances to show thatthe lattice results have already entered into the precisionregime with different fermion discretizations giving consis-tent continuumresults for the pseudocritical temperature andfluctuations of different quantum numbers The continuumresult for the EoS would be available in very near future withconsistency already observed for different discretizationsThelattice community has opened the door for a very activecollaboration between the theorists and experimentalistsWith the EoS as an input one can study the phenomenologyof the hot and dense matter created at the heavy ioncolliders On the hand there is a proposal of nonperturbativedetermination of the freezeout curve using lattice techniquesonce the experimental data on cumulants of the chargedhadrons are available


A good understanding of the QCD phase diagram at zerobaryon density has been achieved from the lattice studiesWhile the early universe transition from the QGP to thehadron phase is now known to be an analytic crossover andnot a real phase transition it is observed that the chiraldynamics will have observable effects in the crossover regionOne of the remnant effects of the chiral symmetry would bethe presence of a critical end-point The search for the stillelusive critical endpoint is one of the focus areas of latticestudies and the important developments made so far in thisarea are reviewed

While QCD at small baryon density is reasonably wellunderstood with lattice techniques the physics of baryonrich systems cannot be formulated satisfactorily on the latticedue to the infamous sign-problem A lot of conceptual workin understanding the severity and consequences of the signproblem as well algorithmic developments in circumventingthis problem is ongoing which is one of the challengingproblems in the field of lattice thermodynamics

Acknowledgments

Sayantan Sharma would like to thank all the members ofthe Theoretical Physics Group at Bielefeld University andin particular Frithjof Karsch Edwin Laermann Olaf Kacz-marek and Christian Schmidt for a lot of discussions thathave enriched the authorrsquos knowledge about QCD thermo-dynamics and lattice QCD The author expresses gratitudeto Edwin Laermann for a careful reading of the paper andhis helpful suggestions and Toru Kojo and Amaresh Jaiswalfor their constructive criticism that has led to a considerableimprovement of this paper The author also acknowledgesRajiv Gavai and Rajamani Narayanan for very enjoyablecollaboration in which the author learnt many aspects of thesubject

References

[1] C Hoelbling ldquoLight hadron spectroscopy and pseudoscalardecay constantsrdquo in International Symposium on Lattice FieldTheory (LATTICE rsquo10) PoS Villasimius Italy June 2010httparxivorgabs11020410

[2] F Karsch ldquoLattice QCD at finite temperature a status reportrdquoZeitschrift Fur Physik C vol 38 no 1 pp 147ndash155 1988httparxivorgabs13023028

[3] P F Kolb and U W HeinzQuark Gluon Plasma 3 Edited by RC Hwa and X-N Wang World Scientific Singapore 2003

[4] M-P Lombardo ldquoHigh Temperature QCDrdquo in InternationalSymposium on Lattice Field Theory (LATTICE rsquo12) PoS CairnsAustralia June 2012

[5] GAarts ldquoComplex Langevin dynamics and other approaches atfinite chemical potentialrdquo in International Symposium on LatticeField Theory (LATTICE rsquo12) PoS Cairns Australia June 2012

[6] S B Nielsen and M Ninomiya ldquoAbsence of neutrinos on alattice (I) Proof by homotopy theoryrdquo Nuclear Physics B vol185 no 1 pp 20ndash40 1981

[7] K G Wilson ldquoConfinement of quarksrdquo Physical Review D vol10 no 8 pp 2445ndash2459 1974

[8] J Kogut and L Susskind ldquoHamiltonian formulation ofWilsonrsquoslattice gauge theoriesrdquo Physical Review D vol 11 no 2 pp 395ndash408 1975

[9] C Morningstar and M J Peardon ldquoAnalytic smearing of SU(3)link variables in lattice QCDrdquo Physical Review D vol 69 no 5Article ID 054501 9 pages 2004

[10] E Follana Q Mason and C Davies ldquoHighly improved stag-gered quarks on the lattice with applications to charm physicsrdquoPhysical Review D vol 75 Article ID 054502 23 pages 2007httparxivorgabshep-lat0610092

[11] J Engels R Joswig F Karsch E Laermann M Lutgemeierand B Petersson ldquoThermodynamics of four-flavour QCD withimproved staggered fermionsrdquo Physics Letters B vol 396 pp210ndash216 1997 httparxivorgabs09112215

[12] K Orginos and D Toussaint ldquoTesting improved actions fordynamical Kogut-Susskind quarksrdquo Physical Review D vol 59no 1 Article ID 014501 7 pages 1998 httparxivorgabshep-lat9805009

[13] J F Lagae andD K Sinclair ldquoImproved staggered quark actionswith reduced flavor symmetry violations for lattice QCDrdquoPhysical Review D vol 59 no 1 Article ID 014511 6 pages 1998httparxivorgabshep-lat9806014

[14] G P Lepage ldquoFlavor-symmetry restoration and Symanzikimprovement for staggered quarksrdquo Physical Review D vol 59no 7 Article ID 074502 4 pages 1999 httparxivorgabshep-lat9809157

[15] R Narayanan and H Neuberger ldquoChiral fermions on thelatticerdquo Physical Review Letters vol 71 no 20 pp 3251ndash32541993

[16] H Neuberger ldquoExactly massless quarks on the latticerdquo PhysicsLetters B vol 417 no 1-2 pp 141ndash144 1998

[17] D B Kaplan ldquoA method for simulating chiral fermions on thelatticerdquo Physics Letters B vol 288 no 3-4 pp 342ndash347 1992

[18] M Cheng S Ejiri P Hegde et al ldquoEquation of state for physicalquark massesrdquo Physical Review D vol 81 no 5 Article ID054504 8 pages 2010 httparxivorgabs09112215

[19] S Borsanyi G Endrodi Z Fodor et al ldquoThe QCD equationof state with dynamical quarksrdquo Journal of High Energy Physicsvol 2010 article 77 2010 httparxivorgabs10072580

[20] G Cossu et al ldquoFinite temperature QCD at fixed Q withoverlap fermionsrdquo in International Symposium on Lattice FieldTheory (LATTICE rsquo10) PoS Villasimius Italy June 2010httparxivorgabs10110257

[21] S Borsanyi Y Delgado S Durr et al ldquoQCD thermodynamicswith dynamical overlap fermionsrdquo Physics Letters B vol 713 no3 pp 342ndash346 2012 httparxivorgabs12044089

[22] M Cheng N H Christ P Hegde et al ldquoThe finite temperatureQCD using 2+1 flavors of domain wall fermions at 119873119905 = 8rdquoPhysical Review D vol 81 no 5 Article ID 054510 14 pages2010

[23] T-W Chiu ldquoSimulation of lattice QCD with domain-wallfermionsrdquo In press httparxivorgabs13026918

[24] A Bazavov C Bernard C DeTar et al ldquoNonperturbative QCDsimulations with 2+1 flavors of improved staggered quarksrdquoReviews of Modern Physics vol 82 no 2 pp 1349ndash1417 2010

[25] G Boyd J Engels F Karsch et al ldquoThermodynamics of SU(3)lattice gauge theoryrdquo Nuclear Physics B vol 469 no 3 pp 419ndash444 1996 httparxivorgabshep-lat9602007

[26] F Karsch ldquoLattice QCD at finite temperature and densityrdquoNuclear Physics B-Proceedings Supplements vol 83 Article ID990900 p 14 2000 httparxivorgabshep-lat9909006


[27] F Karsch E Laermann and C Schmidt ldquoThe chiral criticalpoint in 3-flavour QCDrdquo Physics Letters B vol 520 p 41 2001

[28] P de Forcrand and O Philipsen ldquoThe QCD phase diagramfor three degenerate flavors and small baryon densityrdquo NuclearPhysics B vol 673 pp 170ndash186 2003

[29] D Smith and C Schmidt ldquoOn the universal critical behaviorin 3-flavor QCDrdquo in International Symposium on Lattice FieldTheory (LATTICE rsquo11) vol 216 of PoS 2011

[30] X-Y Jin andRDMawhinney ldquoLatticeQCDwith 12 degeneratequark flavorsrdquo in International Symposium on Lattice FieldTheory (LATTICE rsquo11) PoS 2011

[31] M Cheng et al ldquoStudy of the finite temperature transition in 3-flavorQCDrdquoPhysical ReviewD vol 75 no 3 Article ID 03450611 pages 2007

[32] M DrsquoElia A Di Giacomo and C Pica ldquoTwo flavor QCDand confinementrdquo Physical Review D vol 27 no 11 Article ID114510 27 pages 2005 httparxivorgabshep-lat0503030

[33] C Bonati G Cossu M DrsquoElia A Di Giacomo and C PicaldquoA test of first order scaling in Nf =2 QCD a progress reportrdquoin International Symposium on Lattice Field Theory (LATTICErsquo08) vol 204 of PoS July 2008 httparxivorgabs09013231

[34] S Ejiri F Karsch E Laermann et al ldquoMagnetic equation ofstate in (2+1)-flavor QCDrdquo Physical Review D vol 80 no 9Article ID 094505 16 pages 2009 httparxivorgabs09095122

[35] S Gavin A Gocksch and R D Pisarski ldquoQCD and the chiralcritical pointrdquo Physical Review D vol 49 no 7 pp R3079ndashR3082 1994

[36] C Bernard T Burch C DeTar et al ldquoQCD thermodynamicswith three flavors of improved staggered quarksrdquo PhysicalReview D vol 71 no 3 Article ID 034504 11 pages 2005

[37] M Cheng N Christ S Datta et al ldquoTransition temperature inQCDrdquo Physical Review D vol 74 no 5 Article ID 054507 15pages 2006

[38] Y Aoki G Endrodi Z Fodor S D Katz and K K Szabo ldquoTheorder of the quantum chromodynamics transition predicted bythe standard model of particle physicsrdquo Nature vol 443 no7112 pp 675ndash678 2006

[39] H Saito S Ejiri S Aoki et al ldquoPhase structure of finitetemperature QCD in the heavy quark regionrdquo Physical ReviewD vol 84 no 5 Article ID 054502 9 pages 2011

[40] H-T Ding A Bazavov P Hegde F Karsch S Mukherjeeand P Petreczky ldquoExploring phase diagram of 119873119905 = 3 QCDat 120583 = 0 with HISQ fermionsrdquo in International Symposiumon Lattice Field Theory (LATTICE rsquo11) vol 191 of PoS 2011httparxivorgabs11110185

[41] G Endrodi Z Fodor S D Katz and K K Szabo ldquoThe nature ofthe finite temperatureQCD transition as a function of the quarkmassesrdquo in International Symposium on Lattice Field Theory(LATTICE rsquo07) vol 182 2007

[42] S L Adler ldquoAxial-vector vertex in spinor electrodynamicsrdquoPhysical Review vol 177 no 5 pp 2426ndash2438 1969

[43] J Bell andR Jackiw ldquoAPCACpuzzle120587∘rarr 120574120574 in the120590-modelrdquo

Nuovo Cimento A vol 60 no 1 pp 47ndash61 1969[44] K Fujikawa ldquoPath integral for gauge theories with fermionsrdquo

Physical Review D vol 21 no 10 p 2848 1980[45] R D Pisarski and F Wilczek ldquoRemarks on the chiral phase

transition in chromodynamicsrdquo Physical Review D vol 29 no2 pp 338ndash341 1984

[46] J Engels J Fingberg F Karsch D Miller andMWeber ldquoNon-perturbative thermodynamics of SU(N) gauge theoriesrdquo PhysicsLetters B vol 252 no 4 pp 625ndash630 1990

[47] L Giusti and H B Meyer ldquoThermal momentum distributionfrom path integrals with shifted boundary conditionsrdquo PhysicalReview Letters vol 106 no 13 Article ID 131601 4 pages 2011httparxivorgabs10112727

[48] P Petreczky ldquoOn trace anomaly in 2+1 flavor QCDrdquo in Inter-national Symposium on Lattice Field Theory (LATTICE rsquo12) vol069 of PoS 2012 httparxivorgabs12111678

[49] A Bazavov T Bhattacharya M Cheng et al ldquoThe chiraland deconfinement aspects of the QCD transitionrdquo PhysicalReview D vol 85 no 5 Article ID 054503 37 pages 2012httparxivorgabs11111710

[50] S Borsanyi G Endrodi Z Fodor et al ldquoThe QCD equation ofstate and the effects of the charmrdquo in International Symposiumon Lattice Field Theory (LATTICE rsquo11) vol 201 of PoS July 2011httparxivorgabs12040995

[51] T Umeda S Aoki S Ejiri et al ldquoEquation of state in 2+1flavor QCD with improved Wilson quarks by the fixed scaleapproachrdquo Physical Review D vol 85 no 9 Article ID 09450811 pages 2012 httparxivorgabs12024719

[52] F Burger M Kirchner M Muller-Preussker et al ldquoPseudo-critical temperature and thermal equation of state from119873119891 = 2 twisted mass lattice QCDrdquo in Proceedings ofthe 30th International Symposium on Lattice Field Theory(LATTICE rsquo12) vol 68 of PoS Cairns Australia June 2012httparxivorgabs12120982

[53] P de Forcrand and O Philipsen ldquoThe chiral critical line of119873119891 = 2 + 1 QCD at zero and non-zero baryon densityrdquoJournal of High Energy Physics vol 2007 article 77 2007httparxivorgabshep-lat0607017

[54] M Cheng N Christ S Datta et al ldquoQCD equation of state withalmost physical quark massesrdquo Physical Review D vol 77 no 1Article ID 014511 20 pages 2008

[55] S Borsanyi Z Fodor C Hoelbling et al ldquoIs there still any Tcmystery in lattice QCD Results with physical masses in thecontinuum limit IIIrdquo Journal of High Energy Physics vol 2010article 73 2010

[56] M Creutz PoS CONFINEMENT8 article 016 2008[57] C Bernard M Golterman Y Shamir and S R Sharpe ldquorsquot

Hooft vertices partial quenching and rooted staggered QCDrdquoPhysical Review D vol 77 no 11 Article ID 114504 11 pages2008

[58] S Borsanyi S Durr Z Fodor et al ldquoQCD thermodynam-ics with continuum extrapolated Wilson fermions Irdquo Jour-nal of High Energy Physics vol 2012 article 126 2012httparxivorgabs12050440

[59] Z Fodor S D Katz and K K Szabo ldquoDynamical overlapfermions results with hybrid Monte-Carlo algorithmrdquo Journalof High Energy Physics vol 2004 article 3 2004

[60] S Aoki H Fukaya S Hashimoto and T Onogi ldquoFinite volumeQCD at fixed topological chargerdquo Physical ReviewD vol 76 no5 Article ID 054508 11 pages 2007

[61] A Bazavov T Bhattacharya M I Buchoff et al ldquoThe chiraltransition and 119880(1)119860 symmetry restoration from lattice QCDusing Domain Wall Fermionsrdquo Physical Review D vol 86 no9 Article ID 094503 30 pages 2012 httparxivorgabs12053535

[62] F Karsch E Laermann and A Peikert ldquoThe Pressure in 2 2+1and 3 Flavour QCDrdquo Physics Letters B vol 478 no 4 pp 447ndash455 2000 httparxivorgabshep-lat0002003


[63] P Petreczky ldquoReview of recent highlights in lattice calcu-lations at finite temperature and finite densityrdquo in Proceed-ings of the Xth Quark Confinement and the Hadron Spec-trum conference (ConfinementX rsquo12) vol 28 of PoS 2012httparxivorgabs13016188

[64] M Laine and Y Schroeder ldquoQuark mass thresholds in QCDthermodynamicsrdquo Physical Review D vol 73 no 8 Article ID085009 13 pages 2006 httparxivorgabshep-ph0603048

[65] M Cheng ldquoCharm quarks and the QCD equation of staterdquo inInternational Symposiumon Lattice FieldTheory (LATTICE rsquo07)vol 173 of PoS 2007

[66] C DeTar L Levkova S Gottlieb et al ldquoQCD thermodynamicswith nonzero chemical potential at 119873119905 = 6 and effects fromheavy quarksrdquo Physical Review D vol 81 no 11 Article ID114504 17 pages 2010 httparxivorgabs10035682

[67] M Hindmarsh and O Philipsen ldquoWIMP dark matter and theQCD equation of staterdquo Physical Review D vol 71 no 8 ArticleID 087302 4 pages 2005 httparxivorgabshep-ph0501232

[68] M McGuigan and W Soldner ldquoQCD cosmology from the lat-tice equation of staterdquo In press httparxivorgabs08100265

[69] J Engels S Holtmann T Mendes and T Schulze ldquoEquation ofstate andGoldstone-mode effects of the three-dimensionalO(2)modelrdquo Physics Letters B vol 492 no 1-2 pp 219ndash227 2000

[70] D Toussaint ldquoScaling functions for O(4) in three dimensionsrdquoPhysical Review D vol 55 no 1 pp 362ndash366 1997

[71] J Engels and T Mendes ldquoGoldstone-mode effects and scal-ing function for the three-dimensional O(4) modelrdquo NuclearPhysics B vol 572 no 1-2 pp 289ndash304 2000

[72] P Hasenfratz V Laliena and F Niedermayer ldquoThe indextheorem inQCDwith a finite cut-offrdquo Physics Letters B vol 427no 1-2 pp 125ndash131 1998

[73] Z Lin PoS Lattice article 084 2012[74] S Aoki H Fukaya and Y Taniguchi ldquoChiral symmetry restora-

tion the eigenvalue density of the Dirac operator and the axialU(1) anomaly at finite temperaturerdquo Physical Review D vol 86no 11 Article ID 114512 18 pages 2012

[75] G Cossu S Aoki S Hashimoto et al ldquoTopological susceptibil-ity and axial symmetry at finite temperaturerdquo in InternationalSymposium on Lattice Field Theory (LATTICE rsquo11) vol 188 ofPoS July 2011 httparxivorgabs12044519

[76] G Cossu S Aoki H Fukaya et al ldquoFinite temperature studyof the axial U(1) symmetry on the lattice with overlap fermionformulationrdquo httparxivorgabs13046145

[77] H Ohno U M Heller F Karsch and S Mukherjee ldquoU A(1)breaking at finite temperature from theDirac spectrumwith thedynamical HISQ actionrdquo in The 30th International Symposiumon Lattice Field Theory (LATTICE rsquo12) vol 95 of PoS CairnsAustralia June 2012 httparxivorgabs12112591

[78] B B Brandt A Francis H B Meyer H Wittig and O Phil-ipsen ldquoQCD thermodynamics with two flavours of Wilsonfermions on large latticesrdquo in Proceedings of the 30th Interna-tional Symposium on Lattice Field Theory (LATTICE rsquo12) vol073 of PoS Cairns Australia June 2012 httparxivorgabs12106972

[79] P Hasenfratz and F Karsch ldquoChemical potential on the latticerdquoPhysics Letters B vol 125 no 4 pp 308ndash310 1983

[80] J Kogut H Matsuoka M Stone et al ldquoChiral symmetryrestoration in baryon rich environmentsrdquoNuclear Physics B vol225 no 1 pp 93ndash122 1983

[81] R V Gavai ldquoChemical potential on the latticerdquo Physical ReviewD vol 32 no 2 pp 519ndash521 1985

[82] D T Son and M A Stephanov ldquoQCD at finite isospin densityrdquoPhysical Review Letters vol 86 no 4 pp 592ndash595 2001httparxivorgabshep-ph0005225

[83] K Splittorff ldquoLattice simulations of QCD with 120583119861= 0 versusphase quenched QCDrdquo httparxivorgabshep-lat0505001

[84] J Han and M A Stephanov ldquoA random matrix study of theQCD sign problemrdquo Physical Review D vol 78 no 5 ArticleID 054507 7 pages 2008 httparxivorgabs08051939

[85] M P Lombardo K Splittorff and J J M VerbaarschotldquoDistributions of the phase angle of the fermion determinantin QCDrdquo Physical Review D vol 80 no 5 Article ID 054509 19pages 2009

[86] I M Barbour S E Morrison E G Klepfish J B Kogut andM-P Lombardo ldquoResults on finite density QCDrdquo NuclearPhysics B vol 60 pp 220ndash234 1998 httparxivorgabshep-lat9705042

[87] Z Fodor and S D Katz ldquoA new method to study lattice QCDat finite temperature and chemical potentialrdquo Physics Letters Bvol 534 no 1ndash4 pp 87ndash92 2002 httpxxxlanlgovabshep-lat0104001

[88] Z Fodor S D Katz and K K Szabo ldquoThe QCD equation ofstate at nonzero densities lattice resultrdquo Physics Letters B vol568 no 1-2 pp 73ndash77 2003

[89] F Csikor G I Egri Z Fodor S D Katz K K Szabo and AI Toth ldquoEquation of state at finite temperature and chemicalpotential lattice QCD resultsrdquo Journal of High Energy Physicsvol 2004 article 46 2004

[90] C R Allton S Ejiri S J Hands et al ldquoEquation of state for twoflavor QCD at nonzero chemical potentialrdquo Physical Review Dvol 68 no 1 Article ID 014507 2003

[91] C Allton M Doring S Ejiri et al ldquoThermodynamics of twoflavor QCD to sixth order in quark chemical potentialrdquo PhysicalReview D vol 71 no 5 Article ID 054508 20 pages 2005httparxivorgabshep-lat0501030

[92] R V Gavai and S Gupta ldquoPressure and non-linear sus-ceptibilities in QCD at finite chemical potentialsrdquo PhysicalReview D vol 68 no 3 Article ID 034506 6 pages 2003httparxivorgabshep-lat0303013

[93] J Engels O Kaczmarek F Karsch and E Laermann ldquoThequenched limit of lattice QCD at non-zero baryon num-berrdquo Nuclear Physics B vol 558 no 1-2 pp 307ndash326 1999httparxivorgabshep-lat9903030

[94] K-F Liu ldquoFinite density algorithm in lattice QCDmdasha canon-ical ensemble approachrdquo International Journal of ModernPhysics B vol 16 no 14 Article ID 020202 p 2017 2002httparxivorgabshep-lat0202026

[95] A AlexandruM Faber I Horvath andK-F Liu ldquoLatticeQCDat finite density via a new canonical approachrdquo Physical ReviewD vol 72 no 11 Article ID 114513 13 pages 2005

[96] P de Forcrand and S Kratochvila ldquoFinite density QCD with acanonical approachrdquoNuclear Physics B vol 153 no 1 pp 62ndash672006 httparxivorgabshep-lat0602024

[97] M G Alford A Kapustin and FWilczek ldquoImaginary chemicalpotential and finite fermion density on the latticerdquo PhysicalReview D vol 59 no 5 Article ID 054502 4 pages 1999httparxivorgabshep-lat9807039

[98] M-P Lombardo ldquoFinite density (might well be easier) atfinite temperaturerdquo Nuclear Physics B vol 83 p 375 2000httparxivorgabshep-lat9908006

[99] M DrsquoElia and M-P Lombardo ldquoQCD thermodynamics froman imaginary 120583119861 results on the four flavor lattice modelrdquo


Physical Review D vol 70 no 7 Article ID 074509 11 pages2004

[100] P de Forcrand and O Philipsen ldquoThe QCD phase diagramfor small densities from imaginary chemical potentialrdquoNuclearPhysics B vol 642 no 1-2 pp 290ndash306 2002 httparxivorgabshep-lat0205016

[101] G Parisi ldquoOn complex probabilitiesrdquo Physics Letters B vol 131no 4ndash6 pp 393ndash395 1983

[102] F Karsch and H W Wyld ldquoComplex langevin simulation ofthe SU(3) spinmodel with nonzero chemical potentialrdquoPhysicalReview Letters vol 55 no 21 pp 2242ndash2245 1985

[103] J Ambjorn and S-K Yang ldquoNumerical problems in applyingthe langevin equation to complex effective actionsrdquo PhysicsLetters B vol 165 no 1mdash3 pp 140ndash146 1985

[104] G Aarts F A James E Seiler and I O Stamatescu ldquoAdap-tive stepsize and instabilities in complex Langevin dynam-icsrdquo Physics Letters B vol 687 no 2-3 pp 154ndash159 2010httparxivorgabs09120617

[105] G Aarts E Seiler and I O Stamatescu ldquoComplex Langevinmethod when can it be trustedrdquo Physical Review D vol 81 no5 Article ID 054508 13 pages 2010

