The finite temperature QCD transition for physical quark masses is a
crossover. For smaller quark masses a first-order phase transition is expected.
Using Symanzik improved gauge and stout improved fermion action for 2+1 flavour
staggered QCD we give estimates/bounds for the phase line separating the
first-order region from the crossover one. The calculations are carried out on
two different lattice spacings. Our conclusion for the critical mass is $m_0
\lesssim 0.07 \cdot m_{phys}$ for $N_T=4$ and $m_0 \lesssim 0.12 \cdot
m_{phys}$ for $N_T=6$ lattices.Comment: Talk presented at the XXV International Symposium on Lattice Field
  Theory, July 30 - August 4 2007, Regensburg, Germany. 7 pages, 6 figure

Endrodi, G.

Fodor, Z.

Katz, S. D.

Szabo, K. K.

English

arXiv

Endrodi, G

Fodor, Z

Katz, S D

Szabo, K K

ELTE Digital Institutional Repository (EDIT)

arXiv:0710.0998v1  [hep-lat]  4 Oct 2007The nature of the finite temperature QCD transitionas a function of the quark massesG. Endr ődi ∗1, Z. Fodor 1,2, S.D. Katz1,2, K.K. Szabó 21Institute for Theoretical Physics, Eötvös University, H-1117 Budapest, Hungary.2Department of Physics, University of Wuppertal, D-42097 Wuppertal, Germany.E-mail: endrodi@general.elte.huThe finite temperature QCD transition for physical quark masses is a crossover. For smallerquark masses a first-order phase transition is expected. Using Symanzik improved gauge andstout improved fermion action for 2+1 flavour staggered QCD we give estimates/bounds for thephase line separating the first-order region from the crossover ne. The calculations are carriedout on two different lattice spacings. Our conclusion for the critical mass ism0 . 0.07·mphysforNT = 4 andm0 . 0.12·mphysfor NT = 6 lattices.The XXV International Symposium on Lattice Field TheoryJuly 30 - August 4 2007Regensburg, Germany∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licen. http://pos.sissa.it/The nature of the finite temperature QCD transition as a functio of the quark masses G. Endr̋odi1. IntroductionQuantum chromodynamics (QCD) is the theory of strong interac ions. Due to one of its mostimportant properties, asymptotic freedom, at high temperatures it describes a different phase ofmatter called quark-gluon plasma (QGP). The phase transitio between the hadronic phase of mat-ter and QGP can be investigated by lattice simulations. The transition at zero chemical potential –which represents the case of equal number of quarks and antiqu rks – is of huge importance, sinceit is relevant for both regarding the early Universe and highenergy collisions.The 2+1 flavour QCD transition was recently found to be an analytic crossover [1] (instead ofa first-order phase transition), which usually results in different transition temperatures for differentobservables [2] and to a broadening of the equation of state around the transition temperature [3].These works were carried out using physical quark masses; neverth less different values of thequark masses can also have relevance. For three massless quark we expect from QCD effectivemodels, that a first-order phase transition takes place. Forinfinite quark masses (which describepure gauge theory) lattice results indicate that there is also first-order transition. For two masslessflavours a second-order transition is expected. We can summarize our knowledge on figure 1.mmsudPZ(2)O(4)physical point!cross−over regionregion1st order1st orderFigure 1: The phase diagram of QCD. First-order and crossover regionsare eparated by second-order lines.For the one pointed towards by the arrow we expect a Z(2) universality class. The exact position of the lineis to be determined.There are second-order phase transition lines, that separat the first-order and crossover re-gions. For the 2 flavour case, the universality class of the phase line is predicted to be O(4) [4],while for the 2+1 case, we expect Z(2). However, the exact position of this latter phase line stillneeds to be determined with adequate accuracy. In the work of[5] the phase line is found to be atabout 80% of the physical quark mass onNT = 4 lattices with the unimproved staggered action.The same authors have presented theirNT = 6 study at the present conference [6]. Based on the re-sults about the strength of the transition for different lattice spacings and/or discretization schemes,one expects that reducing the discretization errors results in a weaker transition. In agreement withthis expectation they observed that the first-order region shrinks, and the physical point is fartherfrom the phase line.The location of the second-order line has high importance, since combined with the curva-ture of the phase surface (in them− µ space), it can influence the position [7, 8] (or even theexistence [5]) of the critical endpoint on the QCD phase diagram.2The nature of the finite temperature QCD transition as a functio of the quark masses G. Endr̋odiIn this paper we give estimates and upper bounds for the critical mass by means of analyzingthe behaviour of some quantities that are sensitive to the nature of the phase transition. Thesequantities will be the susceptibility and the Binder cumulant of the chiral condensate. It will turnout, that the latter is more trustworthy in locating the second- rder line, nevertheless