[106] S Chandrasekharan ldquoFermion bag approach to lattice fieldtheoriesrdquo Physical Review D vol 82 no 2 Article ID 02500715 pages 2010

[107] C Gattringer ldquoFlux representation of an effective Polyakov loopmodel for QCD thermodynamicsrdquo Nuclear Physics B vol 850no 2 pp 242ndash252 2011

[108] R V Gavai and S Gupta ldquoLattice QCD predictions for shapesof event distributions along the freezeout curve in heavy-ioncollisionsrdquo Physics Letters B vol 696 no 5 pp 459ndash463 2011

[109] A Bazavov H-T Ding P Hegde et al ldquoFreeze-out conditionsin heavy Ion collisions from QCD thermodynamicsrdquo PhysicalReview Letters vol 109 no 19 Article ID 192302 5 pages 2012httparxivorgabs12081220

[110] M Golterman Y Shamir and B Svetitsky ldquoBreakdown ofstaggered fermions at nonzero chemical potentialrdquo PhysicalReview D vol 74 no 7 Article ID 071501 5 pages 2006httparxivorgabshep-lat0602026

[111] M Luscher ldquoExact chiral symmetry on the lattice and theGinsparg-Wilson relationrdquo Physics Letters B vol 428 no 3-4pp 342ndash345 1998

[112] J Mandula ldquoNote on the lattice fermion chiral symmetrygrouprdquo In press httparxivorgabs07120651

[113] J C R Bloch and TWettig ldquoOverlap dirac operator at nonzerochemical potential and randommatrix theoryrdquo Physical ReviewLetters vol 97 no 1 Article ID 012003 4 pages 2006

[114] D Banerjee R V Gavai and S Sharma ldquoThermodynamics ofthe ideal overlap quarks on the latticerdquo Physical Review D vol78 no 1 Article ID 014506 13 pages 2008 and PoS (LATTICE2008) 177

[115] R Narayanan and S Sharma ldquoIntroduction of the chemicalpotential in the overlap formalismrdquo Journal of High EnergyPhysics vol 2011 article 151 2011

[116] R V Gavai and S Sharma ldquoExact chiral invariance at finitedensity on the latticerdquo Physics Letters B vol 716 no 3ndash5 pp446ndash449 2012

[117] S Jeon and V Koch ldquoCharged particle ratio fluctuation as asignal for quark-gluon plasmardquo Physical Review Letters vol 85no 10 pp 2076ndash2079 2000

[118] M Asakawa UW Heinz and B Muller ldquoFluctuation probes ofquark deconfinementrdquo Physical Review Letters vol 85 no 10pp 2072ndash2075 2000 httparxivorgabshep-ph0003169

[119] S A Gottlieb W Liu D Toussaint R L Renken and RL Sugar ldquoQuark-number susceptibility of high-temperatureQCDrdquo Physical Review Letters vol 59 no 20 pp 2247ndash22501987

[120] A Bazavov T Bhattacharya C E DeTar et al ldquoFluctuationsand Correlations of net baryon number electric charge andstrangeness a comparison of lattice QCD results with thehadron resonance gas modelrdquo Physical Review D vol 86 no3 Article ID 034509 15 pages 2012 httparxivorgabs12030784

[121] S Borsanyi Z Fodor S D Katz et al ldquoFluctuations ofconserved charges at finite temperature from lattice QCDrdquoJournal of High Energy Physics vol 2012 article 138 2012httparxivorgabs11124416

[122] J O Andersen S Mogliacci N Su and A Vuorinen ldquoQuarknumber susceptibilities from resummed perturbation theoryrdquoPhysical ReviewD vol 87 no 7 Article ID 074003 6 pages 2013httparxivorgabs12100912

[123] N Haque M G Mustafa and M Strickland ldquoQuark numbersusceptibilities from two-loop hard thermal loop perturbationtheoryrdquo In presshttparxivorgabs13023228

[124] S Gupta ldquoFinding the critical end point of QCD lat-tice and experimentrdquo in Proceedings of Critical Point andOnset of Deconfinement (CPOD rsquo09) vol 25 of PoS 2009httparxivorgabs09094630

[125] F Karsch and K Redlich ldquoProbing freeze-out conditions inheavy ion collisions with moments of charge fluctuationsrdquoPhysics Letters B vol 695 no 1ndash4 pp 136ndash142 2011

[126] V Koch A Majumder and J Randrup ldquoBaryon-strangenesscorrelations a diagnostic of strongly interacting matterrdquo Phys-ical Review Letters vol 95 no 18 Article ID 182301 4 pages2005

[127] R V Gavai and S Gupta ldquoFluctuations strangeness andquasiquarks in heavy-ion collisions from lattice QCDrdquo PhysicalReview D vol 73 no 1 Article ID 014004 11 pages 2006

[128] J Cleymans H Oeschler K Redlich and S Wheaton ldquoCom-parison of chemical freeze-out criteria in heavy-ion collisionsrdquoPhysical Review C vol 73 no 3 Article ID 034905 10 pages2006

[129] M Kitazawa and M Asakawa ldquoRevealing baryon numberfluctuations from proton number fluctuations in relativisticheavy ion collisionsrdquo Physical Review C vol 85 no 2 ArticleID 021901 5 pages 2012 httparxivorgabs11072755

[130] M Asakawa and K Yazaki ldquoChiral restoration at finite densityand temperaturerdquo Nuclear Physics A vol 504 no 4 pp 668ndash684 1989

[131] J Berges and K Rajagopal ldquoColor superconductivity andchiral symmetry restoration at non-zero baryon density andtemperaturerdquo Nuclear Physics B vol 538 no 1-2 pp 215ndash2321999

[132] A M Halasz A D Jackson R E Shrock M A Stephanov andJ J M Verbaarschot ldquoPhase diagram of QCDrdquo Physical ReviewD vol 58 no 9 Article ID 096007 11 pages 1998

[133] Z Fodor and S D Katz ldquoCritical point of QCD at finiteT and 120583lattice results for physical quark massesrdquo Journal of High EnergyPhysics vol 2004 article 50 2004

[134] M Stephanov K Rajagopal and E Shuryak ldquoSignatures of thetricritical point in QCDrdquo Physical Review Letters vol 81 no 22pp 4816ndash4819 1998

[135] R V Gavai and S Gupta ldquoThe critical end point of QCDrdquoPhysical Review D vol 71 no 11 Article ID 114014 21 pages2005 httparxivorgabshep-lat0412035


[136] R V Gavai and S Gupta ldquoQCD at finite chemical potentialwith six time slicesrdquo Physical Review D vol 78 no 11 ArticleID 114503 15 pages 2008 httparxivorgabs08062233

[137] S Datta R V Gavai and S Gupta ldquoThe QCD Critical Pointmarching towards continuumrdquoNuclear Physics A vol 904-905pp 883cndash886c 2013 httparxivorgabs12106784

[138] A Li A Alexandru and K-F Liu ldquoCritical point of 119873119891 =

3 QCD from lattice simulations in the canonical ensemblerdquoPhysical ReviewD vol 84 no 7 Article ID 071503 5 pages 2011httparxivorgabs11033045

[139] Ph de Forcrand and O Philipsen ldquoThe Chiral critical point of119873119891 = 3 QCD at finite density to the order (120583119879)4rdquo Journal ofHigh Energy Physics vol 2008 article 12 2008

[140] J T Moscicki M Wos M Lamanna P de Forcrand andO Philipsen ldquoLattice QCD thermodynamics on the GridrdquoComputer Physics Communications vol 181 no 10 pp 1715ndash1726 2010 httparxivorgabs09115682

[141] O Philipsen ldquoStatus of the QCD phase diagram from lat-tice calculationsrdquo Acta Physica Polonica vol 5 p 825 2012httparxivorgabs11115370

[142] O Kaczmarek F Karsch E Laermann et al ldquoThe phaseboundary for the chiral transition in (2+1)-flavor QCD at smallvalues of the chemical potentialrdquo Physical Review D vol 83 no1 Article ID 014504 9 pages 2011 httparxivorgabs10113130

[143] G Endrodi Z Fodor S Katz and K Szabo ldquoThe QCD phasediagram at nonzero quark densityrdquo Journal of High EnergyPhysics vol 2011 article 1 2011 httparxivorgabs11021356

[144] M A Stephanov ldquoNon-gaussian fluctuations near the QCDcritical pointrdquo Physical Review Letters vol 102 no 3 Article ID032301 4 pages 2009

[145] M Stephanov K Rajagopal and E Shuryak ldquoEvent-by-eventfluctuations in heavy ion collisions and the QCD critical pointrdquoPhysical Review D vol 60 no 11 Article ID 114028 32 pages1999

[146] S Ejiri F Karsch E Laermann andC Schmidt ldquoThe isentropicequation of state of 2-flavor QCDrdquo Physical Review D vol 73no 5 Article ID 054506 6 pages 2006 httparxivorgabshep-lat0512040

[147] Sz Borsanyi G Endrodi Z Fodor et al ldquoQCD equation of stateat nonzero chemical potential continuum results with physicalquark masses at order 1205832rdquo Journal of High Energy Physics vol1208 article 53 2012 httparxivorgabs12046710

[148] T Takaishi P de Forcrand and A Nakamura ldquoEquation ofState at Finite Density from Imaginary Chemical Potentialrdquo inInternational SymposiumonLattice FieldTheory (LATTICE rsquo09)vol 198 of PoS 2009 httparxivorgabs10020890

[149] R V Gavai and S Sharma ldquoFaster method of computation oflattice quark number susceptibilitiesrdquoPhysical ReviewD vol 85no 5 Article ID 054508 8 pages 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


and how the continuum limit is taken which is essential torelate the lattice data with the real world experiments I havediscussed the current understanding we have of the nature ofQCD phase transition as a function of quark masses inferredfrom lattice studies Subsequently the different aspects of thehot QCD medium for physical quark masses are discussedthe EoS the nature and the temperature of transition andthe behaviour of various thermodynamic observables in thedifferent phases In the study of thermodynamics the contri-bution of the lighter 119906 119889 and 119904 quarks is usually consideredThe effect of heavier charm quarks onQCD thermodynamicsis discussed in this section in view of their relevance forthe heavy ion experiments at LHC where hydrodynamicevolution is expected to set in already at temperatures about500MeV and also for the physics of early universe Therelevance of chiral symmetry for theQCDphase diagram andthe effects of chiral anomaly are discussed in detailThe chiralanomaly is believed to have an important role in shaping thephase diagram and several lattice studies in the recent yearsare trying to understand its effect It is a difficult problemand I have tried to compile the recent results and review thegeneral understanding within the community about how toimprove upon them

The second section is about lattice QCD at finite densitywhere there is an inherent short coming of the lattice algo-rithms due to the so-called sign problem A brief overviewof the different methods used and those being developed bythe lattice practitioners to circumvent this problem is givenIt is an active field of research with a lot of understandingof the origin and the severity of this problem gained inrecent years which is motivating the search for its possiblecure In the regime where the density of baryons is not toolarge which is being probed by the experiments at RHIClattice techniques have been used successfully to producesome interesting results One such important proposal inthe recent time is the first principles determination of thechemical freezeout curve using experimental data on theelectric charge fluctuations This and the lattice results onthe fluctuations of different quantum numbers in the hotmedium and the EoS at finite baryon density are discussedin detail An important feature of the QCD phase diagramis the possible presence of a critical end-point for thechiral first order transition Since critical end-point searchis one of the main objectives at RHIC I have reviewedthe current lattice results on this topic The presence ofthe critical end-point is still not conclusively proven fromlattice studies It is a very challenging problem and I mentionthe further work in progress to address this problem effec-tively Fermions with exact chiral symmetry on the latticeare important in this context I have discussed the recentsuccessful development to construct fermion operators thathave exact chiral symmetry even at finite density whichwould be relevant for future studies on the critical end-pointThe signatures of the critical end-point could be detectedin the experiments if the critical region is not separatedfrom the freezeout curve It is thus crucial to estimate thecurvature of the critical line fromfirst principles and I devotean entire subsection to discuss the lattice results on thistopic

I apologize for my inability to include all the pioneeringworks that have firmly established this subject and also toreview the extensive set of interesting contemporary worksFor a comprehensive review of the current activity in latticethermodynamics at finite temperature and density I refer tothe excellent review talks of the lattice conference 2012 [4 5]

2 QCD at Finite Temperature on the Lattice

The starting point of any thermodynamic study is the parti-tion functionThe QCD partition function for119873119891 flavours ofquarks in the canonical ensemble is given as

ZQCD (119879 119881) = intD119880120583 (119909)

119873119891

prod

119891=1

det119863119891119890minus119878119866 (1)

where 119863119891 is the fermion operator for each flavour ofquark 119891 119880120583 is the gauge link defined as 119880120583(119909) =

exp(minus119894119892 int119909119860120583(119909

1015840)119889119909

1015840) in terms of gauge fields 119860120583 which

are adjoint representation of the SU(3) color group and 119892

is the strength of the gauge coupling 119878119866 is the gluon actionin Euclidean space of finite temporal extent of size denotedby the inverse of the temperature of the system 119879 LatticeQCD involves discretizing the spacetime into a lattice with aspacing denoted by 119886The volume of the lattice is given as119881 =

11987331198863 where119873 are the number of lattice sites along the spatial

directions and the temperature being 119879 = 1(119873120591119886) where119873120591 are the number of sites along the temporal direction Thelattice is usually denoted as 1198733

times 119873120591 The gluon action andthe fermion determinant are discretized on the lattice Thesimplest gluon action known asWilson plaquette action is ofthe form

119878119866 =6

1198922sum

119909120583]120583lt]

(1 minus1

3Tr Re119880120583] (119909))

119880120583] (119909) = 119880120583 (119909)119880] (119909 + 120583)119880dagger120583 (119909 + ]) 119880dagger

] (119909)

(2)

where 119880120583](119909) is called a plaquette The naive discretizationof the continuum Dirac equation on the lattice results in thefermion operator of the form

119863119891 (119909 119910)

= sum

119909119910

[

4

sum

120583=1

1

2120574120583 (119880120583 (119909) 120575119910119909+120583 minus 119880

dagger120583 (119910) 120575119910119909minus120583) + 119886119898119891120575119909119910]

(3)

where in each of the expressions the site index 119909 = 1 minus 1198733times

119873120591 The discretization of the gluon and fermion operatorsare not unique and there are several choices which givethe correct continuum limit Usually discretized operatorswith small finite 119886 corrections are preferred Reducing 119886-dependent corrections by adding suitable ldquoirrelevantrdquo termsin the Renormalization Group (RG) sense is known asthe improvement of the operator Another issue relatedto the discretization of the fermion operator is called theldquofermion doubling problemrdquo It arises because the naive





as

(1199032120597119881119902119902 (119903)

120597119903)

119903=1199031

= 10 (4)



119871 (x) = 1

3Tr 119875

119873120591

prod

1199094=1



⟨120595119891120595119891⟩ = lim119898119891rarr0

lim119881rarrinfin

119879

119881

120597 lnZQCD

120597119898119891

119891 = 1 119873119891

(6)











Crossover

Physical pointms

Nf = 1

Nf = 3

Pure gauge

Firstorder

Firstorder

Second order

infin

infin

Nf = 2

mc = (ms270 ms270)

O(4)Z(2)



mud

mtrics



119868 (119879) = 1198795 120597

120597119879

119901

1198794 (7)


119868 (119879)

1198794= minus119873

4120591 (119886

119889120573

119889119886(⟨119878119866⟩ minus ⟨119878119866⟩0)

+ sum

119891

119886

119889 (119898119891119886)

119889119886(⟨120595119891120595119891⟩ minus ⟨120595119891120595119891⟩0

))

120573 =6

1198922

(8)



1

2

3

4

5

6

150 200 250 300 350

(120576minus3p

)T4

T (MeV)

Stout cont

r1 scale

N120591 = 6

N120591 = 8

N120591 = 10

N120591 = 12

HISQtreeN120591 = 4

(a)

05

1

15

2

25

3

35

4

300 400 500 600 700 800

(120576minus3p

)T4

T (MeV)

N120591 = 6

N120591 = 8

N120591 = 6N120591 = 8


HISQtreeN120591 = 4 p4 N120591 = 4

(b)


5

4

3

2

1

0

(120576minus3p

)T4

100 150 200 250 300 350 400 450 500 550T (MeV)


323 times 10


(a)

4

3

2

1

2

1

I(T)T4

0 200 400 600 800 1000T (MeV)


0 50 100 150

HRG

(b)









119868 (119879)

1198794= 119890


sdot (ℎ0 +1198910 [tanh (1198911119905 + 1198912) + 1]

1 + 1198921119905 + 11989221199052

)

119905 =119879

200MeV

(9)


ℎ0 = 01396 ℎ1 = minus018 ℎ2 = 0035

1198910 = 276 1198911 = 679 1198912 = minus529

1198921 = minus047 1198922 = 104

(10)




Δ 119897119904 (119879) =

⟨120595120595⟩119897119879 minus (119898119897119898119904) ⟨120595120595⟩119904119879

⟨120595120595⟩1198970 minus (119898119897119898119904) ⟨120595120595⟩1199040

119897 = 119906 119889 (11)


120594119877 =1

1198811198793119898

2119897

1205972

1205971198982119897

(lnZ (119879) minus lnZ (0)) (12)



119879119888 =

170 (4) (3) for 119871119877

157 (3) (3) Δ 119897119904

147 (2) (3) 120594119877

(13)





0

5

10

15

20

120576T4

700500 600300 400100 200T (MeV)

3pT4

(120576 minus 3p)T4

(a)

900700500300100

10

8

6

4

2

0

minus2

T (MeV)

Interpolation

N120591 = 4N120591 = 6N120591 = 8

N120591 = 10

N120591 = 12

(120598minus3p

)T4

(b)


1

08

06

04

02

0

Δls

fK scale

T (MeV)120 140 160 180 200

AsqtadN120591 = 8

N120591 = 12

HISQtreeN120591 = 6

N120591 = 8

N120591 = 12

N120591 = 8 ml = 0037ms

Stout cont

(a)

fK scale

T (MeV)120 140 160 180 200

04

035

03

025

02

015

01

005

0

Lre

n(T

)

HISQtreeN120591 = 6

N120591 = 8

N120591 = 12

AsqtadN120591 = 8

N120591 = 12

Stout cont

(b)





125 150 175 200 225 250 275(MeV)

0005

0

minus0005

minus001

minus0015

minus002

minus003

minus0025

minus0035008 01 012 014 016

TmΩ


mR

R120595Rm

1205874

120595

(a)

150 175 200 225 250 275(MeV)

008 01 012 014 016TmΩ


2

15

1

05

0

LR

(b)











001

0

minus001

minus002

minus003

minus004

minus005

minus006

minus007

minus008

120 140 160 180 200 220 240 260T (MeV)

01 012 014 016 018 02 022 024

6 times 123

8 times 163

Staggered

Tw0

mR120595120595Rm

1205874

(a)

0002

00015

0001

00005

0140 150 160 170 180 190 200

T (MeV)

12059511205951T3

Δ lsT3

(b)






200 400 600 800 1000T (MeV)

20

15

10

5

15

10

5

SB

100 150 200 250

s(T)T3

N120591 = 6N120591 = 8N120591 = 10

(a)

200 400 600 800 1000T (MeV)

5

4

3

2

1

SB

100 150 200 250

25215105

p(T

)T4

N120591 = 6N120591 = 8N120591 = 10

(b)

c2 s(T

)

200 400 600 800 1000T (MeV)

035

03

025

02

015

01

035030250201501

SB

100 150 200 250 300

N120591 = 6N120591 = 8N120591 = 10

(c)







119872119887 = 119898119904

⟨120595120595⟩

1198794 (14)



20

15

10

5

0


120576T4

IT4

pT4

150 200 250 300 350 400T (MeV)

(a)

200 300 400 500 600 700 800 900 1000T (MeV)

6

5

4

3

2

1

0

PT

4

Nf = 2 + 1 EOS Nf = 2 + 1 + 1 N120591 = 8

Nf = 2 + 1 + 1 N120591 = 6 Nf = 2 + 1 + 1 N120591 = 10

(b)



119872119887 (119879119867) = ℎ1120575

119891119866 (119905

ℎ1120573120575) + 119891reg (119879119867) (15)



119905 =1

1199050

119879 minus 1198791198880

1198791198880

ℎ =119867

ℎ0

119867 =119898119897

119898119904

(16)


119891reg = 119867(1198860 + 1198861

119879 minus 1198791198880

1198791198880

+ 1198862(119879 minus 1198791198880

1198791198880

)

2

) (17)







120594120587 minus 120594120575 = int1198894119909 [⟨120595 (119909) 12059121205745120595 (119909) 120595 (0) 12059121205745120595 (0)⟩

minus ⟨120595 (119909) 1205912120595 (119909) 120595 (0) 1205912120595 (0)⟩]

(18)



000

050

100

150

200

250

094 096 098 100 102 104 106 108TTc

Mb

120

110

140

180

Chiral limit

mlms

(a)

000

050

100

150

200

All masses

th1120573120575

Mbh

1120575

O(2)

15

25

110

120

140

180


mlms

(b)



⟨120595120595⟩ = int119889120582120588 (120582119898)2119898

1198982 + 1205822

120594120587 minus 120594120575 = int119889120582120588 (120582119898)4119898

2

(1198982 + 1205822)2

(19)





1205822)



2120575(120582) + 1198881120582 for the density




lim119898rarr0

⟨120588 (120582119898)⟩ = lim119898rarr0

⟨120588 (119898)⟩1205823

3+ O (120582

4) (20)



350

300

250

200

150

100

50

0

T (MeV)140 150 160 170 180 190 200

120594disc T2

1205945disc T2 120594top T

2(ml + mres )2

(120594120587 minus 120594120575)T2

(a)

0025

002

0015

001

0005

00

120588(120582)

002 004 006 008 01120582

163 times 8

Min(120582100)ml

ms

(b)


T ⋍ 170MeV

T ⋍ 210MeV

1

05

0

0 100 200 300 400 500

120588(120582)

(MeV

3)

120588(120582)

(MeV

3)

120588(120582)

(MeV

3)

1

05

0

1

05

0

120582 (MeV)

T ≃ 180sim190MeV

120573 = 218 am = 005

120573 = 218 am = 001

120573 = 225 am = 001120573 = 220 am = 001120573 = 220 am = 0025120573 = 220 am = 005

120573 = 240 am = 001

120573 = 230 am = 001

120573 = 230 am = 0025

120573 = 230 am = 005

(a)



0 2 4 6 8 10 12 14 16

times10minus7

35

3

25

2

15

1

Distance

= 001120573 = 225 (Tsim192) ma

(b)






120588(120582)

120582a

0 004 008 012 016


T = 3301MeVmlms = 120

10eminus02

10eminus03

10eminus04

10eminus05

10eminus06

(a)

14

12

1

08

06

04

02

M(2120587

T)

085 09 095 1 105 11 115 12TTc

P

S

V

A

(b)







119873119891

prod

119891=1

det119863119891 (120583) 119890minus119878119866

(21)


119863119891(120583)119909119910 = [

3

sum

119894=1

1

2120574119894 (119880119894 (119909) 120575119910119909+119894 minus 119880

dagger119894 (119910) 120575119910119909minus119894)

+1

21205744 (119890

1205831198861198804 (119909) 120575119910119909+4 minus 119890

minus120583119886119880

dagger120583 (119910) 120575119910119909minus4)

+ 119886119898119891120575119909119910]

(22)



10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816e119894120579(23)





⟨e119894120579⟩ =

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816e119894120579119890minus119878119866

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816119890minus119878119866

= 119890minus119881Δ119865119879

(24)






120594119883119899 =

119879

119881

120597119899 lnZ120597120583

119899119883

119883 equiv 119861 119878 119876 (25)





1198762 120594

1198612 (120583 =



1205941198612 (120583)

1198792=1205941198612 (0)

1198792+

1205832

211987921205941198614 (0) +

1205834

411987941205941198616 (0) 119879

2+ sdot sdot sdot

(26)







120594119861119878119876119894119895119896 =

119879

119881

120597119894+119895+119896 lnZ

120597120583119894119861120597120583

119895

119878120597120583119896119876

(27)


11986111987811 120594

1198782





SB

HRG

1

08

06

04

02

0

minus3120594

BS

11120594

S 2

120 140 160 180 200 220 240T (MeV)

N120591 = 12

N120591 = 8N120591 = 6

fK scale

(a)

HRG

005

004

003

002

001

0

120594BQ

11T

2

fK scale

120 140 160 180 200 220 240T (MeV)

N120591 = 12 N120591 = 6N120591 = 8Cont extrap

(b)




11987711988312 equiv

119872119883

1205902119883

=120583119861

119879(119877

119883112 +

1205832119861

1198792119877119883312 + O (120583

4119861))

11987711988331 equiv

1198781198831205903119883

119872119883

= 119877119883131 +

1205832119861

1198792119877119883331 + O (120583

4119861)

(28)













RQ 31

HRG

30

25

20

15

10

05

00

Free

T (MeV)140 150 160 170 180 190 200 210 220 230 240

120583BT = 1

120583BT = 0

N120591 = 6

N120591 = 8

(a)

0035

0030

0025

0020HRG

HRG

T = 160MeV

T = 160MeV

T = 170MeV

T = 170MeV

T = 150MeV

T = 150MeV

030

025

020

015

120583S120583

B

0 20 40 60 80 100 120 140 160 180 200120583B (MeV)

minus120583Q120583

B(b)



07

08

09

1

11

0 1 2 3 4 5

Freezeout curve

120583BT

TT

c

30 GeV20 GeV

18 GeV (CERN)10 GeV

(a)

HRG

Power law

10

1

01

001

0001

m1

1 10 100 1000 10000radicSNN (GeV)

(b)








119903119899 (119899 = odd) = radic120594119861119899+1

1198792120594119861119899+3

119903119899 (119899 = even) = [1205941198612

119879119899120594119861119899+2

]

1119899

(29)


1198618 This




119861) [140]

119898119888 (120583119861)

119898119888 (0)= 1 minus 39 (8) (

120583119861

3120587119879)

2





119905 =1

1199050

(119879 minus 1198791198880

1198791198880

+ 120581119861

120583119861

3119879) ℎ =

119898119897

ℎ0119898119904

(31)


119872119887 (120583) = 119872119887 (0) +120594119898119861

2119879(120583119861

3119879)

2

+ O(120583119861

3119879)

4

(32)


(1198792119898119904)120597

2119872119887120597(1205831198613119879)


120594119898119861

119879=2120581119861119879

1199050119898119904

ℎminus(1minus120573)120573120575

1198911015840119866 (

119905

ℎ1120573120575) (33)




119879119888 (120583119861)

119879119888 (0)= 1 minus 120581119861(

120583119861

3119879)

2

+ O(120583119861

3119879)

4

997904rArr 119879119888 (120583119861) ≃ 154 (1 minus 00066(120583119861

3119879)

2

) MeV

(34)


150

100

50

Tem

pera

ture

(MeV

)


Freezeout

T120594119904c (120583)

T120595120595c (120583)



(a)

007

006

005

004

003

002

001


z

minus2120581qf998400G(z) 120

140180

minusm

st0h(1minus120573)120573120575120594mqT

2

N120591 = 8 mlms = 120

N120591 = 4 mlms = 110

(b)







119875 (120583119897 120583119904)

1198794=119875 (0)

1198794+sum

119894119895

120594119894119895 (119879)

1198794minus119894minus119895(120583119897

119879)

119894

(120583119904

119879)

119895

(35)



119868 (120583119897 120583119904)

1198794= minus

1198733120591

1198733

119889 lnZ119889 ln 119886

=119868 (0)

1198794+sum

119894119895

119887119894119895 (119879) (120583119897

119879)

119894

(120583119904

119879)

119895

(36)


8

6

4

2

0

IT4

100 200 300 400 500 600T (MeV)

SnB = 30

SnB = infinSnB = 45

SnB = 300

Filled Nt = 6

Empty Nt = 4

(a)

6

4

2

5 10 15

pT

4

120598T4






11990611990411120594









119901 up to O ((120583119861119879)4)

119901 up to O ((120583119861119879)2)

le

11 for120583119861

119879le 2

135 for120583119861

119879le 3

(37)


4 Summary





Acknowledgments


References


































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







as

(1199032120597119881119902119902 (119903)

120597119903)

119903=1199031

= 10 (4)



119871 (x) = 1

3Tr 119875

119873120591

prod

1199094=1



⟨120595119891120595119891⟩ = lim119898119891rarr0

lim119881rarrinfin

119879

119881

120597 lnZQCD

120597119898119891

119891 = 1 119873119891

(6)











Crossover

Physical pointms

Nf = 1

Nf = 3

Pure gauge

Firstorder

Firstorder

Second order

infin

infin

Nf = 2

mc = (ms270 ms270)

O(4)Z(2)



mud

mtrics



119868 (119879) = 1198795 120597

120597119879

119901

1198794 (7)


119868 (119879)

1198794= minus119873

4120591 (119886

119889120573

119889119886(⟨119878119866⟩ minus ⟨119878119866⟩0)

+ sum

119891

119886

119889 (119898119891119886)

119889119886(⟨120595119891120595119891⟩ minus ⟨120595119891120595119891⟩0

))

120573 =6

1198922

(8)



1

2

3

4

5

6

150 200 250 300 350

(120576minus3p

)T4

T (MeV)

Stout cont

r1 scale

N120591 = 6

N120591 = 8

N120591 = 10

N120591 = 12

HISQtreeN120591 = 4

(a)

05

1

15

2

25

3

35

4

300 400 500 600 700 800

(120576minus3p

)T4

T (MeV)

N120591 = 6

N120591 = 8

N120591 = 6N120591 = 8


HISQtreeN120591 = 4 p4 N120591 = 4

(b)


5

4

3

2

1

0

(120576minus3p

)T4

100 150 200 250 300 350 400 450 500 550T (MeV)


323 times 10


(a)

4

3

2

1

2

1

I(T)T4

0 200 400 600 800 1000T (MeV)


0 50 100 150

HRG

(b)









119868 (119879)

1198794= 119890


sdot (ℎ0 +1198910 [tanh (1198911119905 + 1198912) + 1]

1 + 1198921119905 + 11989221199052

)

119905 =119879

200MeV

(9)


ℎ0 = 01396 ℎ1 = minus018 ℎ2 = 0035

1198910 = 276 1198911 = 679 1198912 = minus529

1198921 = minus047 1198922 = 104

(10)




Δ 119897119904 (119879) =

⟨120595120595⟩119897119879 minus (119898119897119898119904) ⟨120595120595⟩119904119879

⟨120595120595⟩1198970 minus (119898119897119898119904) ⟨120595120595⟩1199040

119897 = 119906 119889 (11)


120594119877 =1

1198811198793119898

2119897

1205972

1205971198982119897

(lnZ (119879) minus lnZ (0)) (12)



119879119888 =

170 (4) (3) for 119871119877

157 (3) (3) Δ 119897119904

147 (2) (3) 120594119877

(13)





0

5

10

15

20

120576T4

700500 600300 400100 200T (MeV)

3pT4

(120576 minus 3p)T4

(a)

900700500300100

10

8

6

4

2

0

minus2

T (MeV)

Interpolation

N120591 = 4N120591 = 6N120591 = 8

N120591 = 10

N120591 = 12

(120598minus3p

)T4

(b)


1

08

06

04

02

0

Δls

fK scale

T (MeV)120 140 160 180 200

AsqtadN120591 = 8

N120591 = 12

HISQtreeN120591 = 6

N120591 = 8

N120591 = 12

N120591 = 8 ml = 0037ms

Stout cont

(a)

fK scale

T (MeV)120 140 160 180 200

04

035

03

025

02

015

01

005

0

Lre

n(T

)

HISQtreeN120591 = 6

N120591 = 8

N120591 = 12

AsqtadN120591 = 8

N120591 = 12

Stout cont

(b)





125 150 175 200 225 250 275(MeV)

0005

0

minus0005

minus001

minus0015

minus002

minus003

minus0025

minus0035008 01 012 014 016

TmΩ


mR

R120595Rm

1205874

120595

(a)

150 175 200 225 250 275(MeV)

008 01 012 014 016TmΩ


2

15

1

05

0

LR

(b)











001

0

minus001

minus002

minus003

minus004

minus005

minus006

minus007

minus008

120 140 160 180 200 220 240 260T (MeV)

01 012 014 016 018 02 022 024

6 times 123

8 times 163

Staggered

Tw0

mR120595120595Rm

1205874

(a)

0002

00015

0001

00005

0140 150 160 170 180 190 200

T (MeV)

12059511205951T3

Δ lsT3

(b)






200 400 600 800 1000T (MeV)

20

15

10

5

15

10

5

SB

100 150 200 250

s(T)T3

N120591 = 6N120591 = 8N120591 = 10

(a)

200 400 600 800 1000T (MeV)

5

4

3

2

1

SB

100 150 200 250

25215105

p(T

)T4

N120591 = 6N120591 = 8N120591 = 10

(b)

c2 s(T

)

200 400 600 800 1000T (MeV)

035

03

025

02

015

01

035030250201501

SB

100 150 200 250 300

N120591 = 6N120591 = 8N120591 = 10

(c)







119872119887 = 119898119904

⟨120595120595⟩

1198794 (14)



20

15

10

5

0


120576T4

IT4

pT4

150 200 250 300 350 400T (MeV)

(a)

200 300 400 500 600 700 800 900 1000T (MeV)

6

5

4

3

2

1

0

PT

4

Nf = 2 + 1 EOS Nf = 2 + 1 + 1 N120591 = 8

Nf = 2 + 1 + 1 N120591 = 6 Nf = 2 + 1 + 1 N120591 = 10

(b)



119872119887 (119879119867) = ℎ1120575

119891119866 (119905

ℎ1120573120575) + 119891reg (119879119867) (15)



119905 =1

1199050

119879 minus 1198791198880

1198791198880

ℎ =119867

ℎ0

119867 =119898119897

119898119904

(16)


119891reg = 119867(1198860 + 1198861

119879 minus 1198791198880

1198791198880

+ 1198862(119879 minus 1198791198880

1198791198880

)

2

) (17)







120594120587 minus 120594120575 = int1198894119909 [⟨120595 (119909) 12059121205745120595 (119909) 120595 (0) 12059121205745120595 (0)⟩

minus ⟨120595 (119909) 1205912120595 (119909) 120595 (0) 1205912120595 (0)⟩]

(18)



000

050

100

150

200

250

094 096 098 100 102 104 106 108TTc

Mb

120

110

140

180

Chiral limit

mlms

(a)

000

050

100

150

200

All masses

th1120573120575

Mbh

1120575

O(2)

15

25

110

120

140

180


mlms

(b)



⟨120595120595⟩ = int119889120582120588 (120582119898)2119898

1198982 + 1205822

120594120587 minus 120594120575 = int119889120582120588 (120582119898)4119898

2

(1198982 + 1205822)2

(19)





1205822)



2120575(120582) + 1198881120582 for the density




lim119898rarr0

⟨120588 (120582119898)⟩ = lim119898rarr0

⟨120588 (119898)⟩1205823

3+ O (120582

4) (20)



350

300

250

200

150

100

50

0

T (MeV)140 150 160 170 180 190 200

120594disc T2

1205945disc T2 120594top T

2(ml + mres )2

(120594120587 minus 120594120575)T2

(a)

0025

002

0015

001

0005

00

120588(120582)

002 004 006 008 01120582

163 times 8

Min(120582100)ml

ms

(b)


T ⋍ 170MeV

T ⋍ 210MeV

1

05

0

0 100 200 300 400 500

120588(120582)

(MeV

3)

120588(120582)

(MeV

3)

120588(120582)

(MeV

3)

1

05

0

1

05

0

120582 (MeV)

T ≃ 180sim190MeV

120573 = 218 am = 005

120573 = 218 am = 001

120573 = 225 am = 001120573 = 220 am = 001120573 = 220 am = 0025120573 = 220 am = 005

120573 = 240 am = 001

120573 = 230 am = 001

120573 = 230 am = 0025

120573 = 230 am = 005

(a)



0 2 4 6 8 10 12 14 16

times10minus7

35

3

25

2

15

1

Distance

= 001120573 = 225 (Tsim192) ma

(b)






120588(120582)

120582a

0 004 008 012 016


T = 3301MeVmlms = 120

10eminus02

10eminus03

10eminus04

10eminus05

10eminus06

(a)

14

12

1

08

06

04

02

M(2120587

T)

085 09 095 1 105 11 115 12TTc

P

S

V

A

(b)







119873119891

prod

119891=1

det119863119891 (120583) 119890minus119878119866

(21)


119863119891(120583)119909119910 = [

3

sum

119894=1

1

2120574119894 (119880119894 (119909) 120575119910119909+119894 minus 119880

dagger119894 (119910) 120575119910119909minus119894)

+1

21205744 (119890

1205831198861198804 (119909) 120575119910119909+4 minus 119890

minus120583119886119880

dagger120583 (119910) 120575119910119909minus4)

+ 119886119898119891120575119909119910]

(22)



10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816e119894120579(23)





⟨e119894120579⟩ =

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816e119894120579119890minus119878119866

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816119890minus119878119866

= 119890minus119881Δ119865119879

(24)






120594119883119899 =

119879

119881

120597119899 lnZ120597120583

119899119883

119883 equiv 119861 119878 119876 (25)





1198762 120594

1198612 (120583 =



1205941198612 (120583)

1198792=1205941198612 (0)

1198792+

1205832

211987921205941198614 (0) +

1205834

411987941205941198616 (0) 119879

2+ sdot sdot sdot

(26)







120594119861119878119876119894119895119896 =

119879

119881

120597119894+119895+119896 lnZ

120597120583119894119861120597120583

119895

119878120597120583119896119876

(27)


11986111987811 120594

1198782





SB

HRG

1

08

06

04

02

0

minus3120594

BS

11120594

S 2

120 140 160 180 200 220 240T (MeV)

N120591 = 12

N120591 = 8N120591 = 6

fK scale

(a)

HRG

005

004

003

002

001

0

120594BQ

11T

2

fK scale

120 140 160 180 200 220 240T (MeV)

N120591 = 12 N120591 = 6N120591 = 8Cont extrap

(b)




11987711988312 equiv

119872119883

1205902119883

=120583119861

119879(119877

119883112 +

1205832119861

1198792119877119883312 + O (120583

4119861))

11987711988331 equiv

1198781198831205903119883

119872119883

= 119877119883131 +

1205832119861

1198792119877119883331 + O (120583

4119861)

(28)













RQ 31

HRG

30

25

20

15

10

05

00

Free

T (MeV)140 150 160 170 180 190 200 210 220 230 240

120583BT = 1

120583BT = 0

N120591 = 6

N120591 = 8

(a)

0035

0030

0025

0020HRG

HRG

T = 160MeV

T = 160MeV

T = 170MeV

T = 170MeV

T = 150MeV

T = 150MeV

030

025

020

015

120583S120583

B

0 20 40 60 80 100 120 140 160 180 200120583B (MeV)

minus120583Q120583

B(b)



07

08

09

1

11

0 1 2 3 4 5

Freezeout curve

120583BT

TT

c

30 GeV20 GeV

18 GeV (CERN)10 GeV

(a)

HRG

Power law

10

1

01

001

0001

m1

1 10 100 1000 10000radicSNN (GeV)

(b)








119903119899 (119899 = odd) = radic120594119861119899+1

1198792120594119861119899+3

119903119899 (119899 = even) = [1205941198612

119879119899120594119861119899+2

]

1119899

(29)


1198618 This




119861) [140]

119898119888 (120583119861)

119898119888 (0)= 1 minus 39 (8) (

120583119861

3120587119879)

2





119905 =1

1199050

(119879 minus 1198791198880

1198791198880

+ 120581119861

120583119861

3119879) ℎ =

119898119897

ℎ0119898119904

(31)


119872119887 (120583) = 119872119887 (0) +120594119898119861

2119879(120583119861

3119879)

2

+ O(120583119861

3119879)

4

(32)


(1198792119898119904)120597

2119872119887120597(1205831198613119879)


120594119898119861

119879=2120581119861119879

1199050119898119904

ℎminus(1minus120573)120573120575

1198911015840119866 (

119905

ℎ1120573120575) (33)




119879119888 (120583119861)

119879119888 (0)= 1 minus 120581119861(

120583119861

3119879)

2

+ O(120583119861

3119879)

4

997904rArr 119879119888 (120583119861) ≃ 154 (1 minus 00066(120583119861

3119879)

2

) MeV

(34)


150

100

50

Tem

pera

ture

(MeV

)


Freezeout

T120594119904c (120583)

T120595120595c (120583)



(a)

007

006

005

004

003

002

001


z

minus2120581qf998400G(z) 120

140180

minusm

st0h(1minus120573)120573120575120594mqT

2

N120591 = 8 mlms = 120

N120591 = 4 mlms = 110

(b)







119875 (120583119897 120583119904)

1198794=119875 (0)

1198794+sum

119894119895

120594119894119895 (119879)

1198794minus119894minus119895(120583119897

119879)

119894

(120583119904

119879)

119895

(35)



119868 (120583119897 120583119904)

1198794= minus

1198733120591

1198733

119889 lnZ119889 ln 119886

=119868 (0)

1198794+sum

119894119895

119887119894119895 (119879) (120583119897

119879)

119894

(120583119904

119879)

119895

(36)


8

6

4

2

0

IT4

100 200 300 400 500 600T (MeV)

SnB = 30

SnB = infinSnB = 45

SnB = 300

Filled Nt = 6

Empty Nt = 4

(a)

6

4

2

5 10 15

pT

4

120598T4






11990611990411120594









119901 up to O ((120583119861119879)4)

119901 up to O ((120583119861119879)2)

le

11 for120583119861

119879le 2

135 for120583119861

119879le 3

(37)


4 Summary





Acknowledgments


References


































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics












Crossover

Physical pointms

Nf = 1

Nf = 3

Pure gauge

Firstorder

Firstorder

Second order

infin

infin

Nf = 2

mc = (ms270 ms270)

O(4)Z(2)



mud

mtrics



119868 (119879) = 1198795 120597

120597119879

119901

1198794 (7)


119868 (119879)

1198794= minus119873

4120591 (119886

119889120573

119889119886(⟨119878119866⟩ minus ⟨119878119866⟩0)

+ sum

119891

119886

119889 (119898119891119886)

119889119886(⟨120595119891120595119891⟩ minus ⟨120595119891120595119891⟩0

))

120573 =6

1198922

(8)



1

2

3

4

5

6

150 200 250 300 350

(120576minus3p

)T4

T (MeV)

Stout cont

r1 scale

N120591 = 6

N120591 = 8

N120591 = 10

N120591 = 12

HISQtreeN120591 = 4

(a)

05

1

15

2

25

3

35

4

300 400 500 600 700 800

(120576minus3p

)T4

T (MeV)

N120591 = 6

N120591 = 8

N120591 = 6N120591 = 8


HISQtreeN120591 = 4 p4 N120591 = 4

(b)


5

4

3

2

1

0

(120576minus3p

)T4

100 150 200 250 300 350 400 450 500 550T (MeV)


323 times 10


(a)

4

3

2

1

2

1

I(T)T4

0 200 400 600 800 1000T (MeV)


0 50 100 150

HRG

(b)









119868 (119879)

1198794= 119890


sdot (ℎ0 +1198910 [tanh (1198911119905 + 1198912) + 1]

1 + 1198921119905 + 11989221199052

)

119905 =119879

200MeV

(9)


ℎ0 = 01396 ℎ1 = minus018 ℎ2 = 0035

1198910 = 276 1198911 = 679 1198912 = minus529

1198921 = minus047 1198922 = 104

(10)




Δ 119897119904 (119879) =

⟨120595120595⟩119897119879 minus (119898119897119898119904) ⟨120595120595⟩119904119879

⟨120595120595⟩1198970 minus (119898119897119898119904) ⟨120595120595⟩1199040

119897 = 119906 119889 (11)


120594119877 =1

1198811198793119898

2119897

1205972

1205971198982119897

(lnZ (119879) minus lnZ (0)) (12)



119879119888 =

170 (4) (3) for 119871119877

157 (3) (3) Δ 119897119904

147 (2) (3) 120594119877

(13)





0

5

10

15

20

120576T4

700500 600300 400100 200T (MeV)

3pT4

(120576 minus 3p)T4

(a)

900700500300100

10

8

6

4

2

0

minus2

T (MeV)

Interpolation

N120591 = 4N120591 = 6N120591 = 8

N120591 = 10

N120591 = 12

(120598minus3p

)T4

(b)


1

08

06

04

02

0

Δls

fK scale

T (MeV)120 140 160 180 200

AsqtadN120591 = 8

N120591 = 12

HISQtreeN120591 = 6

N120591 = 8

N120591 = 12

N120591 = 8 ml = 0037ms

Stout cont

(a)

fK scale

T (MeV)120 140 160 180 200

04

035

03

025

02

015

01

005

0

Lre

n(T

)

HISQtreeN120591 = 6

N120591 = 8

N120591 = 12

AsqtadN120591 = 8

N120591 = 12

Stout cont

(b)





125 150 175 200 225 250 275(MeV)

0005

0

minus0005

minus001

minus0015

minus002

minus003

minus0025

minus0035008 01 012 014 016

TmΩ


mR

R120595Rm

1205874

120595

(a)

150 175 200 225 250 275(MeV)

008 01 012 014 016TmΩ


2

15

1

05

0

LR

(b)











001

0

minus001

minus002

minus003

minus004

minus005

minus006

minus007

minus008

120 140 160 180 200 220 240 260T (MeV)

01 012 014 016 018 02 022 024

6 times 123

8 times 163

Staggered

Tw0

mR120595120595Rm

1205874

(a)

0002

00015

0001

00005

0140 150 160 170 180 190 200

T (MeV)

12059511205951T3

Δ lsT3

(b)






200 400 600 800 1000T (MeV)

20

15

10

5

15

10

5

SB

100 150 200 250

s(T)T3

N120591 = 6N120591 = 8N120591 = 10

(a)

200 400 600 800 1000T (MeV)

5

4

3

2

1

SB

100 150 200 250

25215105

p(T

)T4

N120591 = 6N120591 = 8N120591 = 10

(b)

c2 s(T

)

200 400 600 800 1000T (MeV)

035

03

025

02

015

01

035030250201501

SB

100 150 200 250 300

N120591 = 6N120591 = 8N120591 = 10

(c)







119872119887 = 119898119904

⟨120595120595⟩

1198794 (14)



20

15

10

5

0


120576T4

IT4

pT4

150 200 250 300 350 400T (MeV)

(a)

200 300 400 500 600 700 800 900 1000T (MeV)

6

5

4

3

2

1

0

PT

4

Nf = 2 + 1 EOS Nf = 2 + 1 + 1 N120591 = 8

Nf = 2 + 1 + 1 N120591 = 6 Nf = 2 + 1 + 1 N120591 = 10

(b)



119872119887 (119879119867) = ℎ1120575

119891119866 (119905

ℎ1120573120575) + 119891reg (119879119867) (15)



119905 =1

1199050

119879 minus 1198791198880

1198791198880

ℎ =119867

ℎ0

119867 =119898119897

119898119904

(16)


119891reg = 119867(1198860 + 1198861

119879 minus 1198791198880

1198791198880

+ 1198862(119879 minus 1198791198880

1198791198880

)

2

) (17)







120594120587 minus 120594120575 = int1198894119909 [⟨120595 (119909) 12059121205745120595 (119909) 120595 (0) 12059121205745120595 (0)⟩

minus ⟨120595 (119909) 1205912120595 (119909) 120595 (0) 1205912120595 (0)⟩]

(18)



000

050

100

150

200

250

094 096 098 100 102 104 106 108TTc

Mb

120

110

140

180

Chiral limit

mlms

(a)

000

050

100

150

200

All masses

th1120573120575

Mbh

1120575

O(2)

15

25

110

120

140

180


mlms

(b)



⟨120595120595⟩ = int119889120582120588 (120582119898)2119898

1198982 + 1205822

120594120587 minus 120594120575 = int119889120582120588 (120582119898)4119898

2

(1198982 + 1205822)2

(19)





1205822)



2120575(120582) + 1198881120582 for the density




lim119898rarr0

⟨120588 (120582119898)⟩ = lim119898rarr0

⟨120588 (119898)⟩1205823

3+ O (120582

4) (20)



350

300

250

200

150

100

50

0

T (MeV)140 150 160 170 180 190 200

120594disc T2

1205945disc T2 120594top T

2(ml + mres )2

(120594120587 minus 120594120575)T2

(a)

0025

002

0015

001

0005

00

120588(120582)

002 004 006 008 01120582

163 times 8

Min(120582100)ml

ms

(b)


T ⋍ 170MeV

T ⋍ 210MeV

1

05

0

0 100 200 300 400 500

120588(120582)

(MeV

3)

120588(120582)

(MeV

3)

120588(120582)

(MeV

3)

1

05

0

1

05

0

120582 (MeV)

T ≃ 180sim190MeV

120573 = 218 am = 005

120573 = 218 am = 001

120573 = 225 am = 001120573 = 220 am = 001120573 = 220 am = 0025120573 = 220 am = 005

120573 = 240 am = 001

120573 = 230 am = 001

120573 = 230 am = 0025

120573 = 230 am = 005

(a)



0 2 4 6 8 10 12 14 16

times10minus7

35

3

25

2

15

1

Distance

= 001120573 = 225 (Tsim192) ma

(b)






120588(120582)

120582a

0 004 008 012 016


T = 3301MeVmlms = 120

10eminus02

10eminus03

10eminus04

10eminus05

10eminus06

(a)

14

12

1

08

06

04

02

M(2120587

T)

085 09 095 1 105 11 115 12TTc

P

S

V

A

(b)







119873119891

prod

119891=1

det119863119891 (120583) 119890minus119878119866

(21)


119863119891(120583)119909119910 = [

3

sum

119894=1

1

2120574119894 (119880119894 (119909) 120575119910119909+119894 minus 119880

dagger119894 (119910) 120575119910119909minus119894)

+1

21205744 (119890

1205831198861198804 (119909) 120575119910119909+4 minus 119890

minus120583119886119880

dagger120583 (119910) 120575119910119909minus4)

+ 119886119898119891120575119909119910]

(22)



10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816e119894120579(23)





⟨e119894120579⟩ =

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816e119894120579119890minus119878119866

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816119890minus119878119866

= 119890minus119881Δ119865119879

(24)






120594119883119899 =

119879

119881

120597119899 lnZ120597120583

119899119883

119883 equiv 119861 119878 119876 (25)





1198762 120594

1198612 (120583 =



1205941198612 (120583)

1198792=1205941198612 (0)

1198792+

1205832

211987921205941198614 (0) +

1205834

411987941205941198616 (0) 119879

2+ sdot sdot sdot

(26)







120594119861119878119876119894119895119896 =

119879

119881

120597119894+119895+119896 lnZ

120597120583119894119861120597120583

119895

119878120597120583119896119876

(27)


11986111987811 120594

1198782





SB

HRG

1

08

06

04

02

0

minus3120594

BS

11120594

S 2

120 140 160 180 200 220 240T (MeV)

N120591 = 12

N120591 = 8N120591 = 6

fK scale

(a)

HRG

005

004

003

002

001

0

120594BQ

11T

2

fK scale

120 140 160 180 200 220 240T (MeV)

N120591 = 12 N120591 = 6N120591 = 8Cont extrap

(b)




11987711988312 equiv

119872119883

1205902119883

=120583119861

119879(119877

119883112 +

1205832119861

1198792119877119883312 + O (120583

4119861))

11987711988331 equiv

1198781198831205903119883

119872119883

= 119877119883131 +

1205832119861

1198792119877119883331 + O (120583

4119861)

(28)













RQ 31

HRG

30

25

20

15

10

05

00

Free

T (MeV)140 150 160 170 180 190 200 210 220 230 240

120583BT = 1

120583BT = 0

N120591 = 6

N120591 = 8

(a)

0035

0030

0025

0020HRG

HRG

T = 160MeV

T = 160MeV

T = 170MeV

T = 170MeV

T = 150MeV

T = 150MeV

030

025

020

015

120583S120583

B

0 20 40 60 80 100 120 140 160 180 200120583B (MeV)

minus120583Q120583

B(b)



07

08

09

1

11

0 1 2 3 4 5

Freezeout curve

120583BT

TT

c

30 GeV20 GeV

18 GeV (CERN)10 GeV

(a)

HRG

Power law

10

1

01

001

0001

m1

1 10 100 1000 10000radicSNN (GeV)

(b)








119903119899 (119899 = odd) = radic120594119861119899+1

1198792120594119861119899+3

119903119899 (119899 = even) = [1205941198612

119879119899120594119861119899+2

]

1119899

(29)


1198618 This




119861) [140]

119898119888 (120583119861)

119898119888 (0)= 1 minus 39 (8) (

120583119861

3120587119879)

2





119905 =1

1199050

(119879 minus 1198791198880

1198791198880

+ 120581119861

120583119861

3119879) ℎ =

119898119897

ℎ0119898119904

(31)


119872119887 (120583) = 119872119887 (0) +120594119898119861

2119879(120583119861

3119879)

2

+ O(120583119861

3119879)

4

(32)


(1198792119898119904)120597

2119872119887120597(1205831198613119879)


120594119898119861

119879=2120581119861119879

1199050119898119904

ℎminus(1minus120573)120573120575

1198911015840119866 (

119905

ℎ1120573120575) (33)




119879119888 (120583119861)

119879119888 (0)= 1 minus 120581119861(

120583119861

3119879)

2

+ O(120583119861

3119879)

4

997904rArr 119879119888 (120583119861) ≃ 154 (1 minus 00066(120583119861

3119879)

2

) MeV

(34)


150

100

50

Tem

pera

ture

(MeV

)


Freezeout

T120594119904c (120583)

T120595120595c (120583)



(a)

007

006

005

004

003

002

001


z

minus2120581qf998400G(z) 120

140180

minusm

st0h(1minus120573)120573120575120594mqT

2

N120591 = 8 mlms = 120

N120591 = 4 mlms = 110

(b)







119875 (120583119897 120583119904)

1198794=119875 (0)

1198794+sum

119894119895

120594119894119895 (119879)

1198794minus119894minus119895(120583119897

119879)

119894

(120583119904

119879)

119895

(35)



119868 (120583119897 120583119904)

1198794= minus

1198733120591

1198733

119889 lnZ119889 ln 119886

=119868 (0)

1198794+sum

119894119895

119887119894119895 (119879) (120583119897

119879)

119894

(120583119904

119879)

119895

(36)


8

6

4

2

0

IT4

100 200 300 400 500 600T (MeV)

SnB = 30

SnB = infinSnB = 45

SnB = 300

Filled Nt = 6

Empty Nt = 4

(a)

6

4

2

5 10 15

pT

4

120598T4






11990611990411120594









119901 up to O ((120583119861119879)4)

119901 up to O ((120583119861119879)2)

le

11 for120583119861

119879le 2

135 for120583119861

119879le 3

(37)


4 Summary





Acknowledgments


References


































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




1

2

3

4

5

6

150 200 250 300 350

(120576minus3p

)T4

T (MeV)

Stout cont

r1 scale

N120591 = 6

N120591 = 8

N120591 = 10

N120591 = 12

HISQtreeN120591 = 4

(a)

05

1

15

2

25

3

35

4

300 400 500 600 700 800

(120576minus3p

)T4

T (MeV)

N120591 = 6

N120591 = 8

N120591 = 6N120591 = 8


HISQtreeN120591 = 4 p4 N120591 = 4

(b)


5

4

3

2

1

0

(120576minus3p

)T4

100 150 200 250 300 350 400 450 500 550T (MeV)


323 times 10


(a)

4

3

2

1

2

1

I(T)T4

0 200 400 600 800 1000T (MeV)


0 50 100 150

HRG

(b)









119868 (119879)

1198794= 119890


sdot (ℎ0 +1198910 [tanh (1198911119905 + 1198912) + 1]

1 + 1198921119905 + 11989221199052

)

119905 =119879

200MeV

(9)


ℎ0 = 01396 ℎ1 = minus018 ℎ2 = 0035

1198910 = 276 1198911 = 679 1198912 = minus529

1198921 = minus047 1198922 = 104

(10)




Δ 119897119904 (119879) =

⟨120595120595⟩119897119879 minus (119898119897119898119904) ⟨120595120595⟩119904119879

⟨120595120595⟩1198970 minus (119898119897119898119904) ⟨120595120595⟩1199040

119897 = 119906 119889 (11)


120594119877 =1

1198811198793119898

2119897

1205972

1205971198982119897

(lnZ (119879) minus lnZ (0)) (12)



119879119888 =

170 (4) (3) for 119871119877

157 (3) (3) Δ 119897119904

147 (2) (3) 120594119877

(13)





0

5

10

15

20

120576T4

700500 600300 400100 200T (MeV)

3pT4

(120576 minus 3p)T4

(a)

900700500300100

10

8

6

4

2

0

minus2

T (MeV)

Interpolation

N120591 = 4N120591 = 6N120591 = 8

N120591 = 10

N120591 = 12

(120598minus3p

)T4

(b)


1

08

06

04

02

0

Δls

fK scale

T (MeV)120 140 160 180 200

AsqtadN120591 = 8

N120591 = 12

HISQtreeN120591 = 6

N120591 = 8

N120591 = 12

N120591 = 8 ml = 0037ms

Stout cont

(a)

fK scale

T (MeV)120 140 160 180 200

04

035

03

025

02

015

01

005

0

Lre

n(T

)

HISQtreeN120591 = 6

N120591 = 8

N120591 = 12

AsqtadN120591 = 8

N120591 = 12

Stout cont

(b)





125 150 175 200 225 250 275(MeV)

0005

0

minus0005

minus001

minus0015

minus002

minus003

minus0025

minus0035008 01 012 014 016

TmΩ


mR

R120595Rm

1205874

120595

(a)

150 175 200 225 250 275(MeV)

008 01 012 014 016TmΩ


2

15

1

05

0

LR

(b)











001

0

minus001

minus002

minus003

minus004

minus005

minus006

minus007

minus008

120 140 160 180 200 220 240 260T (MeV)

01 012 014 016 018 02 022 024

6 times 123

8 times 163

Staggered

Tw0

mR120595120595Rm

1205874

(a)

0002

00015

0001

00005

0140 150 160 170 180 190 200

T (MeV)

12059511205951T3

Δ lsT3

(b)






200 400 600 800 1000T (MeV)

20

15

10

5

15

10

5

SB

100 150 200 250

s(T)T3

N120591 = 6N120591 = 8N120591 = 10

(a)

200 400 600 800 1000T (MeV)

5

4

3

2

1

SB

100 150 200 250

25215105

p(T

)T4

N120591 = 6N120591 = 8N120591 = 10

(b)

c2 s(T

)

200 400 600 800 1000T (MeV)

035

03

025

02

015

01

035030250201501

SB

100 150 200 250 300

N120591 = 6N120591 = 8N120591 = 10

(c)







119872119887 = 119898119904

⟨120595120595⟩

1198794 (14)



20

15

10

5

0


120576T4

IT4

pT4

150 200 250 300 350 400T (MeV)

(a)

200 300 400 500 600 700 800 900 1000T (MeV)

6

5

4

3

2

1

0

PT

4

Nf = 2 + 1 EOS Nf = 2 + 1 + 1 N120591 = 8

Nf = 2 + 1 + 1 N120591 = 6 Nf = 2 + 1 + 1 N120591 = 10

(b)



119872119887 (119879119867) = ℎ1120575

119891119866 (119905

ℎ1120573120575) + 119891reg (119879119867) (15)



119905 =1

1199050

119879 minus 1198791198880

1198791198880

ℎ =119867

ℎ0

119867 =119898119897

119898119904

(16)


119891reg = 119867(1198860 + 1198861

119879 minus 1198791198880

1198791198880

+ 1198862(119879 minus 1198791198880

1198791198880

)

2

) (17)







120594120587 minus 120594120575 = int1198894119909 [⟨120595 (119909) 12059121205745120595 (119909) 120595 (0) 12059121205745120595 (0)⟩

minus ⟨120595 (119909) 1205912120595 (119909) 120595 (0) 1205912120595 (0)⟩]

(18)



000

050

100

150

200

250

094 096 098 100 102 104 106 108TTc

Mb

120

110

140

180

Chiral limit

mlms

(a)

000

050

100

150

200

All masses

th1120573120575

Mbh

1120575

O(2)

15

25

110

120

140

180


mlms

(b)



⟨120595120595⟩ = int119889120582120588 (120582119898)2119898

1198982 + 1205822

120594120587 minus 120594120575 = int119889120582120588 (120582119898)4119898

2

(1198982 + 1205822)2

(19)





1205822)



2120575(120582) + 1198881120582 for the density




lim119898rarr0

⟨120588 (120582119898)⟩ = lim119898rarr0

⟨120588 (119898)⟩1205823

3+ O (120582

4) (20)



350

300

250

200

150

100

50

0

T (MeV)140 150 160 170 180 190 200

120594disc T2

1205945disc T2 120594top T

2(ml + mres )2

(120594120587 minus 120594120575)T2

(a)

0025

002

0015

001

0005

00

120588(120582)

002 004 006 008 01120582

163 times 8

Min(120582100)ml

ms

(b)


T ⋍ 170MeV

T ⋍ 210MeV

1

05

0

0 100 200 300 400 500

120588(120582)

(MeV

3)

120588(120582)

(MeV

3)

120588(120582)

(MeV

3)

1

05

0

1

05

0

120582 (MeV)

T ≃ 180sim190MeV

120573 = 218 am = 005

120573 = 218 am = 001

120573 = 225 am = 001120573 = 220 am = 001120573 = 220 am = 0025120573 = 220 am = 005

120573 = 240 am = 001

120573 = 230 am = 001

120573 = 230 am = 0025

120573 = 230 am = 005

(a)



0 2 4 6 8 10 12 14 16

times10minus7

35

3

25

2

15

1

Distance

= 001120573 = 225 (Tsim192) ma

(b)






120588(120582)

120582a

0 004 008 012 016


T = 3301MeVmlms = 120

10eminus02

10eminus03

10eminus04

10eminus05

10eminus06

(a)

14

12

1

08

06

04

02

M(2120587

T)

085 09 095 1 105 11 115 12TTc

P

S

V

A

(b)







119873119891

prod

119891=1

det119863119891 (120583) 119890minus119878119866

(21)


119863119891(120583)119909119910 = [

3

sum

119894=1

1

2120574119894 (119880119894 (119909) 120575119910119909+119894 minus 119880

dagger119894 (119910) 120575119910119909minus119894)

+1

21205744 (119890

1205831198861198804 (119909) 120575119910119909+4 minus 119890

minus120583119886119880

dagger120583 (119910) 120575119910119909minus4)

+ 119886119898119891120575119909119910]

(22)



10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816e119894120579(23)





⟨e119894120579⟩ =

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816e119894120579119890minus119878119866

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816119890minus119878119866

= 119890minus119881Δ119865119879

(24)






120594119883119899 =

119879

119881

120597119899 lnZ120597120583

119899119883

119883 equiv 119861 119878 119876 (25)





1198762 120594

1198612 (120583 =



1205941198612 (120583)

1198792=1205941198612 (0)

1198792+

1205832

211987921205941198614 (0) +

1205834

411987941205941198616 (0) 119879

2+ sdot sdot sdot

(26)







120594119861119878119876119894119895119896 =

119879

119881

120597119894+119895+119896 lnZ

120597120583119894119861120597120583

119895

119878120597120583119896119876

(27)


11986111987811 120594

1198782





SB

HRG

1

08

06

04

02

0

minus3120594

BS

11120594

S 2

120 140 160 180 200 220 240T (MeV)

N120591 = 12

N120591 = 8N120591 = 6

fK scale

(a)

HRG

005

004

003

002

001

0

120594BQ

11T

2

fK scale

120 140 160 180 200 220 240T (MeV)

N120591 = 12 N120591 = 6N120591 = 8Cont extrap

(b)




11987711988312 equiv

119872119883

1205902119883

=120583119861

119879(119877

119883112 +

1205832119861

1198792119877119883312 + O (120583

4119861))

11987711988331 equiv

1198781198831205903119883

119872119883

= 119877119883131 +

1205832119861

1198792119877119883331 + O (120583

4119861)

(28)













RQ 31

HRG

30

25

20

15

10

05

00

Free

T (MeV)140 150 160 170 180 190 200 210 220 230 240

120583BT = 1

120583BT = 0

N120591 = 6

N120591 = 8

(a)

0035

0030

0025

0020HRG

HRG

T = 160MeV

T = 160MeV

T = 170MeV

T = 170MeV

T = 150MeV

T = 150MeV

030

025

020

015

120583S120583

B

0 20 40 60 80 100 120 140 160 180 200120583B (MeV)

minus120583Q120583

B(b)



07

08

09

1

11

0 1 2 3 4 5

Freezeout curve

120583BT

TT

c

30 GeV20 GeV

18 GeV (CERN)10 GeV

(a)

HRG

Power law

10

1

01

001

0001

m1

1 10 100 1000 10000radicSNN (GeV)

(b)








119903119899 (119899 = odd) = radic120594119861119899+1

1198792120594119861119899+3

119903119899 (119899 = even) = [1205941198612

119879119899120594119861119899+2

]

1119899

(29)


1198618 This




119861) [140]

119898119888 (120583119861)

119898119888 (0)= 1 minus 39 (8) (

120583119861

3120587119879)

2





119905 =1

1199050

(119879 minus 1198791198880

1198791198880

+ 120581119861

120583119861

3119879) ℎ =

119898119897

ℎ0119898119904

(31)


119872119887 (120583) = 119872119887 (0) +120594119898119861

2119879(120583119861

3119879)

2

+ O(120583119861

3119879)

4

(32)


(1198792119898119904)120597

2119872119887120597(1205831198613119879)


120594119898119861

119879=2120581119861119879

1199050119898119904

ℎminus(1minus120573)120573120575

1198911015840119866 (

119905

ℎ1120573120575) (33)




119879119888 (120583119861)

119879119888 (0)= 1 minus 120581119861(

120583119861

3119879)

2

+ O(120583119861

3119879)

4

997904rArr 119879119888 (120583119861) ≃ 154 (1 minus 00066(120583119861

3119879)

2

) MeV

(34)


150

100

50

Tem

pera

ture

(MeV

)


Freezeout

T120594119904c (120583)

T120595120595c (120583)



(a)

007

006

005

004

003

002

001


z

minus2120581qf998400G(z) 120

140180

minusm

st0h(1minus120573)120573120575120594mqT

2

N120591 = 8 mlms = 120

N120591 = 4 mlms = 110

(b)







119875 (120583119897 120583119904)

1198794=119875 (0)

1198794+sum

119894119895

120594119894119895 (119879)

1198794minus119894minus119895(120583119897

119879)

119894

(120583119904

119879)

119895

(35)



119868 (120583119897 120583119904)

1198794= minus

1198733120591

1198733

119889 lnZ119889 ln 119886

=119868 (0)

1198794+sum

119894119895

119887119894119895 (119879) (120583119897

119879)

119894

(120583119904

119879)

119895

(36)


8

6

4

2

0

IT4

100 200 300 400 500 600T (MeV)

SnB = 30

SnB = infinSnB = 45

SnB = 300

Filled Nt = 6

Empty Nt = 4

(a)

6

4

2

5 10 15

pT

4

120598T4






11990611990411120594









119901 up to O ((120583119861119879)4)

119901 up to O ((120583119861119879)2)

le

11 for120583119861

119879le 2

135 for120583119861

119879le 3

(37)


4 Summary





Acknowledgments


References


































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







119868 (119879)

1198794= 119890


sdot (ℎ0 +1198910 [tanh (1198911119905 + 1198912) + 1]

1 + 1198921119905 + 11989221199052

)

119905 =119879

200MeV

(9)


ℎ0 = 01396 ℎ1 = minus018 ℎ2 = 0035

1198910 = 276 1198911 = 679 1198912 = minus529

1198921 = minus047 1198922 = 104

(10)




Δ 119897119904 (119879) =

⟨120595120595⟩119897119879 minus (119898119897119898119904) ⟨120595120595⟩119904119879

⟨120595120595⟩1198970 minus (119898119897119898119904) ⟨120595120595⟩1199040

119897 = 119906 119889 (11)


120594119877 =1

1198811198793119898

2119897

1205972

1205971198982119897

(lnZ (119879) minus lnZ (0)) (12)



119879119888 =

170 (4) (3) for 119871119877

157 (3) (3) Δ 119897119904

147 (2) (3) 120594119877

(13)





0

5

10

15

20

120576T4

700500 600300 400100 200T (MeV)

3pT4

(120576 minus 3p)T4

(a)

900700500300100

10

8

6

4

2

0

minus2

T (MeV)

Interpolation

N120591 = 4N120591 = 6N120591 = 8

N120591 = 10

N120591 = 12

(120598minus3p

)T4

(b)


1

08

06

04

02

0

Δls

fK scale

T (MeV)120 140 160 180 200

AsqtadN120591 = 8

N120591 = 12

HISQtreeN120591 = 6

N120591 = 8

N120591 = 12

N120591 = 8 ml = 0037ms

Stout cont

(a)

fK scale

T (MeV)120 140 160 180 200

04

035

03

025

02

015

01

005

0

Lre

n(T

)

HISQtreeN120591 = 6

N120591 = 8

N120591 = 12

AsqtadN120591 = 8

N120591 = 12

Stout cont

(b)





125 150 175 200 225 250 275(MeV)

0005

0

minus0005

minus001

minus0015

minus002

minus003

minus0025

minus0035008 01 012 014 016

TmΩ


mR

R120595Rm

1205874

120595

(a)

150 175 200 225 250 275(MeV)

008 01 012 014 016TmΩ


2

15

1

05

0

LR

(b)











001

0

minus001

minus002

minus003

minus004

minus005

minus006

minus007

minus008

120 140 160 180 200 220 240 260T (MeV)

01 012 014 016 018 02 022 024

6 times 123

8 times 163

Staggered

Tw0

mR120595120595Rm

1205874

(a)

0002

00015

0001

00005

0140 150 160 170 180 190 200

T (MeV)

12059511205951T3

Δ lsT3

(b)






200 400 600 800 1000T (MeV)

20

15

10

5

15

10

5

SB

100 150 200 250

s(T)T3

N120591 = 6N120591 = 8N120591 = 10

(a)

200 400 600 800 1000T (MeV)

5

4

3

2

1

SB

100 150 200 250

25215105

p(T

)T4

N120591 = 6N120591 = 8N120591 = 10

(b)

c2 s(T

)

200 400 600 800 1000T (MeV)

035

03

025

02

015

01

035030250201501

SB

100 150 200 250 300

N120591 = 6N120591 = 8N120591 = 10

(c)







119872119887 = 119898119904

⟨120595120595⟩

1198794 (14)



20

15

10

5

0


120576T4

IT4

pT4

150 200 250 300 350 400T (MeV)

(a)

200 300 400 500 600 700 800 900 1000T (MeV)

6

5

4

3

2

1

0

PT

4

Nf = 2 + 1 EOS Nf = 2 + 1 + 1 N120591 = 8

Nf = 2 + 1 + 1 N120591 = 6 Nf = 2 + 1 + 1 N120591 = 10

(b)



119872119887 (119879119867) = ℎ1120575

119891119866 (119905

ℎ1120573120575) + 119891reg (119879119867) (15)



119905 =1

1199050

119879 minus 1198791198880

1198791198880

ℎ =119867

ℎ0

119867 =119898119897

119898119904

(16)


119891reg = 119867(1198860 + 1198861

119879 minus 1198791198880

1198791198880

+ 1198862(119879 minus 1198791198880

1198791198880

)

2

) (17)







120594120587 minus 120594120575 = int1198894119909 [⟨120595 (119909) 12059121205745120595 (119909) 120595 (0) 12059121205745120595 (0)⟩

minus ⟨120595 (119909) 1205912120595 (119909) 120595 (0) 1205912120595 (0)⟩]

(18)



000

050

100

150

200

250

094 096 098 100 102 104 106 108TTc

Mb

120

110

140

180

Chiral limit

mlms

(a)

000

050

100

150

200

All masses

th1120573120575

Mbh

1120575

O(2)

15

25

110

120

140

180


mlms

(b)



⟨120595120595⟩ = int119889120582120588 (120582119898)2119898

1198982 + 1205822

120594120587 minus 120594120575 = int119889120582120588 (120582119898)4119898

2

(1198982 + 1205822)2

(19)





1205822)



2120575(120582) + 1198881120582 for the density




lim119898rarr0

⟨120588 (120582119898)⟩ = lim119898rarr0

⟨120588 (119898)⟩1205823

3+ O (120582

4) (20)



350

300

250

200

150

100

50

0

T (MeV)140 150 160 170 180 190 200

120594disc T2

1205945disc T2 120594top T

2(ml + mres )2

(120594120587 minus 120594120575)T2

(a)

0025

002

0015

001

0005

00

120588(120582)

002 004 006 008 01120582

163 times 8

Min(120582100)ml

ms

(b)


T ⋍ 170MeV

T ⋍ 210MeV

1

05

0

0 100 200 300 400 500

120588(120582)

(MeV

3)

120588(120582)

(MeV

3)

120588(120582)

(MeV

3)

1

05

0

1

05

0

120582 (MeV)

T ≃ 180sim190MeV

120573 = 218 am = 005

120573 = 218 am = 001

120573 = 225 am = 001120573 = 220 am = 001120573 = 220 am = 0025120573 = 220 am = 005

120573 = 240 am = 001

120573 = 230 am = 001

120573 = 230 am = 0025

120573 = 230 am = 005

(a)



0 2 4 6 8 10 12 14 16

times10minus7

35

3

25

2

15

1

Distance

= 001120573 = 225 (Tsim192) ma

(b)






120588(120582)

120582a

0 004 008 012 016


T = 3301MeVmlms = 120

10eminus02

10eminus03

10eminus04

10eminus05

10eminus06

(a)

14

12

1

08

06

04

02

M(2120587

T)

085 09 095 1 105 11 115 12TTc

P

S

V

A

(b)







119873119891

prod

119891=1

det119863119891 (120583) 119890minus119878119866

(21)


119863119891(120583)119909119910 = [

3

sum

119894=1

1

2120574119894 (119880119894 (119909) 120575119910119909+119894 minus 119880

dagger119894 (119910) 120575119910119909minus119894)

+1

21205744 (119890

1205831198861198804 (119909) 120575119910119909+4 minus 119890

minus120583119886119880

dagger120583 (119910) 120575119910119909minus4)

+ 119886119898119891120575119909119910]

(22)



10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816e119894120579(23)





⟨e119894120579⟩ =

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816e119894120579119890minus119878119866

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816119890minus119878119866

= 119890minus119881Δ119865119879

(24)






120594119883119899 =

119879

119881

120597119899 lnZ120597120583

119899119883

119883 equiv 119861 119878 119876 (25)





1198762 120594

1198612 (120583 =



1205941198612 (120583)

1198792=1205941198612 (0)

1198792+

1205832

211987921205941198614 (0) +

1205834

411987941205941198616 (0) 119879

2+ sdot sdot sdot

(26)







120594119861119878119876119894119895119896 =

119879

119881

120597119894+119895+119896 lnZ

120597120583119894119861120597120583

119895

119878120597120583119896119876

(27)


11986111987811 120594

1198782





SB

HRG

1

08

06

04

02

0

minus3120594

BS

11120594

S 2

120 140 160 180 200 220 240T (MeV)

N120591 = 12

N120591 = 8N120591 = 6

fK scale

(a)

HRG

005

004

003

002

001

0

120594BQ

11T

2

fK scale

120 140 160 180 200 220 240T (MeV)

N120591 = 12 N120591 = 6N120591 = 8Cont extrap

(b)




11987711988312 equiv

119872119883

1205902119883

=120583119861

119879(119877

119883112 +

1205832119861

1198792119877119883312 + O (120583

4119861))

11987711988331 equiv

1198781198831205903119883

119872119883

= 119877119883131 +

1205832119861

1198792119877119883331 + O (120583

4119861)

(28)













RQ 31

HRG

30

25

20

15

10

05

00

Free

T (MeV)140 150 160 170 180 190 200 210 220 230 240

120583BT = 1

120583BT = 0

N120591 = 6

N120591 = 8

(a)

0035

0030

0025

0020HRG

HRG

T = 160MeV

T = 160MeV

T = 170MeV

T = 170MeV

T = 150MeV

T = 150MeV

030

025

020

015

120583S120583

B

0 20 40 60 80 100 120 140 160 180 200120583B (MeV)

minus120583Q120583

B(b)



07

08

09

1

11

0 1 2 3 4 5

Freezeout curve

120583BT

TT

c

30 GeV20 GeV

18 GeV (CERN)10 GeV

(a)

HRG

Power law

10

1

01

001

0001

m1

1 10 100 1000 10000radicSNN (GeV)

(b)








119903119899 (119899 = odd) = radic120594119861119899+1

1198792120594119861119899+3

119903119899 (119899 = even) = [1205941198612

119879119899120594119861119899+2

]

1119899

(29)


1198618 This




119861) [140]

119898119888 (120583119861)

119898119888 (0)= 1 minus 39 (8) (

120583119861

3120587119879)

2





119905 =1

1199050

(119879 minus 1198791198880

1198791198880

+ 120581119861

120583119861

3119879) ℎ =

119898119897

ℎ0119898119904

(31)


119872119887 (120583) = 119872119887 (0) +120594119898119861

2119879(120583119861

3119879)

2

+ O(120583119861

3119879)

4

(32)


(1198792119898119904)120597

2119872119887120597(1205831198613119879)


120594119898119861

119879=2120581119861119879

1199050119898119904

ℎminus(1minus120573)120573120575

1198911015840119866 (

119905

ℎ1120573120575) (33)




119879119888 (120583119861)

119879119888 (0)= 1 minus 120581119861(

120583119861

3119879)

2

+ O(120583119861

3119879)

4

997904rArr 119879119888 (120583119861) ≃ 154 (1 minus 00066(120583119861

3119879)

2

) MeV

(34)


150

100

50

Tem

pera

ture

(MeV

)


Freezeout

T120594119904c (120583)

T120595120595c (120583)



(a)

007

006

005

004

003

002

001


z

minus2120581qf998400G(z) 120

140180

minusm

st0h(1minus120573)120573120575120594mqT

2

N120591 = 8 mlms = 120

N120591 = 4 mlms = 110

(b)







119875 (120583119897 120583119904)

1198794=119875 (0)

1198794+sum

119894119895

120594119894119895 (119879)

1198794minus119894minus119895(120583119897

119879)

119894

(120583119904

119879)

119895

(35)



119868 (120583119897 120583119904)

1198794= minus

1198733120591

1198733

119889 lnZ119889 ln 119886

=119868 (0)

1198794+sum

119894119895

119887119894119895 (119879) (120583119897

119879)

119894

(120583119904

119879)

119895

(36)


8

6

4

2

0

IT4

100 200 300 400 500 600T (MeV)

SnB = 30

SnB = infinSnB = 45

SnB = 300

Filled Nt = 6

Empty Nt = 4

(a)

6

4

2

5 10 15

pT

4

120598T4






11990611990411120594









119901 up to O ((120583119861119879)4)

119901 up to O ((120583119861119879)2)

le

11 for120583119861

119879le 2

135 for120583119861

119879le 3

(37)


4 Summary





Acknowledgments


References


































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0

5

10

15

20

120576T4

700500 600300 400100 200T (MeV)

3pT4

(120576 minus 3p)T4

(a)

900700500300100

10

8

6

4

2

0

minus2

T (MeV)

Interpolation

N120591 = 4N120591 = 6N120591 = 8

N120591 = 10

N120591 = 12

(120598minus3p

)T4

(b)


1

08

06

04

02

0

Δls

fK scale

T (MeV)120 140 160 180 200

AsqtadN120591 = 8

N120591 = 12

HISQtreeN120591 = 6

N120591 = 8

N120591 = 12

N120591 = 8 ml = 0037ms

Stout cont

(a)

fK scale

T (MeV)120 140 160 180 200

04

035

03

025

02

015

01

005

0

Lre

n(T

)

HISQtreeN120591 = 6

N120591 = 8

N120591 = 12

AsqtadN120591 = 8

N120591 = 12

Stout cont

(b)





125 150 175 200 225 250 275(MeV)

0005

0

minus0005

minus001

minus0015

minus002

minus003

minus0025

minus0035008 01 012 014 016

TmΩ


mR

R120595Rm

1205874

120595

(a)

150 175 200 225 250 275(MeV)

008 01 012 014 016TmΩ


2

15

1

05

0

LR

(b)











001

0

minus001

minus002

minus003

minus004

minus005

minus006

minus007

minus008

120 140 160 180 200 220 240 260T (MeV)

01 012 014 016 018 02 022 024

6 times 123

8 times 163

Staggered

Tw0

mR120595120595Rm

1205874

(a)

0002

00015

0001

00005

0140 150 160 170 180 190 200

T (MeV)

12059511205951T3

Δ lsT3

(b)






200 400 600 800 1000T (MeV)

20

15

10

5

15

10

5

SB

100 150 200 250

s(T)T3

N120591 = 6N120591 = 8N120591 = 10

(a)

200 400 600 800 1000T (MeV)

5

4

3

2

1

SB

100 150 200 250

25215105

p(T

)T4

N120591 = 6N120591 = 8N120591 = 10

(b)

c2 s(T

)

200 400 600 800 1000T (MeV)

035

03

025

02

015

01

035030250201501

SB

100 150 200 250 300

N120591 = 6N120591 = 8N120591 = 10

(c)







119872119887 = 119898119904

⟨120595120595⟩

1198794 (14)



20

15

10

5

0


120576T4

IT4

pT4

150 200 250 300 350 400T (MeV)

(a)

200 300 400 500 600 700 800 900 1000T (MeV)

6

5

4

3

2

1

0

PT

4

Nf = 2 + 1 EOS Nf = 2 + 1 + 1 N120591 = 8

Nf = 2 + 1 + 1 N120591 = 6 Nf = 2 + 1 + 1 N120591 = 10

(b)



119872119887 (119879119867) = ℎ1120575

119891119866 (119905

ℎ1120573120575) + 119891reg (119879119867) (15)



119905 =1

1199050

119879 minus 1198791198880

1198791198880

ℎ =119867

ℎ0

119867 =119898119897

119898119904

(16)


119891reg = 119867(1198860 + 1198861

119879 minus 1198791198880

1198791198880

+ 1198862(119879 minus 1198791198880

1198791198880

)

2

) (17)







120594120587 minus 120594120575 = int1198894119909 [⟨120595 (119909) 12059121205745120595 (119909) 120595 (0) 12059121205745120595 (0)⟩

minus ⟨120595 (119909) 1205912120595 (119909) 120595 (0) 1205912120595 (0)⟩]

(18)



000

050

100

150

200

250

094 096 098 100 102 104 106 108TTc

Mb

120

110

140

180

Chiral limit

mlms

(a)

000

050

100

150

200

All masses

th1120573120575

Mbh

1120575

O(2)

15

25

110

120

140

180


mlms

(b)



⟨120595120595⟩ = int119889120582120588 (120582119898)2119898

1198982 + 1205822

120594120587 minus 120594120575 = int119889120582120588 (120582119898)4119898

2

(1198982 + 1205822)2

(19)





1205822)



2120575(120582) + 1198881120582 for the density




lim119898rarr0

⟨120588 (120582119898)⟩ = lim119898rarr0

⟨120588 (119898)⟩1205823

3+ O (120582

4) (20)



350

300

250

200

150

100

50

0

T (MeV)140 150 160 170 180 190 200

120594disc T2

1205945disc T2 120594top T

2(ml + mres )2

(120594120587 minus 120594120575)T2

(a)

0025

002

0015

001

0005

00

120588(120582)

002 004 006 008 01120582

163 times 8

Min(120582100)ml

ms

(b)


T ⋍ 170MeV

T ⋍ 210MeV

1

05

0

0 100 200 300 400 500

120588(120582)

(MeV

3)

120588(120582)

(MeV

3)

120588(120582)

(MeV

3)

1

05

0

1

05

0

120582 (MeV)

T ≃ 180sim190MeV

120573 = 218 am = 005

120573 = 218 am = 001

120573 = 225 am = 001120573 = 220 am = 001120573 = 220 am = 0025120573 = 220 am = 005

120573 = 240 am = 001

120573 = 230 am = 001

120573 = 230 am = 0025

120573 = 230 am = 005

(a)



0 2 4 6 8 10 12 14 16

times10minus7

35

3

25

2

15

1

Distance

= 001120573 = 225 (Tsim192) ma

(b)






120588(120582)

120582a

0 004 008 012 016


T = 3301MeVmlms = 120

10eminus02

10eminus03

10eminus04

10eminus05

10eminus06

(a)

14

12

1

08

06

04

02

M(2120587

T)

085 09 095 1 105 11 115 12TTc

P

S

V

A

(b)







119873119891

prod

119891=1

det119863119891 (120583) 119890minus119878119866

(21)


119863119891(120583)119909119910 = [

3

sum

119894=1

1

2120574119894 (119880119894 (119909) 120575119910119909+119894 minus 119880

dagger119894 (119910) 120575119910119909minus119894)

+1

21205744 (119890

1205831198861198804 (119909) 120575119910119909+4 minus 119890

minus120583119886119880

dagger120583 (119910) 120575119910119909minus4)

+ 119886119898119891120575119909119910]

(22)



10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816e119894120579(23)





⟨e119894120579⟩ =

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816e119894120579119890minus119878119866

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816119890minus119878119866

= 119890minus119881Δ119865119879

(24)






120594119883119899 =

119879

119881

120597119899 lnZ120597120583

119899119883

119883 equiv 119861 119878 119876 (25)





1198762 120594

1198612 (120583 =



1205941198612 (120583)

1198792=1205941198612 (0)

1198792+

1205832

211987921205941198614 (0) +

1205834

411987941205941198616 (0) 119879

2+ sdot sdot sdot

(26)







120594119861119878119876119894119895119896 =

119879

119881

120597119894+119895+119896 lnZ

120597120583119894119861120597120583

119895

119878120597120583119896119876

(27)


11986111987811 120594

1198782





SB

HRG

1

08

06

04

02

0

minus3120594

BS

11120594

S 2

120 140 160 180 200 220 240T (MeV)

N120591 = 12

N120591 = 8N120591 = 6

fK scale

(a)

HRG

005

004

003

002

001

0

120594BQ

11T

2

fK scale

120 140 160 180 200 220 240T (MeV)

N120591 = 12 N120591 = 6N120591 = 8Cont extrap

(b)




11987711988312 equiv

119872119883

1205902119883

=120583119861

119879(119877

119883112 +

1205832119861

1198792119877119883312 + O (120583

4119861))

11987711988331 equiv

1198781198831205903119883

119872119883

= 119877119883131 +

1205832119861

1198792119877119883331 + O (120583

4119861)

(28)













RQ 31

HRG

30

25

20

15

10

05

00

Free

T (MeV)140 150 160 170 180 190 200 210 220 230 240

120583BT = 1

120583BT = 0

N120591 = 6

N120591 = 8

(a)

0035

0030

0025

0020HRG

HRG

T = 160MeV

T = 160MeV

T = 170MeV

T = 170MeV

T = 150MeV

T = 150MeV

030

025

020

015

120583S120583

B

0 20 40 60 80 100 120 140 160 180 200120583B (MeV)

minus120583Q120583

B(b)



07

08

09

1

11

0 1 2 3 4 5

Freezeout curve

120583BT

TT

c

30 GeV20 GeV

18 GeV (CERN)10 GeV

(a)

HRG

Power law

10

1

01

001

0001

m1

1 10 100 1000 10000radicSNN (GeV)

(b)








119903119899 (119899 = odd) = radic120594119861119899+1

1198792120594119861119899+3

119903119899 (119899 = even) = [1205941198612

119879119899120594119861119899+2

]

1119899

(29)


1198618 This




119861) [140]

119898119888 (120583119861)

119898119888 (0)= 1 minus 39 (8) (

120583119861

3120587119879)

2





119905 =1

1199050

(119879 minus 1198791198880

1198791198880

+ 120581119861

120583119861

3119879) ℎ =

119898119897

ℎ0119898119904

(31)


119872119887 (120583) = 119872119887 (0) +120594119898119861

2119879(120583119861

3119879)

2

+ O(120583119861

3119879)

4

(32)


(1198792119898119904)120597

2119872119887120597(1205831198613119879)


120594119898119861

119879=2120581119861119879

1199050119898119904

ℎminus(1minus120573)120573120575

1198911015840119866 (

119905

ℎ1120573120575) (33)




119879119888 (120583119861)

119879119888 (0)= 1 minus 120581119861(

120583119861

3119879)

2

+ O(120583119861

3119879)

4

997904rArr 119879119888 (120583119861) ≃ 154 (1 minus 00066(120583119861

3119879)

2

) MeV

(34)


150

100

50

Tem

pera

ture

(MeV

)


Freezeout

T120594119904c (120583)

T120595120595c (120583)



(a)

007

006

005

004

003

002

001


z

minus2120581qf998400G(z) 120

140180

minusm

st0h(1minus120573)120573120575120594mqT

2

N120591 = 8 mlms = 120

N120591 = 4 mlms = 110

(b)







119875 (120583119897 120583119904)

1198794=119875 (0)

1198794+sum

119894119895

120594119894119895 (119879)

1198794minus119894minus119895(120583119897

119879)

119894

(120583119904

119879)

119895

(35)



119868 (120583119897 120583119904)

1198794= minus

1198733120591

1198733

119889 lnZ119889 ln 119886

=119868 (0)

1198794+sum

119894119895

119887119894119895 (119879) (120583119897

119879)

119894

(120583119904

119879)

119895

(36)


8

6

4

2

0

IT4

100 200 300 400 500 600T (MeV)

SnB = 30

SnB = infinSnB = 45

SnB = 300

Filled Nt = 6

Empty Nt = 4

(a)

6

4

2

5 10 15

pT

4

120598T4






11990611990411120594









119901 up to O ((120583119861119879)4)

119901 up to O ((120583119861119879)2)

le

11 for120583119861

119879le 2

135 for120583119861

119879le 3

(37)


4 Summary





Acknowledgments


References


































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




125 150 175 200 225 250 275(MeV)

0005

0

minus0005

minus001

minus0015

minus002

minus003

minus0025

minus0035008 01 012 014 016

TmΩ


mR

R120595Rm

1205874

120595

(a)

150 175 200 225 250 275(MeV)

008 01 012 014 016TmΩ


2

15

1

05

0

LR

(b)











001

0

minus001

minus002

minus003

minus004

minus005

minus006

minus007

minus008

120 140 160 180 200 220 240 260T (MeV)

01 012 014 016 018 02 022 024

6 times 123

8 times 163

Staggered

Tw0

mR120595120595Rm

1205874

(a)

0002

00015

0001

00005

0140 150 160 170 180 190 200

T (MeV)

12059511205951T3

Δ lsT3

(b)






200 400 600 800 1000T (MeV)

20

15

10

5

15

10

5

SB

100 150 200 250

s(T)T3

N120591 = 6N120591 = 8N120591 = 10

(a)

200 400 600 800 1000T (MeV)

5

4

3

2

1

SB

100 150 200 250

25215105

p(T

)T4

N120591 = 6N120591 = 8N120591 = 10

(b)

c2 s(T

)

200 400 600 800 1000T (MeV)

035

03

025

02

015

01

035030250201501

SB

100 150 200 250 300

N120591 = 6N120591 = 8N120591 = 10

(c)







119872119887 = 119898119904

⟨120595120595⟩

1198794 (14)



20

15

10

5

0


120576T4

IT4

pT4

150 200 250 300 350 400T (MeV)

(a)

200 300 400 500 600 700 800 900 1000T (MeV)

6

5

4

3

2

1

0

PT

4

Nf = 2 + 1 EOS Nf = 2 + 1 + 1 N120591 = 8

Nf = 2 + 1 + 1 N120591 = 6 Nf = 2 + 1 + 1 N120591 = 10

(b)



119872119887 (119879119867) = ℎ1120575

119891119866 (119905

ℎ1120573120575) + 119891reg (119879119867) (15)



119905 =1

1199050

119879 minus 1198791198880

1198791198880

ℎ =119867

ℎ0

119867 =119898119897

119898119904

(16)


119891reg = 119867(1198860 + 1198861

119879 minus 1198791198880

1198791198880

+ 1198862(119879 minus 1198791198880

1198791198880

)

2

) (17)







120594120587 minus 120594120575 = int1198894119909 [⟨120595 (119909) 12059121205745120595 (119909) 120595 (0) 12059121205745120595 (0)⟩

minus ⟨120595 (119909) 1205912120595 (119909) 120595 (0) 1205912120595 (0)⟩]

(18)



000

050

100

150

200

250

094 096 098 100 102 104 106 108TTc

Mb

120

110

140

180

Chiral limit

mlms

(a)

000

050

100

150

200

All masses

th1120573120575

Mbh

1120575

O(2)

15

25

110

120

140

180


mlms

(b)



⟨120595120595⟩ = int119889120582120588 (120582119898)2119898

1198982 + 1205822

120594120587 minus 120594120575 = int119889120582120588 (120582119898)4119898

2

(1198982 + 1205822)2

(19)





1205822)



2120575(120582) + 1198881120582 for the density




lim119898rarr0

⟨120588 (120582119898)⟩ = lim119898rarr0

⟨120588 (119898)⟩1205823

3+ O (120582

4) (20)



350

300

250

200

150

100

50

0

T (MeV)140 150 160 170 180 190 200

120594disc T2

1205945disc T2 120594top T

2(ml + mres )2

(120594120587 minus 120594120575)T2

(a)

0025

002

0015

001

0005

00

120588(120582)

002 004 006 008 01120582

163 times 8

Min(120582100)ml

ms

(b)


T ⋍ 170MeV

T ⋍ 210MeV

1

05

0

0 100 200 300 400 500

120588(120582)

(MeV

3)

120588(120582)

(MeV

3)

120588(120582)

(MeV

3)

1

05

0

1

05

0

120582 (MeV)

T ≃ 180sim190MeV

120573 = 218 am = 005

120573 = 218 am = 001

120573 = 225 am = 001120573 = 220 am = 001120573 = 220 am = 0025120573 = 220 am = 005

120573 = 240 am = 001

120573 = 230 am = 001

120573 = 230 am = 0025

120573 = 230 am = 005

(a)



0 2 4 6 8 10 12 14 16

times10minus7

35

3

25

2

15

1

Distance

= 001120573 = 225 (Tsim192) ma

(b)






120588(120582)

120582a

0 004 008 012 016


T = 3301MeVmlms = 120

10eminus02

10eminus03

10eminus04

10eminus05

10eminus06

(a)

14

12

1

08

06

04

02

M(2120587

T)

085 09 095 1 105 11 115 12TTc

P

S

V

A

(b)







119873119891

prod

119891=1

det119863119891 (120583) 119890minus119878119866

(21)


119863119891(120583)119909119910 = [

3

sum

119894=1

1

2120574119894 (119880119894 (119909) 120575119910119909+119894 minus 119880

dagger119894 (119910) 120575119910119909minus119894)

+1

21205744 (119890

1205831198861198804 (119909) 120575119910119909+4 minus 119890

minus120583119886119880

dagger120583 (119910) 120575119910119909minus4)

+ 119886119898119891120575119909119910]

(22)



10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816e119894120579(23)





⟨e119894120579⟩ =

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816e119894120579119890minus119878119866

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816119890minus119878119866

= 119890minus119881Δ119865119879

(24)






120594119883119899 =

119879

119881

120597119899 lnZ120597120583

119899119883

119883 equiv 119861 119878 119876 (25)





1198762 120594

1198612 (120583 =



1205941198612 (120583)

1198792=1205941198612 (0)

1198792+

1205832

211987921205941198614 (0) +

1205834

411987941205941198616 (0) 119879

2+ sdot sdot sdot

(26)







120594119861119878119876119894119895119896 =

119879

119881

120597119894+119895+119896 lnZ

120597120583119894119861120597120583

119895

119878120597120583119896119876

(27)


11986111987811 120594

1198782





SB

HRG

1

08

06

04

02

0

minus3120594

BS

11120594

S 2

120 140 160 180 200 220 240T (MeV)

N120591 = 12

N120591 = 8N120591 = 6

fK scale

(a)

HRG

005

004

003

002

001

0

120594BQ

11T

2

fK scale

120 140 160 180 200 220 240T (MeV)

N120591 = 12 N120591 = 6N120591 = 8Cont extrap

(b)




11987711988312 equiv

119872119883

1205902119883

=120583119861

119879(119877

119883112 +

1205832119861

1198792119877119883312 + O (120583

4119861))

11987711988331 equiv

1198781198831205903119883

119872119883

= 119877119883131 +

1205832119861

1198792119877119883331 + O (120583

4119861)

(28)













RQ 31

HRG

30

25

20

15

10

05

00

Free

T (MeV)140 150 160 170 180 190 200 210 220 230 240

120583BT = 1

120583BT = 0

N120591 = 6

N120591 = 8

(a)

0035

0030

0025

0020HRG

HRG

T = 160MeV

T = 160MeV

T = 170MeV

T = 170MeV

T = 150MeV

T = 150MeV

030

025

020

015

120583S120583

B

0 20 40 60 80 100 120 140 160 180 200120583B (MeV)

minus120583Q120583

B(b)



07

08

09

1

11

0 1 2 3 4 5

Freezeout curve

120583BT

TT

c

30 GeV20 GeV

18 GeV (CERN)10 GeV

(a)

HRG

Power law

10

1

01

001

0001

m1

1 10 100 1000 10000radicSNN (GeV)

(b)








119903119899 (119899 = odd) = radic120594119861119899+1

1198792120594119861119899+3

119903119899 (119899 = even) = [1205941198612

119879119899120594119861119899+2

]

1119899

(29)


1198618 This




119861) [140]

119898119888 (120583119861)

119898119888 (0)= 1 minus 39 (8) (

120583119861

3120587119879)

2





119905 =1

1199050

(119879 minus 1198791198880

1198791198880

+ 120581119861

120583119861

3119879) ℎ =

119898119897

ℎ0119898119904

(31)


119872119887 (120583) = 119872119887 (0) +120594119898119861

2119879(120583119861

3119879)

2

+ O(120583119861

3119879)

4

(32)


(1198792119898119904)120597

2119872119887120597(1205831198613119879)


120594119898119861

119879=2120581119861119879

1199050119898119904

ℎminus(1minus120573)120573120575

1198911015840119866 (

119905

ℎ1120573120575) (33)




119879119888 (120583119861)

119879119888 (0)= 1 minus 120581119861(

120583119861

3119879)

2

+ O(120583119861

3119879)

4

997904rArr 119879119888 (120583119861) ≃ 154 (1 minus 00066(120583119861

3119879)

2

) MeV

(34)


150

100

50

Tem

pera

ture

(MeV

)


Freezeout

T120594119904c (120583)

T120595120595c (120583)



(a)

007

006

005

004

003

002

001


z

minus2120581qf998400G(z) 120

140180

minusm

st0h(1minus120573)120573120575120594mqT

2

N120591 = 8 mlms = 120

N120591 = 4 mlms = 110

(b)







119875 (120583119897 120583119904)

1198794=119875 (0)

1198794+sum

119894119895

120594119894119895 (119879)

1198794minus119894minus119895(120583119897

119879)

119894

(120583119904

119879)

119895

(35)



119868 (120583119897 120583119904)

1198794= minus

1198733120591

1198733

119889 lnZ119889 ln 119886

=119868 (0)

1198794+sum

119894119895

119887119894119895 (119879) (120583119897

119879)

119894

(120583119904

119879)

119895

(36)


8

6

4

2

0

IT4

100 200 300 400 500 600T (MeV)

SnB = 30

SnB = infinSnB = 45

SnB = 300

Filled Nt = 6

Empty Nt = 4

(a)

6

4

2

5 10 15

pT

4

120598T4






11990611990411120594









119901 up to O ((120583119861119879)4)

119901 up to O ((120583119861119879)2)

le

11 for120583119861

119879le 2

135 for120583119861

119879le 3

(37)


4 Summary





Acknowledgments


References


































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




001

0

minus001

minus002

minus003

minus004

minus005

minus006

minus007

minus008

120 140 160 180 200 220 240 260T (MeV)

01 012 014 016 018 02 022 024

6 times 123

8 times 163

Staggered

Tw0

mR120595120595Rm

1205874

(a)

0002

00015

0001

00005

0140 150 160 170 180 190 200

T (MeV)

12059511205951T3

Δ lsT3

(b)






200 400 600 800 1000T (MeV)

20

15

10

5

15

10

5

SB

100 150 200 250

s(T)T3

N120591 = 6N120591 = 8N120591 = 10

(a)

200 400 600 800 1000T (MeV)

5

4

3

2

1

SB

100 150 200 250

25215105

p(T

)T4

N120591 = 6N120591 = 8N120591 = 10

(b)

c2 s(T

)

200 400 600 800 1000T (MeV)

035

03

025

02

015

01

035030250201501

SB

100 150 200 250 300

N120591 = 6N120591 = 8N120591 = 10

(c)







119872119887 = 119898119904

⟨120595120595⟩

1198794 (14)



20

15

10

5

0


120576T4

IT4

pT4

150 200 250 300 350 400T (MeV)

(a)

200 300 400 500 600 700 800 900 1000T (MeV)

6

5

4

3

2

1

0

PT

4

Nf = 2 + 1 EOS Nf = 2 + 1 + 1 N120591 = 8

Nf = 2 + 1 + 1 N120591 = 6 Nf = 2 + 1 + 1 N120591 = 10

(b)



119872119887 (119879119867) = ℎ1120575

119891119866 (119905

ℎ1120573120575) + 119891reg (119879119867) (15)



119905 =1

1199050

119879 minus 1198791198880

1198791198880

ℎ =119867

ℎ0

119867 =119898119897

119898119904

(16)


119891reg = 119867(1198860 + 1198861

119879 minus 1198791198880

1198791198880

+ 1198862(119879 minus 1198791198880

1198791198880

)

2

) (17)







120594120587 minus 120594120575 = int1198894119909 [⟨120595 (119909) 12059121205745120595 (119909) 120595 (0) 12059121205745120595 (0)⟩

minus ⟨120595 (119909) 1205912120595 (119909) 120595 (0) 1205912120595 (0)⟩]

(18)



000

050

100

150

200

250

094 096 098 100 102 104 106 108TTc

Mb

120

110

140

180

Chiral limit

mlms

(a)

000

050

100

150

200

All masses

th1120573120575

Mbh

1120575

O(2)

15

25

110

120

140

180


mlms

(b)



⟨120595120595⟩ = int119889120582120588 (120582119898)2119898

1198982 + 1205822

120594120587 minus 120594120575 = int119889120582120588 (120582119898)4119898

2

(1198982 + 1205822)2

(19)





1205822)



2120575(120582) + 1198881120582 for the density




lim119898rarr0

⟨120588 (120582119898)⟩ = lim119898rarr0

⟨120588 (119898)⟩1205823

3+ O (120582

4) (20)



350

300

250

200

150

100

50

0

T (MeV)140 150 160 170 180 190 200

120594disc T2

1205945disc T2 120594top T

2(ml + mres )2

(120594120587 minus 120594120575)T2

(a)

0025

002

0015

001

0005

00

120588(120582)

002 004 006 008 01120582

163 times 8

Min(120582100)ml

ms

(b)


T ⋍ 170MeV

T ⋍ 210MeV

1

05

0

0 100 200 300 400 500

120588(120582)

(MeV

3)

120588(120582)

(MeV

3)

120588(120582)

(MeV

3)

1

05

0

1

05

0

120582 (MeV)

T ≃ 180sim190MeV

120573 = 218 am = 005

120573 = 218 am = 001

120573 = 225 am = 001120573 = 220 am = 001120573 = 220 am = 0025120573 = 220 am = 005

120573 = 240 am = 001

120573 = 230 am = 001

120573 = 230 am = 0025

120573 = 230 am = 005

(a)



0 2 4 6 8 10 12 14 16

times10minus7

35

3

25

2

15

1

Distance

= 001120573 = 225 (Tsim192) ma

(b)






120588(120582)

120582a

0 004 008 012 016


T = 3301MeVmlms = 120

10eminus02

10eminus03

10eminus04

10eminus05

10eminus06

(a)

14

12

1

08

06

04

02

M(2120587

T)

085 09 095 1 105 11 115 12TTc

P

S

V

A

(b)







119873119891

prod

119891=1

det119863119891 (120583) 119890minus119878119866

(21)


119863119891(120583)119909119910 = [

3

sum

119894=1

1

2120574119894 (119880119894 (119909) 120575119910119909+119894 minus 119880

dagger119894 (119910) 120575119910119909minus119894)

+1

21205744 (119890

1205831198861198804 (119909) 120575119910119909+4 minus 119890

minus120583119886119880

dagger120583 (119910) 120575119910119909minus4)

+ 119886119898119891120575119909119910]

(22)



10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816e119894120579(23)





⟨e119894120579⟩ =

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816e119894120579119890minus119878119866

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816119890minus119878119866

= 119890minus119881Δ119865119879

(24)






120594119883119899 =

119879

119881

120597119899 lnZ120597120583

119899119883

119883 equiv 119861 119878 119876 (25)





1198762 120594

1198612 (120583 =



1205941198612 (120583)

1198792=1205941198612 (0)

1198792+

1205832

211987921205941198614 (0) +

1205834

411987941205941198616 (0) 119879

2+ sdot sdot sdot

(26)







120594119861119878119876119894119895119896 =

119879

119881

120597119894+119895+119896 lnZ

120597120583119894119861120597120583

119895

119878120597120583119896119876

(27)


11986111987811 120594

1198782





SB

HRG

1

08

06

04

02

0

minus3120594

BS

11120594

S 2

120 140 160 180 200 220 240T (MeV)

N120591 = 12

N120591 = 8N120591 = 6

fK scale

(a)

HRG

005

004

003

002

001

0

120594BQ

11T

2

fK scale

120 140 160 180 200 220 240T (MeV)

N120591 = 12 N120591 = 6N120591 = 8Cont extrap

(b)




11987711988312 equiv

119872119883

1205902119883

=120583119861

119879(119877

119883112 +

1205832119861

1198792119877119883312 + O (120583

4119861))

11987711988331 equiv

1198781198831205903119883

119872119883

= 119877119883131 +

1205832119861

1198792119877119883331 + O (120583

4119861)

(28)













RQ 31

HRG

30

25

20

15

10

05

00

Free

T (MeV)140 150 160 170 180 190 200 210 220 230 240

120583BT = 1

120583BT = 0

N120591 = 6

N120591 = 8

(a)

0035

0030

0025

0020HRG

HRG

T = 160MeV

T = 160MeV

T = 170MeV

T = 170MeV

T = 150MeV

T = 150MeV

030

025

020

015

120583S120583

B

0 20 40 60 80 100 120 140 160 180 200120583B (MeV)

minus120583Q120583

B(b)



07

08

09

1

11

0 1 2 3 4 5

Freezeout curve

120583BT

TT

c

30 GeV20 GeV

18 GeV (CERN)10 GeV

(a)

HRG

Power law

10

1

01

001

0001

m1

1 10 100 1000 10000radicSNN (GeV)

(b)








119903119899 (119899 = odd) = radic120594119861119899+1

1198792120594119861119899+3

119903119899 (119899 = even) = [1205941198612

119879119899120594119861119899+2

]

1119899

(29)


1198618 This




119861) [140]

119898119888 (120583119861)

119898119888 (0)= 1 minus 39 (8) (

120583119861

3120587119879)

2





119905 =1

1199050

(119879 minus 1198791198880

1198791198880

+ 120581119861

120583119861

3119879) ℎ =

119898119897

ℎ0119898119904

(31)


119872119887 (120583) = 119872119887 (0) +120594119898119861

2119879(120583119861

3119879)

2

+ O(120583119861

3119879)

4

(32)


(1198792119898119904)120597

2119872119887120597(1205831198613119879)


120594119898119861

119879=2120581119861119879

1199050119898119904

ℎminus(1minus120573)120573120575

1198911015840119866 (

119905

ℎ1120573120575) (33)




119879119888 (120583119861)

119879119888 (0)= 1 minus 120581119861(

120583119861

3119879)

2

+ O(120583119861

3119879)

4

997904rArr 119879119888 (120583119861) ≃ 154 (1 minus 00066(120583119861

3119879)

2

) MeV

(34)


150

100

50

Tem

pera

ture

(MeV

)


Freezeout

T120594119904c (120583)

T120595120595c (120583)



(a)

007

006

005

004

003

002

001


z

minus2120581qf998400G(z) 120

140180

minusm

st0h(1minus120573)120573120575120594mqT

2

N120591 = 8 mlms = 120

N120591 = 4 mlms = 110

(b)







119875 (120583119897 120583119904)

1198794=119875 (0)

1198794+sum

119894119895

120594119894119895 (119879)

1198794minus119894minus119895(120583119897

119879)

119894

(120583119904

119879)

119895

(35)



119868 (120583119897 120583119904)

1198794= minus

1198733120591

1198733

119889 lnZ119889 ln 119886

=119868 (0)

1198794+sum

119894119895

119887119894119895 (119879) (120583119897

119879)

119894

(120583119904

119879)

119895

(36)


8

6

4

2

0

IT4

100 200 300 400 500 600T (MeV)

SnB = 30

SnB = infinSnB = 45

SnB = 300

Filled Nt = 6

Empty Nt = 4

(a)

6

4

2

5 10 15

pT

4

120598T4






11990611990411120594









119901 up to O ((120583119861119879)4)

119901 up to O ((120583119861119879)2)

le

11 for120583119861

119879le 2

135 for120583119861

119879le 3

(37)


4 Summary





Acknowledgments


References


































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




200 400 600 800 1000T (MeV)

20

15

10

5

15

10

5

SB

100 150 200 250

s(T)T3

N120591 = 6N120591 = 8N120591 = 10

(a)

200 400 600 800 1000T (MeV)

5

4

3

2

1

SB

100 150 200 250

25215105

p(T

)T4

N120591 = 6N120591 = 8N120591 = 10

(b)

c2 s(T

)

200 400 600 800 1000T (MeV)

035

03

025

02

015

01

035030250201501

SB

100 150 200 250 300

N120591 = 6N120591 = 8N120591 = 10

(c)







119872119887 = 119898119904

⟨120595120595⟩

1198794 (14)



20

15

10

5

0


120576T4

IT4

pT4

150 200 250 300 350 400T (MeV)

(a)

200 300 400 500 600 700 800 900 1000T (MeV)

6

5

4

3

2

1

0

PT

4

Nf = 2 + 1 EOS Nf = 2 + 1 + 1 N120591 = 8

Nf = 2 + 1 + 1 N120591 = 6 Nf = 2 + 1 + 1 N120591 = 10

(b)



119872119887 (119879119867) = ℎ1120575

119891119866 (119905

ℎ1120573120575) + 119891reg (119879119867) (15)



119905 =1

1199050

119879 minus 1198791198880

1198791198880

ℎ =119867

ℎ0

119867 =119898119897

119898119904

(16)


119891reg = 119867(1198860 + 1198861

119879 minus 1198791198880

1198791198880

+ 1198862(119879 minus 1198791198880

1198791198880

)

2

) (17)







120594120587 minus 120594120575 = int1198894119909 [⟨120595 (119909) 12059121205745120595 (119909) 120595 (0) 12059121205745120595 (0)⟩

minus ⟨120595 (119909) 1205912120595 (119909) 120595 (0) 1205912120595 (0)⟩]

(18)



000

050

100

150

200

250

094 096 098 100 102 104 106 108TTc

Mb

120

110

140

180

Chiral limit

mlms

(a)

000

050

100

150

200

All masses

th1120573120575

Mbh

1120575

O(2)

15

25

110

120

140

180


mlms

(b)



⟨120595120595⟩ = int119889120582120588 (120582119898)2119898

1198982 + 1205822

120594120587 minus 120594120575 = int119889120582120588 (120582119898)4119898

2

(1198982 + 1205822)2

(19)





1205822)



2120575(120582) + 1198881120582 for the density




lim119898rarr0

⟨120588 (120582119898)⟩ = lim119898rarr0

⟨120588 (119898)⟩1205823

3+ O (120582

4) (20)



350

300

250

200

150

100

50

0

T (MeV)140 150 160 170 180 190 200

120594disc T2

1205945disc T2 120594top T

2(ml + mres )2

(120594120587 minus 120594120575)T2

(a)

0025

002

0015

001

0005

00

120588(120582)

002 004 006 008 01120582

163 times 8

Min(120582100)ml

ms

(b)


T ⋍ 170MeV

T ⋍ 210MeV

1

05

0

0 100 200 300 400 500

120588(120582)

(MeV

3)

120588(120582)

(MeV

3)

120588(120582)

(MeV

3)

1

05

0

1

05

0

120582 (MeV)

T ≃ 180sim190MeV

120573 = 218 am = 005

120573 = 218 am = 001

120573 = 225 am = 001120573 = 220 am = 001120573 = 220 am = 0025120573 = 220 am = 005

120573 = 240 am = 001

120573 = 230 am = 001

120573 = 230 am = 0025

120573 = 230 am = 005

(a)



0 2 4 6 8 10 12 14 16

times10minus7

35

3

25

2

15

1

Distance

= 001120573 = 225 (Tsim192) ma

(b)






120588(120582)

120582a

0 004 008 012 016


T = 3301MeVmlms = 120

10eminus02

10eminus03

10eminus04

10eminus05

10eminus06

(a)

14

12

1

08

06

04

02

M(2120587

T)

085 09 095 1 105 11 115 12TTc

P

S

V

A

(b)







119873119891

prod

119891=1

det119863119891 (120583) 119890minus119878119866

(21)


119863119891(120583)119909119910 = [

3

sum

119894=1

1

2120574119894 (119880119894 (119909) 120575119910119909+119894 minus 119880

dagger119894 (119910) 120575119910119909minus119894)

+1

21205744 (119890

1205831198861198804 (119909) 120575119910119909+4 minus 119890

minus120583119886119880

dagger120583 (119910) 120575119910119909minus4)

+ 119886119898119891120575119909119910]

(22)



10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816e119894120579(23)





⟨e119894120579⟩ =

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816e119894120579119890minus119878119866

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816119890minus119878119866

= 119890minus119881Δ119865119879

(24)






120594119883119899 =

119879

119881

120597119899 lnZ120597120583

119899119883

119883 equiv 119861 119878 119876 (25)





1198762 120594

1198612 (120583 =



1205941198612 (120583)

1198792=1205941198612 (0)

1198792+

1205832

211987921205941198614 (0) +

1205834

411987941205941198616 (0) 119879

2+ sdot sdot sdot

(26)







120594119861119878119876119894119895119896 =

119879

119881

120597119894+119895+119896 lnZ

120597120583119894119861120597120583

119895

119878120597120583119896119876

(27)


11986111987811 120594

1198782





SB

HRG

1

08

06

04

02

0

minus3120594

BS

11120594

S 2

120 140 160 180 200 220 240T (MeV)

N120591 = 12

N120591 = 8N120591 = 6

fK scale

(a)

HRG

005

004

003

002

001

0

120594BQ

11T

2

fK scale

120 140 160 180 200 220 240T (MeV)

N120591 = 12 N120591 = 6N120591 = 8Cont extrap

(b)




11987711988312 equiv

119872119883

1205902119883

=120583119861

119879(119877

119883112 +

1205832119861

1198792119877119883312 + O (120583

4119861))

11987711988331 equiv

1198781198831205903119883

119872119883

= 119877119883131 +

1205832119861

1198792119877119883331 + O (120583

4119861)

(28)













RQ 31

HRG

30

25

20

15

10

05

00

Free

T (MeV)140 150 160 170 180 190 200 210 220 230 240

120583BT = 1

120583BT = 0

N120591 = 6

N120591 = 8

(a)

0035

0030

0025

0020HRG

HRG

T = 160MeV

T = 160MeV

T = 170MeV

T = 170MeV

T = 150MeV

T = 150MeV

030

025

020

015

120583S120583

B

0 20 40 60 80 100 120 140 160 180 200120583B (MeV)

minus120583Q120583

B(b)



07

08

09

1

11

0 1 2 3 4 5

Freezeout curve

120583BT

TT

c

30 GeV20 GeV

18 GeV (CERN)10 GeV

(a)

HRG

Power law

10

1

01

001

0001

m1

1 10 100 1000 10000radicSNN (GeV)

(b)








119903119899 (119899 = odd) = radic120594119861119899+1

1198792120594119861119899+3

119903119899 (119899 = even) = [1205941198612

119879119899120594119861119899+2

]

1119899

(29)


1198618 This




119861) [140]

119898119888 (120583119861)

119898119888 (0)= 1 minus 39 (8) (

120583119861

3120587119879)

2





119905 =1

1199050

(119879 minus 1198791198880

1198791198880

+ 120581119861

120583119861

3119879) ℎ =

119898119897

ℎ0119898119904

(31)


119872119887 (120583) = 119872119887 (0) +120594119898119861

2119879(120583119861

3119879)

2

+ O(120583119861

3119879)

4

(32)


(1198792119898119904)120597

2119872119887120597(1205831198613119879)


120594119898119861

119879=2120581119861119879

1199050119898119904

ℎminus(1minus120573)120573120575

1198911015840119866 (

119905

ℎ1120573120575) (33)




119879119888 (120583119861)

119879119888 (0)= 1 minus 120581119861(

120583119861

3119879)

2

+ O(120583119861

3119879)

4

997904rArr 119879119888 (120583119861) ≃ 154 (1 minus 00066(120583119861

3119879)

2

) MeV

(34)


150

100

50

Tem

pera

ture

(MeV

)


Freezeout

T120594119904c (120583)

T120595120595c (120583)



(a)

007

006

005

004

003

002

001


z

minus2120581qf998400G(z) 120

140180

minusm

st0h(1minus120573)120573120575120594mqT

2

N120591 = 8 mlms = 120

N120591 = 4 mlms = 110

(b)







119875 (120583119897 120583119904)

1198794=119875 (0)

1198794+sum

119894119895

120594119894119895 (119879)

1198794minus119894minus119895(120583119897

119879)

119894

(120583119904

119879)

119895

(35)



119868 (120583119897 120583119904)

1198794= minus

1198733120591

1198733

119889 lnZ119889 ln 119886

=119868 (0)

1198794+sum

119894119895

119887119894119895 (119879) (120583119897

119879)

119894

(120583119904

119879)

119895

(36)


8

6

4

2

0

IT4

100 200 300 400 500 600T (MeV)

SnB = 30

SnB = infinSnB = 45

SnB = 300

Filled Nt = 6

Empty Nt = 4

(a)

6

4

2

5 10 15

pT

4

120598T4






11990611990411120594









119901 up to O ((120583119861119879)4)

119901 up to O ((120583119861119879)2)

le

11 for120583119861

119879le 2

135 for120583119861

119879le 3

(37)


4 Summary





Acknowledgments


References


































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




20

15

10

5

0


120576T4

IT4

pT4

150 200 250 300 350 400T (MeV)

(a)

200 300 400 500 600 700 800 900 1000T (MeV)

6

5

4

3

2

1

0

PT

4

Nf = 2 + 1 EOS Nf = 2 + 1 + 1 N120591 = 8

Nf = 2 + 1 + 1 N120591 = 6 Nf = 2 + 1 + 1 N120591 = 10

(b)



119872119887 (119879119867) = ℎ1120575

119891119866 (119905

ℎ1120573120575) + 119891reg (119879119867) (15)



119905 =1

1199050

119879 minus 1198791198880

1198791198880

ℎ =119867

ℎ0

119867 =119898119897

119898119904

(16)


119891reg = 119867(1198860 + 1198861

119879 minus 1198791198880

1198791198880

+ 1198862(119879 minus 1198791198880

1198791198880

)

2

) (17)







120594120587 minus 120594120575 = int1198894119909 [⟨120595 (119909) 12059121205745120595 (119909) 120595 (0) 12059121205745120595 (0)⟩

minus ⟨120595 (119909) 1205912120595 (119909) 120595 (0) 1205912120595 (0)⟩]

(18)



000

050

100

150

200

250

094 096 098 100 102 104 106 108TTc

Mb

120

110

140

180

Chiral limit

mlms

(a)

000

050

100

150

200

All masses

th1120573120575

Mbh

1120575

O(2)

15

25

110

120

140

180


mlms

(b)



⟨120595120595⟩ = int119889120582120588 (120582119898)2119898

1198982 + 1205822

120594120587 minus 120594120575 = int119889120582120588 (120582119898)4119898

2

(1198982 + 1205822)2

(19)





1205822)



2120575(120582) + 1198881120582 for the density




lim119898rarr0

⟨120588 (120582119898)⟩ = lim119898rarr0

⟨120588 (119898)⟩1205823

3+ O (120582

4) (20)



350

300

250

200

150

100

50

0

T (MeV)140 150 160 170 180 190 200

120594disc T2

1205945disc T2 120594top T

2(ml + mres )2

(120594120587 minus 120594120575)T2

(a)

0025

002

0015

001

0005

00

120588(120582)

002 004 006 008 01120582

163 times 8

Min(120582100)ml

ms

(b)


T ⋍ 170MeV

T ⋍ 210MeV

1

05

0

0 100 200 300 400 500

120588(120582)

(MeV

3)

120588(120582)

(MeV

3)

120588(120582)

(MeV

3)

1

05

0

1

05

0

120582 (MeV)

T ≃ 180sim190MeV

120573 = 218 am = 005

120573 = 218 am = 001

120573 = 225 am = 001120573 = 220 am = 001120573 = 220 am = 0025120573 = 220 am = 005

120573 = 240 am = 001

120573 = 230 am = 001

120573 = 230 am = 0025

120573 = 230 am = 005

(a)



0 2 4 6 8 10 12 14 16

times10minus7

35

3

25

2

15

1

Distance

= 001120573 = 225 (Tsim192) ma

(b)






120588(120582)

120582a

0 004 008 012 016


T = 3301MeVmlms = 120

10eminus02

10eminus03

10eminus04

10eminus05

10eminus06

(a)

14

12

1

08

06

04

02

M(2120587

T)

085 09 095 1 105 11 115 12TTc

P

S

V

A

(b)







119873119891

prod

119891=1

det119863119891 (120583) 119890minus119878119866

(21)


119863119891(120583)119909119910 = [

3

sum

119894=1

1

2120574119894 (119880119894 (119909) 120575119910119909+119894 minus 119880

dagger119894 (119910) 120575119910119909minus119894)

+1

21205744 (119890

1205831198861198804 (119909) 120575119910119909+4 minus 119890

minus120583119886119880

dagger120583 (119910) 120575119910119909minus4)

+ 119886119898119891120575119909119910]

(22)



10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816e119894120579(23)





⟨e119894120579⟩ =

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816e119894120579119890minus119878119866

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816119890minus119878119866

= 119890minus119881Δ119865119879

(24)






120594119883119899 =

119879

119881

120597119899 lnZ120597120583

119899119883

119883 equiv 119861 119878 119876 (25)





1198762 120594

1198612 (120583 =



1205941198612 (120583)

1198792=1205941198612 (0)

1198792+

1205832

211987921205941198614 (0) +

1205834

411987941205941198616 (0) 119879

2+ sdot sdot sdot

(26)







120594119861119878119876119894119895119896 =

119879

119881

120597119894+119895+119896 lnZ

120597120583119894119861120597120583

119895

119878120597120583119896119876

(27)


11986111987811 120594

1198782





SB

HRG

1

08

06

04

02

0

minus3120594

BS

11120594

S 2

120 140 160 180 200 220 240T (MeV)

N120591 = 12

N120591 = 8N120591 = 6

fK scale

(a)

HRG

005

004

003

002

001

0

120594BQ

11T

2

fK scale

120 140 160 180 200 220 240T (MeV)

N120591 = 12 N120591 = 6N120591 = 8Cont extrap

(b)




11987711988312 equiv

119872119883

1205902119883

=120583119861

119879(119877

119883112 +

1205832119861

1198792119877119883312 + O (120583

4119861))

11987711988331 equiv

1198781198831205903119883

119872119883

= 119877119883131 +

1205832119861

1198792119877119883331 + O (120583

4119861)

(28)













RQ 31

HRG

30

25

20

15

10

05

00

Free

T (MeV)140 150 160 170 180 190 200 210 220 230 240

120583BT = 1

120583BT = 0

N120591 = 6

N120591 = 8

(a)

0035

0030

0025

0020HRG

HRG

T = 160MeV

T = 160MeV

T = 170MeV

T = 170MeV

T = 150MeV

T = 150MeV

030

025

020

015

120583S120583

B

0 20 40 60 80 100 120 140 160 180 200120583B (MeV)

minus120583Q120583

B(b)



07

08

09

1

11

0 1 2 3 4 5

Freezeout curve

120583BT

TT

c

30 GeV20 GeV

18 GeV (CERN)10 GeV

(a)

HRG

Power law

10

1

01

001

0001

m1

1 10 100 1000 10000radicSNN (GeV)

(b)








119903119899 (119899 = odd) = radic120594119861119899+1

1198792120594119861119899+3

119903119899 (119899 = even) = [1205941198612

119879119899120594119861119899+2

]

1119899

(29)


1198618 This




119861) [140]

119898119888 (120583119861)

119898119888 (0)= 1 minus 39 (8) (

120583119861

3120587119879)

2





119905 =1

1199050

(119879 minus 1198791198880

1198791198880

+ 120581119861

120583119861

3119879) ℎ =

119898119897

ℎ0119898119904

(31)


119872119887 (120583) = 119872119887 (0) +120594119898119861

2119879(120583119861

3119879)

2

+ O(120583119861

3119879)

4

(32)


(1198792119898119904)120597

2119872119887120597(1205831198613119879)


120594119898119861

119879=2120581119861119879

1199050119898119904

ℎminus(1minus120573)120573120575

1198911015840119866 (

119905

ℎ1120573120575) (33)




119879119888 (120583119861)

119879119888 (0)= 1 minus 120581119861(

120583119861

3119879)

2

+ O(120583119861

3119879)

4

997904rArr 119879119888 (120583119861) ≃ 154 (1 minus 00066(120583119861

3119879)

2

) MeV

(34)


150

100

50

Tem

pera

ture

(MeV

)


Freezeout

T120594119904c (120583)

T120595120595c (120583)



(a)

007

006

005

004

003

002

001


z

minus2120581qf998400G(z) 120

140180

minusm

st0h(1minus120573)120573120575120594mqT

2

N120591 = 8 mlms = 120

N120591 = 4 mlms = 110

(b)







119875 (120583119897 120583119904)

1198794=119875 (0)

1198794+sum

119894119895

120594119894119895 (119879)

1198794minus119894minus119895(120583119897

119879)

119894

(120583119904

119879)

119895

(35)



119868 (120583119897 120583119904)

1198794= minus

1198733120591

1198733

119889 lnZ119889 ln 119886

=119868 (0)

1198794+sum

119894119895

119887119894119895 (119879) (120583119897

119879)

119894

(120583119904

119879)

119895

(36)


8

6

4

2

0

IT4

100 200 300 400 500 600T (MeV)

SnB = 30

SnB = infinSnB = 45

SnB = 300

Filled Nt = 6

Empty Nt = 4

(a)

6

4

2

5 10 15

pT

4

120598T4






11990611990411120594









119901 up to O ((120583119861119879)4)

119901 up to O ((120583119861119879)2)

le

11 for120583119861

119879le 2

135 for120583119861

119879le 3

(37)


4 Summary





Acknowledgments


References


































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




000

050

100

150

200

250

094 096 098 100 102 104 106 108TTc

Mb

120

110

140

180

Chiral limit

mlms

(a)

000

050

100

150

200

All masses

th1120573120575

Mbh

1120575

O(2)

15

25

110

120

140

180


mlms

(b)



⟨120595120595⟩ = int119889120582120588 (120582119898)2119898

1198982 + 1205822

120594120587 minus 120594120575 = int119889120582120588 (120582119898)4119898

2

(1198982 + 1205822)2

(19)





1205822)



2120575(120582) + 1198881120582 for the density




lim119898rarr0

⟨120588 (120582119898)⟩ = lim119898rarr0

⟨120588 (119898)⟩1205823

3+ O (120582

4) (20)



350

300

250

200

150

100

50

0

T (MeV)140 150 160 170 180 190 200

120594disc T2

1205945disc T2 120594top T

2(ml + mres )2

(120594120587 minus 120594120575)T2

(a)

0025

002

0015

001

0005

00

120588(120582)

002 004 006 008 01120582

163 times 8

Min(120582100)ml

ms

(b)


T ⋍ 170MeV

T ⋍ 210MeV

1

05

0

0 100 200 300 400 500

120588(120582)

(MeV

3)

120588(120582)

(MeV

3)

120588(120582)

(MeV

3)

1

05

0

1

05

0

120582 (MeV)

T ≃ 180sim190MeV

120573 = 218 am = 005

120573 = 218 am = 001

120573 = 225 am = 001120573 = 220 am = 001120573 = 220 am = 0025120573 = 220 am = 005

120573 = 240 am = 001

120573 = 230 am = 001

120573 = 230 am = 0025

120573 = 230 am = 005

(a)



0 2 4 6 8 10 12 14 16

times10minus7

35

3

25

2

15

1

Distance

= 001120573 = 225 (Tsim192) ma

(b)






120588(120582)

120582a

0 004 008 012 016


T = 3301MeVmlms = 120

10eminus02

10eminus03

10eminus04

10eminus05

10eminus06

(a)

14

12

1

08

06

04

02

M(2120587

T)

085 09 095 1 105 11 115 12TTc

P

S

V

A

(b)







119873119891

prod

119891=1

det119863119891 (120583) 119890minus119878119866

(21)


119863119891(120583)119909119910 = [

3

sum

119894=1

1

2120574119894 (119880119894 (119909) 120575119910119909+119894 minus 119880

dagger119894 (119910) 120575119910119909minus119894)

+1

21205744 (119890

1205831198861198804 (119909) 120575119910119909+4 minus 119890

minus120583119886119880

dagger120583 (119910) 120575119910119909minus4)

+ 119886119898119891120575119909119910]

(22)



10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816e119894120579(23)





⟨e119894120579⟩ =

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816e119894120579119890minus119878119866

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816119890minus119878119866

= 119890minus119881Δ119865119879

(24)






120594119883119899 =

119879

119881

120597119899 lnZ120597120583

119899119883

119883 equiv 119861 119878 119876 (25)





1198762 120594

1198612 (120583 =



1205941198612 (120583)

1198792=1205941198612 (0)

1198792+

1205832

211987921205941198614 (0) +

1205834

411987941205941198616 (0) 119879

2+ sdot sdot sdot

(26)







120594119861119878119876119894119895119896 =

119879

119881

120597119894+119895+119896 lnZ

120597120583119894119861120597120583

119895

119878120597120583119896119876

(27)


11986111987811 120594

1198782





SB

HRG

1

08

06

04

02

0

minus3120594

BS

11120594

S 2

120 140 160 180 200 220 240T (MeV)

N120591 = 12

N120591 = 8N120591 = 6

fK scale

(a)

HRG

005

004

003

002

001

0

120594BQ

11T

2

fK scale

120 140 160 180 200 220 240T (MeV)

N120591 = 12 N120591 = 6N120591 = 8Cont extrap

(b)




11987711988312 equiv

119872119883

1205902119883

=120583119861

119879(119877

119883112 +

1205832119861

1198792119877119883312 + O (120583

4119861))

11987711988331 equiv

1198781198831205903119883

119872119883

= 119877119883131 +

1205832119861

1198792119877119883331 + O (120583

4119861)

(28)













RQ 31

HRG

30

25

20

15

10

05

00

Free

T (MeV)140 150 160 170 180 190 200 210 220 230 240

120583BT = 1

120583BT = 0

N120591 = 6

N120591 = 8

(a)

0035

0030

0025

0020HRG

HRG

T = 160MeV

T = 160MeV

T = 170MeV

T = 170MeV

T = 150MeV

T = 150MeV

030

025

020

015

120583S120583

B

0 20 40 60 80 100 120 140 160 180 200120583B (MeV)

minus120583Q120583

B(b)



07

08

09

1

11

0 1 2 3 4 5

Freezeout curve

120583BT

TT

c

30 GeV20 GeV

18 GeV (CERN)10 GeV

(a)

HRG

Power law

10

1

01

001

0001

m1

1 10 100 1000 10000radicSNN (GeV)

(b)








119903119899 (119899 = odd) = radic120594119861119899+1

1198792120594119861119899+3

119903119899 (119899 = even) = [1205941198612

119879119899120594119861119899+2

]

1119899

(29)


1198618 This




119861) [140]

119898119888 (120583119861)

119898119888 (0)= 1 minus 39 (8) (

120583119861

3120587119879)

2





119905 =1

1199050

(119879 minus 1198791198880

1198791198880

+ 120581119861

120583119861

3119879) ℎ =

119898119897

ℎ0119898119904

(31)


119872119887 (120583) = 119872119887 (0) +120594119898119861

2119879(120583119861

3119879)

2

+ O(120583119861

3119879)

4

(32)


(1198792119898119904)120597

2119872119887120597(1205831198613119879)


120594119898119861

119879=2120581119861119879

1199050119898119904

ℎminus(1minus120573)120573120575

1198911015840119866 (

119905

ℎ1120573120575) (33)




119879119888 (120583119861)

119879119888 (0)= 1 minus 120581119861(

120583119861

3119879)

2

+ O(120583119861

3119879)

4

997904rArr 119879119888 (120583119861) ≃ 154 (1 minus 00066(120583119861

3119879)

2

) MeV

(34)


150

100

50

Tem

pera

ture

(MeV

)


Freezeout

T120594119904c (120583)

T120595120595c (120583)



(a)

007

006

005

004

003

002

001


z

minus2120581qf998400G(z) 120

140180

minusm

st0h(1minus120573)120573120575120594mqT

2

N120591 = 8 mlms = 120

N120591 = 4 mlms = 110

(b)







119875 (120583119897 120583119904)

1198794=119875 (0)

1198794+sum

119894119895

120594119894119895 (119879)

1198794minus119894minus119895(120583119897

119879)

119894

(120583119904

119879)

119895

(35)



119868 (120583119897 120583119904)

1198794= minus

1198733120591

1198733

119889 lnZ119889 ln 119886

=119868 (0)

1198794+sum

119894119895

119887119894119895 (119879) (120583119897

119879)

119894

(120583119904

119879)

119895

(36)


8

6

4

2

0

IT4

100 200 300 400 500 600T (MeV)

SnB = 30

SnB = infinSnB = 45

SnB = 300

Filled Nt = 6

Empty Nt = 4

(a)

6

4

2

5 10 15

pT

4

120598T4






11990611990411120594









119901 up to O ((120583119861119879)4)

119901 up to O ((120583119861119879)2)

le

11 for120583119861

119879le 2

135 for120583119861

119879le 3

(37)


4 Summary





Acknowledgments


References


































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




350

300

250

200

150

100

50

0

T (MeV)140 150 160 170 180 190 200

120594disc T2

1205945disc T2 120594top T

2(ml + mres )2

(120594120587 minus 120594120575)T2

(a)

0025

002

0015

001

0005

00

120588(120582)

002 004 006 008 01120582

163 times 8

Min(120582100)ml

ms

(b)


T ⋍ 170MeV

T ⋍ 210MeV

1

05

0

0 100 200 300 400 500

120588(120582)

(MeV

3)

120588(120582)

(MeV

3)

120588(120582)

(MeV

3)

1

05

0

1

05

0

120582 (MeV)

T ≃ 180sim190MeV

120573 = 218 am = 005

120573 = 218 am = 001

120573 = 225 am = 001120573 = 220 am = 001120573 = 220 am = 0025120573 = 220 am = 005

120573 = 240 am = 001

120573 = 230 am = 001

120573 = 230 am = 0025

120573 = 230 am = 005

(a)



0 2 4 6 8 10 12 14 16

times10minus7

35

3

25

2

15

1

Distance

= 001120573 = 225 (Tsim192) ma

(b)






120588(120582)

120582a

0 004 008 012 016


T = 3301MeVmlms = 120

10eminus02

10eminus03

10eminus04

10eminus05

10eminus06

(a)

14

12

1

08

06

04

02

M(2120587

T)

085 09 095 1 105 11 115 12TTc

P

S

V

A

(b)







119873119891

prod

119891=1

det119863119891 (120583) 119890minus119878119866

(21)


119863119891(120583)119909119910 = [

3

sum

119894=1

1

2120574119894 (119880119894 (119909) 120575119910119909+119894 minus 119880

dagger119894 (119910) 120575119910119909minus119894)

+1

21205744 (119890

1205831198861198804 (119909) 120575119910119909+4 minus 119890

minus120583119886119880

dagger120583 (119910) 120575119910119909minus4)

+ 119886119898119891120575119909119910]

(22)



10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816e119894120579(23)





⟨e119894120579⟩ =

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816e119894120579119890minus119878119866

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816119890minus119878119866

= 119890minus119881Δ119865119879

(24)






120594119883119899 =

119879

119881

120597119899 lnZ120597120583

119899119883

119883 equiv 119861 119878 119876 (25)





1198762 120594

1198612 (120583 =



1205941198612 (120583)

1198792=1205941198612 (0)

1198792+

1205832

211987921205941198614 (0) +

1205834

411987941205941198616 (0) 119879

2+ sdot sdot sdot

(26)







120594119861119878119876119894119895119896 =

119879

119881

120597119894+119895+119896 lnZ

120597120583119894119861120597120583

119895

119878120597120583119896119876

(27)


11986111987811 120594

1198782





SB

HRG

1

08

06

04

02

0

minus3120594

BS

11120594

S 2

120 140 160 180 200 220 240T (MeV)

N120591 = 12

N120591 = 8N120591 = 6

fK scale

(a)

HRG

005

004

003

002

001

0

120594BQ

11T

2

fK scale

120 140 160 180 200 220 240T (MeV)

N120591 = 12 N120591 = 6N120591 = 8Cont extrap

(b)




11987711988312 equiv

119872119883

1205902119883

=120583119861

119879(119877

119883112 +

1205832119861

1198792119877119883312 + O (120583

4119861))

11987711988331 equiv

1198781198831205903119883

119872119883

= 119877119883131 +

1205832119861

1198792119877119883331 + O (120583

4119861)

(28)













RQ 31

HRG

30

25

20

15

10

05

00

Free

T (MeV)140 150 160 170 180 190 200 210 220 230 240

120583BT = 1

120583BT = 0

N120591 = 6

N120591 = 8

(a)

0035

0030

0025

0020HRG

HRG

T = 160MeV

T = 160MeV

T = 170MeV

T = 170MeV

T = 150MeV

T = 150MeV

030

025

020

015

120583S120583

B

0 20 40 60 80 100 120 140 160 180 200120583B (MeV)

minus120583Q120583

B(b)



07

08

09

1

11

0 1 2 3 4 5

Freezeout curve

120583BT

TT

c

30 GeV20 GeV

18 GeV (CERN)10 GeV

(a)

HRG

Power law

10

1

01

001

0001

m1

1 10 100 1000 10000radicSNN (GeV)

(b)








119903119899 (119899 = odd) = radic120594119861119899+1

1198792120594119861119899+3

119903119899 (119899 = even) = [1205941198612

119879119899120594119861119899+2

]

1119899

(29)


1198618 This




119861) [140]

119898119888 (120583119861)

119898119888 (0)= 1 minus 39 (8) (

120583119861

3120587119879)

2





119905 =1

1199050

(119879 minus 1198791198880

1198791198880

+ 120581119861

120583119861

3119879) ℎ =

119898119897

ℎ0119898119904

(31)


119872119887 (120583) = 119872119887 (0) +120594119898119861

2119879(120583119861

3119879)

2

+ O(120583119861

3119879)

4

(32)


(1198792119898119904)120597

2119872119887120597(1205831198613119879)


120594119898119861

119879=2120581119861119879

1199050119898119904

ℎminus(1minus120573)120573120575

1198911015840119866 (

119905

ℎ1120573120575) (33)




119879119888 (120583119861)

119879119888 (0)= 1 minus 120581119861(

120583119861

3119879)

2

+ O(120583119861

3119879)

4

997904rArr 119879119888 (120583119861) ≃ 154 (1 minus 00066(120583119861

3119879)

2

) MeV

(34)


150

100

50

Tem

pera

ture

(MeV

)


Freezeout

T120594119904c (120583)

T120595120595c (120583)



(a)

007

006

005

004

003

002

001


z

minus2120581qf998400G(z) 120

140180

minusm

st0h(1minus120573)120573120575120594mqT

2

N120591 = 8 mlms = 120

N120591 = 4 mlms = 110

(b)







119875 (120583119897 120583119904)

1198794=119875 (0)

1198794+sum

119894119895

120594119894119895 (119879)

1198794minus119894minus119895(120583119897

119879)

119894

(120583119904

119879)

119895

(35)



119868 (120583119897 120583119904)

1198794= minus

1198733120591

1198733

119889 lnZ119889 ln 119886

=119868 (0)

1198794+sum

119894119895

119887119894119895 (119879) (120583119897

119879)

119894

(120583119904

119879)

119895

(36)


8

6

4

2

0

IT4

100 200 300 400 500 600T (MeV)

SnB = 30

SnB = infinSnB = 45

SnB = 300

Filled Nt = 6

Empty Nt = 4

(a)

6

4

2

5 10 15

pT

4

120598T4






11990611990411120594









119901 up to O ((120583119861119879)4)

119901 up to O ((120583119861119879)2)

le

11 for120583119861

119879le 2

135 for120583119861

119879le 3

(37)


4 Summary





Acknowledgments


References


































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




120588(120582)

120582a

0 004 008 012 016


T = 3301MeVmlms = 120

10eminus02

10eminus03

10eminus04

10eminus05

10eminus06

(a)

14

12

1

08

06

04

02

M(2120587

T)

085 09 095 1 105 11 115 12TTc

P

S

V

A

(b)







119873119891

prod

119891=1

det119863119891 (120583) 119890minus119878119866

(21)


119863119891(120583)119909119910 = [

3

sum

119894=1

1

2120574119894 (119880119894 (119909) 120575119910119909+119894 minus 119880

dagger119894 (119910) 120575119910119909minus119894)

+1

21205744 (119890

1205831198861198804 (119909) 120575119910119909+4 minus 119890

minus120583119886119880

dagger120583 (119910) 120575119910119909minus4)

+ 119886119898119891120575119909119910]

(22)



10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816e119894120579(23)





⟨e119894120579⟩ =

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816e119894120579119890minus119878119866

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816119890minus119878119866

= 119890minus119881Δ119865119879

(24)






120594119883119899 =

119879

119881

120597119899 lnZ120597120583

119899119883

119883 equiv 119861 119878 119876 (25)





1198762 120594

1198612 (120583 =



1205941198612 (120583)

1198792=1205941198612 (0)

1198792+

1205832

211987921205941198614 (0) +

1205834

411987941205941198616 (0) 119879

2+ sdot sdot sdot

(26)







120594119861119878119876119894119895119896 =

119879

119881

120597119894+119895+119896 lnZ

120597120583119894119861120597120583

119895

119878120597120583119896119876

(27)


11986111987811 120594

1198782





SB

HRG

1

08

06

04

02

0

minus3120594

BS

11120594

S 2

120 140 160 180 200 220 240T (MeV)

N120591 = 12

N120591 = 8N120591 = 6

fK scale

(a)

HRG

005

004

003

002

001

0

120594BQ

11T

2

fK scale

120 140 160 180 200 220 240T (MeV)

N120591 = 12 N120591 = 6N120591 = 8Cont extrap

(b)




11987711988312 equiv

119872119883

1205902119883

=120583119861

119879(119877

119883112 +

1205832119861

1198792119877119883312 + O (120583

4119861))

11987711988331 equiv

1198781198831205903119883

119872119883

= 119877119883131 +

1205832119861

1198792119877119883331 + O (120583

4119861)

(28)













RQ 31

HRG

30

25

20

15

10

05

00

Free

T (MeV)140 150 160 170 180 190 200 210 220 230 240

120583BT = 1

120583BT = 0

N120591 = 6

N120591 = 8

(a)

0035

0030

0025

0020HRG

HRG

T = 160MeV

T = 160MeV

T = 170MeV

T = 170MeV

T = 150MeV

T = 150MeV

030

025

020

015

120583S120583

B

0 20 40 60 80 100 120 140 160 180 200120583B (MeV)

minus120583Q120583

B(b)



07

08

09

1

11

0 1 2 3 4 5

Freezeout curve

120583BT

TT

c

30 GeV20 GeV

18 GeV (CERN)10 GeV

(a)

HRG

Power law

10

1

01

001

0001

m1

1 10 100 1000 10000radicSNN (GeV)

(b)








119903119899 (119899 = odd) = radic120594119861119899+1

1198792120594119861119899+3

119903119899 (119899 = even) = [1205941198612

119879119899120594119861119899+2

]

1119899

(29)


1198618 This




119861) [140]

119898119888 (120583119861)

119898119888 (0)= 1 minus 39 (8) (

120583119861

3120587119879)

2





119905 =1

1199050

(119879 minus 1198791198880

1198791198880

+ 120581119861

120583119861

3119879) ℎ =

119898119897

ℎ0119898119904

(31)


119872119887 (120583) = 119872119887 (0) +120594119898119861

2119879(120583119861

3119879)

2

+ O(120583119861

3119879)

4

(32)


(1198792119898119904)120597

2119872119887120597(1205831198613119879)


120594119898119861

119879=2120581119861119879

1199050119898119904

ℎminus(1minus120573)120573120575

1198911015840119866 (

119905

ℎ1120573120575) (33)




119879119888 (120583119861)

119879119888 (0)= 1 minus 120581119861(

120583119861

3119879)

2

+ O(120583119861

3119879)

4

997904rArr 119879119888 (120583119861) ≃ 154 (1 minus 00066(120583119861

3119879)

2

) MeV

(34)


150

100

50

Tem

pera

ture

(MeV

)


Freezeout

T120594119904c (120583)

T120595120595c (120583)



(a)

007

006

005

004

003

002

001


z

minus2120581qf998400G(z) 120

140180

minusm

st0h(1minus120573)120573120575120594mqT

2

N120591 = 8 mlms = 120

N120591 = 4 mlms = 110

(b)







119875 (120583119897 120583119904)

1198794=119875 (0)

1198794+sum

119894119895

120594119894119895 (119879)

1198794minus119894minus119895(120583119897

119879)

119894

(120583119904

119879)

119895

(35)



119868 (120583119897 120583119904)

1198794= minus

1198733120591

1198733

119889 lnZ119889 ln 119886

=119868 (0)

1198794+sum

119894119895

119887119894119895 (119879) (120583119897

119879)

119894

(120583119904

119879)

119895

(36)


8

6

4

2

0

IT4

100 200 300 400 500 600T (MeV)

SnB = 30

SnB = infinSnB = 45

SnB = 300

Filled Nt = 6

Empty Nt = 4

(a)

6

4

2

5 10 15

pT

4

120598T4






11990611990411120594









119901 up to O ((120583119861119879)4)

119901 up to O ((120583119861119879)2)

le

11 for120583119861

119879le 2

135 for120583119861

119879le 3

(37)


4 Summary





Acknowledgments


References


































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





⟨e119894120579⟩ =

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816e119894120579119890minus119878119866

intD119880prod119873119891119891=1

10038161003816100381610038161003816det119863119891 (120583)

10038161003816100381610038161003816119890minus119878119866

= 119890minus119881Δ119865119879

(24)






120594119883119899 =

119879

119881

120597119899 lnZ120597120583

119899119883

119883 equiv 119861 119878 119876 (25)





1198762 120594

1198612 (120583 =



1205941198612 (120583)

1198792=1205941198612 (0)

1198792+

1205832

211987921205941198614 (0) +

1205834

411987941205941198616 (0) 119879

2+ sdot sdot sdot

(26)







120594119861119878119876119894119895119896 =

119879

119881

120597119894+119895+119896 lnZ

120597120583119894119861120597120583

119895

119878120597120583119896119876

(27)


11986111987811 120594

1198782





SB

HRG

1

08

06

04

02

0

minus3120594

BS

11120594

S 2

120 140 160 180 200 220 240T (MeV)

N120591 = 12

N120591 = 8N120591 = 6

fK scale

(a)

HRG

005

004

003

002

001

0

120594BQ

11T

2

fK scale

120 140 160 180 200 220 240T (MeV)

N120591 = 12 N120591 = 6N120591 = 8Cont extrap

(b)




11987711988312 equiv

119872119883

1205902119883

=120583119861

119879(119877

119883112 +

1205832119861

1198792119877119883312 + O (120583

4119861))

11987711988331 equiv

1198781198831205903119883

119872119883

= 119877119883131 +

1205832119861

1198792119877119883331 + O (120583

4119861)

(28)













RQ 31

HRG

30

25

20

15

10

05

00

Free

T (MeV)140 150 160 170 180 190 200 210 220 230 240

120583BT = 1

120583BT = 0

N120591 = 6

N120591 = 8

(a)

0035

0030

0025

0020HRG

HRG

T = 160MeV

T = 160MeV

T = 170MeV

T = 170MeV

T = 150MeV

T = 150MeV

030

025

020

015

120583S120583

B

0 20 40 60 80 100 120 140 160 180 200120583B (MeV)

minus120583Q120583

B(b)



07

08

09

1

11

0 1 2 3 4 5

Freezeout curve

120583BT

TT

c

30 GeV20 GeV

18 GeV (CERN)10 GeV

(a)

HRG

Power law

10

1

01

001

0001

m1

1 10 100 1000 10000radicSNN (GeV)

(b)








119903119899 (119899 = odd) = radic120594119861119899+1

1198792120594119861119899+3

119903119899 (119899 = even) = [1205941198612

119879119899120594119861119899+2

]

1119899

(29)


1198618 This




119861) [140]

119898119888 (120583119861)

119898119888 (0)= 1 minus 39 (8) (

120583119861

3120587119879)

2





119905 =1

1199050

(119879 minus 1198791198880

1198791198880

+ 120581119861

120583119861

3119879) ℎ =

119898119897

ℎ0119898119904

(31)


119872119887 (120583) = 119872119887 (0) +120594119898119861

2119879(120583119861

3119879)

2

+ O(120583119861

3119879)

4

(32)


(1198792119898119904)120597

2119872119887120597(1205831198613119879)


120594119898119861

119879=2120581119861119879

1199050119898119904

ℎminus(1minus120573)120573120575

1198911015840119866 (

119905

ℎ1120573120575) (33)




119879119888 (120583119861)

119879119888 (0)= 1 minus 120581119861(

120583119861

3119879)

2

+ O(120583119861

3119879)

4

997904rArr 119879119888 (120583119861) ≃ 154 (1 minus 00066(120583119861

3119879)

2

) MeV

(34)


150

100

50

Tem

pera

ture

(MeV

)


Freezeout

T120594119904c (120583)

T120595120595c (120583)



(a)

007

006

005

004

003

002

001


z

minus2120581qf998400G(z) 120

140180

minusm

st0h(1minus120573)120573120575120594mqT

2

N120591 = 8 mlms = 120

N120591 = 4 mlms = 110

(b)







119875 (120583119897 120583119904)

1198794=119875 (0)

1198794+sum

119894119895

120594119894119895 (119879)

1198794minus119894minus119895(120583119897

119879)

119894

(120583119904

119879)

119895

(35)



119868 (120583119897 120583119904)

1198794= minus

1198733120591

1198733

119889 lnZ119889 ln 119886

=119868 (0)

1198794+sum

119894119895

119887119894119895 (119879) (120583119897

119879)

119894

(120583119904

119879)

119895

(36)


8

6

4

2

0

IT4

100 200 300 400 500 600T (MeV)

SnB = 30

SnB = infinSnB = 45

SnB = 300

Filled Nt = 6

Empty Nt = 4

(a)

6

4

2

5 10 15

pT

4

120598T4






11990611990411120594









119901 up to O ((120583119861119879)4)

119901 up to O ((120583119861119879)2)

le

11 for120583119861

119879le 2

135 for120583119861

119879le 3

(37)


4 Summary





Acknowledgments


References


































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






1198762 120594

1198612 (120583 =



1205941198612 (120583)

1198792=1205941198612 (0)

1198792+

1205832

211987921205941198614 (0) +

1205834

411987941205941198616 (0) 119879

2+ sdot sdot sdot

(26)







120594119861119878119876119894119895119896 =

119879

119881

120597119894+119895+119896 lnZ

120597120583119894119861120597120583

119895

119878120597120583119896119876

(27)


11986111987811 120594

1198782





SB

HRG

1

08

06

04

02

0

minus3120594

BS

11120594

S 2

120 140 160 180 200 220 240T (MeV)

N120591 = 12

N120591 = 8N120591 = 6

fK scale

(a)

HRG

005

004

003

002

001

0

120594BQ

11T

2

fK scale

120 140 160 180 200 220 240T (MeV)

N120591 = 12 N120591 = 6N120591 = 8Cont extrap

(b)




11987711988312 equiv

119872119883

1205902119883

=120583119861

119879(119877

119883112 +

1205832119861

1198792119877119883312 + O (120583

4119861))

11987711988331 equiv

1198781198831205903119883

119872119883

= 119877119883131 +

1205832119861

1198792119877119883331 + O (120583

4119861)

(28)













RQ 31

HRG

30

25

20

15

10

05

00

Free

T (MeV)140 150 160 170 180 190 200 210 220 230 240

120583BT = 1

120583BT = 0

N120591 = 6

N120591 = 8

(a)

0035

0030

0025

0020HRG

HRG

T = 160MeV

T = 160MeV

T = 170MeV

T = 170MeV

T = 150MeV

T = 150MeV

030

025

020

015

120583S120583

B

0 20 40 60 80 100 120 140 160 180 200120583B (MeV)

minus120583Q120583

B(b)



07

08

09

1

11

0 1 2 3 4 5

Freezeout curve

120583BT

TT

c

30 GeV20 GeV

18 GeV (CERN)10 GeV

(a)

HRG

Power law

10

1

01

001

0001

m1

1 10 100 1000 10000radicSNN (GeV)

(b)








119903119899 (119899 = odd) = radic120594119861119899+1

1198792120594119861119899+3

119903119899 (119899 = even) = [1205941198612

119879119899120594119861119899+2

]

1119899

(29)


1198618 This




119861) [140]

119898119888 (120583119861)

119898119888 (0)= 1 minus 39 (8) (

120583119861

3120587119879)

2





119905 =1

1199050

(119879 minus 1198791198880

1198791198880

+ 120581119861

120583119861

3119879) ℎ =

119898119897

ℎ0119898119904

(31)


119872119887 (120583) = 119872119887 (0) +120594119898119861

2119879(120583119861

3119879)

2

+ O(120583119861

3119879)

4

(32)


(1198792119898119904)120597

2119872119887120597(1205831198613119879)


120594119898119861

119879=2120581119861119879

1199050119898119904

ℎminus(1minus120573)120573120575

1198911015840119866 (

119905

ℎ1120573120575) (33)




119879119888 (120583119861)

119879119888 (0)= 1 minus 120581119861(

120583119861

3119879)

2

+ O(120583119861

3119879)

4

997904rArr 119879119888 (120583119861) ≃ 154 (1 minus 00066(120583119861

3119879)

2

) MeV

(34)


150

100

50

Tem

pera

ture

(MeV

)


Freezeout

T120594119904c (120583)

T120595120595c (120583)



(a)

007

006

005

004

003

002

001


z

minus2120581qf998400G(z) 120

140180

minusm

st0h(1minus120573)120573120575120594mqT

2

N120591 = 8 mlms = 120

N120591 = 4 mlms = 110

(b)







119875 (120583119897 120583119904)

1198794=119875 (0)

1198794+sum

119894119895

120594119894119895 (119879)

1198794minus119894minus119895(120583119897

119879)

119894

(120583119904

119879)

119895

(35)



119868 (120583119897 120583119904)

1198794= minus

1198733120591

1198733

119889 lnZ119889 ln 119886

=119868 (0)

1198794+sum

119894119895

119887119894119895 (119879) (120583119897

119879)

119894

(120583119904

119879)

119895

(36)


8

6

4

2

0

IT4

100 200 300 400 500 600T (MeV)

SnB = 30

SnB = infinSnB = 45

SnB = 300

Filled Nt = 6

Empty Nt = 4

(a)

6

4

2

5 10 15

pT

4

120598T4






11990611990411120594









119901 up to O ((120583119861119879)4)

119901 up to O ((120583119861119879)2)

le

11 for120583119861

119879le 2

135 for120583119861

119879le 3

(37)


4 Summary





Acknowledgments


References


































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




SB

HRG

1

08

06

04

02

0

minus3120594

BS

11120594

S 2

120 140 160 180 200 220 240T (MeV)

N120591 = 12

N120591 = 8N120591 = 6

fK scale

(a)

HRG

005

004

003

002

001

0

120594BQ

11T

2

fK scale

120 140 160 180 200 220 240T (MeV)

N120591 = 12 N120591 = 6N120591 = 8Cont extrap

(b)




11987711988312 equiv

119872119883

1205902119883

=120583119861

119879(119877

119883112 +

1205832119861

1198792119877119883312 + O (120583

4119861))

11987711988331 equiv

1198781198831205903119883

119872119883

= 119877119883131 +

1205832119861

1198792119877119883331 + O (120583

4119861)

(28)













RQ 31

HRG

30

25

20

15

10

05

00

Free

T (MeV)140 150 160 170 180 190 200 210 220 230 240

120583BT = 1

120583BT = 0

N120591 = 6

N120591 = 8

(a)

0035

0030

0025

0020HRG

HRG

T = 160MeV

T = 160MeV

T = 170MeV

T = 170MeV

T = 150MeV

T = 150MeV

030

025

020

015

120583S120583

B

0 20 40 60 80 100 120 140 160 180 200120583B (MeV)

minus120583Q120583

B(b)



07

08

09

1

11

0 1 2 3 4 5

Freezeout curve

120583BT

TT

c

30 GeV20 GeV

18 GeV (CERN)10 GeV

(a)

HRG

Power law

10

1

01

001

0001

m1

1 10 100 1000 10000radicSNN (GeV)

(b)








119903119899 (119899 = odd) = radic120594119861119899+1

1198792120594119861119899+3

119903119899 (119899 = even) = [1205941198612

119879119899120594119861119899+2

]

1119899

(29)


1198618 This




119861) [140]

119898119888 (120583119861)

119898119888 (0)= 1 minus 39 (8) (

120583119861

3120587119879)

2





119905 =1

1199050

(119879 minus 1198791198880

1198791198880

+ 120581119861

120583119861

3119879) ℎ =

119898119897

ℎ0119898119904

(31)


119872119887 (120583) = 119872119887 (0) +120594119898119861

2119879(120583119861

3119879)

2

+ O(120583119861

3119879)

4

(32)


(1198792119898119904)120597

2119872119887120597(1205831198613119879)


120594119898119861

119879=2120581119861119879

1199050119898119904

ℎminus(1minus120573)120573120575

1198911015840119866 (

119905

ℎ1120573120575) (33)




119879119888 (120583119861)

119879119888 (0)= 1 minus 120581119861(

120583119861

3119879)

2

+ O(120583119861

3119879)

4

997904rArr 119879119888 (120583119861) ≃ 154 (1 minus 00066(120583119861

3119879)

2

) MeV

(34)


150

100

50

Tem

pera

ture

(MeV

)


Freezeout

T120594119904c (120583)

T120595120595c (120583)



(a)

007

006

005

004

003

002

001


z

minus2120581qf998400G(z) 120

140180

minusm

st0h(1minus120573)120573120575120594mqT

2

N120591 = 8 mlms = 120

N120591 = 4 mlms = 110

(b)







119875 (120583119897 120583119904)

1198794=119875 (0)

1198794+sum

119894119895

120594119894119895 (119879)

1198794minus119894minus119895(120583119897

119879)

119894

(120583119904

119879)

119895

(35)



119868 (120583119897 120583119904)

1198794= minus

1198733120591

1198733

119889 lnZ119889 ln 119886

=119868 (0)

1198794+sum

119894119895

119887119894119895 (119879) (120583119897

119879)

119894

(120583119904

119879)

119895

(36)


8

6

4

2

0

IT4

100 200 300 400 500 600T (MeV)

SnB = 30

SnB = infinSnB = 45

SnB = 300

Filled Nt = 6

Empty Nt = 4

(a)

6

4

2

5 10 15

pT

4

120598T4






11990611990411120594









119901 up to O ((120583119861119879)4)

119901 up to O ((120583119861119879)2)

le

11 for120583119861

119879le 2

135 for120583119861

119879le 3

(37)


4 Summary





Acknowledgments


References


































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




RQ 31

HRG

30

25

20

15

10

05

00

Free

T (MeV)140 150 160 170 180 190 200 210 220 230 240

120583BT = 1

120583BT = 0

N120591 = 6

N120591 = 8

(a)

0035

0030

0025

0020HRG

HRG

T = 160MeV

T = 160MeV

T = 170MeV

T = 170MeV

T = 150MeV

T = 150MeV

030

025

020

015

120583S120583

B

0 20 40 60 80 100 120 140 160 180 200120583B (MeV)

minus120583Q120583

B(b)



07

08

09

1

11

0 1 2 3 4 5

Freezeout curve

120583BT

TT

c

30 GeV20 GeV

18 GeV (CERN)10 GeV

(a)

HRG

Power law

10

1

01

001

0001

m1

1 10 100 1000 10000radicSNN (GeV)

(b)








119903119899 (119899 = odd) = radic120594119861119899+1

1198792120594119861119899+3

119903119899 (119899 = even) = [1205941198612

119879119899120594119861119899+2

]

1119899

(29)


1198618 This




119861) [140]

119898119888 (120583119861)

119898119888 (0)= 1 minus 39 (8) (

120583119861

3120587119879)

2





119905 =1

1199050

(119879 minus 1198791198880

1198791198880

+ 120581119861

120583119861

3119879) ℎ =

119898119897

ℎ0119898119904

(31)


119872119887 (120583) = 119872119887 (0) +120594119898119861

2119879(120583119861

3119879)

2

+ O(120583119861

3119879)

4

(32)


(1198792119898119904)120597

2119872119887120597(1205831198613119879)


120594119898119861

119879=2120581119861119879

1199050119898119904

ℎminus(1minus120573)120573120575

1198911015840119866 (

119905

ℎ1120573120575) (33)




119879119888 (120583119861)

119879119888 (0)= 1 minus 120581119861(

120583119861

3119879)

2

+ O(120583119861

3119879)

4

997904rArr 119879119888 (120583119861) ≃ 154 (1 minus 00066(120583119861

3119879)

2

) MeV

(34)


150

100

50

Tem

pera

ture

(MeV

)


Freezeout

T120594119904c (120583)

T120595120595c (120583)



(a)

007

006

005

004

003

002

001


z

minus2120581qf998400G(z) 120

140180

minusm

st0h(1minus120573)120573120575120594mqT

2

N120591 = 8 mlms = 120

N120591 = 4 mlms = 110

(b)







119875 (120583119897 120583119904)

1198794=119875 (0)

1198794+sum

119894119895

120594119894119895 (119879)

1198794minus119894minus119895(120583119897

119879)

119894

(120583119904

119879)

119895

(35)



119868 (120583119897 120583119904)

1198794= minus

1198733120591

1198733

119889 lnZ119889 ln 119886

=119868 (0)

1198794+sum

119894119895

119887119894119895 (119879) (120583119897

119879)

119894

(120583119904

119879)

119895

(36)


8

6

4

2

0

IT4

100 200 300 400 500 600T (MeV)

SnB = 30

SnB = infinSnB = 45

SnB = 300

Filled Nt = 6

Empty Nt = 4

(a)

6

4

2

5 10 15

pT

4

120598T4






11990611990411120594









119901 up to O ((120583119861119879)4)

119901 up to O ((120583119861119879)2)

le

11 for120583119861

119879le 2

135 for120583119861

119879le 3

(37)


4 Summary





Acknowledgments


References


































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119903119899 (119899 = odd) = radic120594119861119899+1

1198792120594119861119899+3

119903119899 (119899 = even) = [1205941198612

119879119899120594119861119899+2

]

1119899

(29)


1198618 This




119861) [140]

119898119888 (120583119861)

119898119888 (0)= 1 minus 39 (8) (

120583119861

3120587119879)

2





119905 =1

1199050

(119879 minus 1198791198880

1198791198880

+ 120581119861

120583119861

3119879) ℎ =

119898119897

ℎ0119898119904

(31)


119872119887 (120583) = 119872119887 (0) +120594119898119861

2119879(120583119861

3119879)

2

+ O(120583119861

3119879)

4

(32)


(1198792119898119904)120597

2119872119887120597(1205831198613119879)


120594119898119861

119879=2120581119861119879

1199050119898119904

ℎminus(1minus120573)120573120575

1198911015840119866 (

119905

ℎ1120573120575) (33)




119879119888 (120583119861)

119879119888 (0)= 1 minus 120581119861(

120583119861

3119879)

2

+ O(120583119861

3119879)

4

997904rArr 119879119888 (120583119861) ≃ 154 (1 minus 00066(120583119861

3119879)

2

) MeV

(34)


150

100

50

Tem

pera

ture

(MeV

)


Freezeout

T120594119904c (120583)

T120595120595c (120583)



(a)

007

006

005

004

003

002

001


z

minus2120581qf998400G(z) 120

140180

minusm

st0h(1minus120573)120573120575120594mqT

2

N120591 = 8 mlms = 120

N120591 = 4 mlms = 110

(b)







119875 (120583119897 120583119904)

1198794=119875 (0)

1198794+sum

119894119895

120594119894119895 (119879)

1198794minus119894minus119895(120583119897

119879)

119894

(120583119904

119879)

119895

(35)



119868 (120583119897 120583119904)

1198794= minus

1198733120591

1198733

119889 lnZ119889 ln 119886

=119868 (0)

1198794+sum

119894119895

119887119894119895 (119879) (120583119897

119879)

119894

(120583119904

119879)

119895

(36)


8

6

4

2

0

IT4

100 200 300 400 500 600T (MeV)

SnB = 30

SnB = infinSnB = 45

SnB = 300

Filled Nt = 6

Empty Nt = 4

(a)

6

4

2

5 10 15

pT

4

120598T4






11990611990411120594









119901 up to O ((120583119861119879)4)

119901 up to O ((120583119861119879)2)

le

11 for120583119861

119879le 2

135 for120583119861

119879le 3

(37)


4 Summary





Acknowledgments


References


































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




150

100

50

Tem

pera

ture

(MeV

)


Freezeout

T120594119904c (120583)

T120595120595c (120583)



(a)

007

006

005

004

003

002

001


z

minus2120581qf998400G(z) 120

140180

minusm

st0h(1minus120573)120573120575120594mqT

2

N120591 = 8 mlms = 120

N120591 = 4 mlms = 110

(b)







119875 (120583119897 120583119904)

1198794=119875 (0)

1198794+sum

119894119895

120594119894119895 (119879)

1198794minus119894minus119895(120583119897

119879)

119894

(120583119904

119879)

119895

(35)



119868 (120583119897 120583119904)

1198794= minus

1198733120591

1198733

119889 lnZ119889 ln 119886

=119868 (0)

1198794+sum

119894119895

119887119894119895 (119879) (120583119897

119879)

119894

(120583119904

119879)

119895

(36)


8

6

4

2

0

IT4

100 200 300 400 500 600T (MeV)

SnB = 30

SnB = infinSnB = 45

SnB = 300

Filled Nt = 6

Empty Nt = 4

(a)

6

4

2

5 10 15

pT

4

120598T4






11990611990411120594









119901 up to O ((120583119861119879)4)

119901 up to O ((120583119861119879)2)

le

11 for120583119861

119879le 2

135 for120583119861

119879le 3

(37)


4 Summary





Acknowledgments


References


































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




8

6

4

2

0

IT4

100 200 300 400 500 600T (MeV)

SnB = 30

SnB = infinSnB = 45

SnB = 300

Filled Nt = 6

Empty Nt = 4

(a)

6

4

2

5 10 15

pT

4

120598T4






11990611990411120594









119901 up to O ((120583119861119879)4)

119901 up to O ((120583119861119879)2)

le

11 for120583119861

119879le 2

135 for120583119861

119879le 3

(37)


4 Summary





Acknowledgments


References


































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






Acknowledgments


References


































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics










































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



review article qcd thermodynamics on the...

Documents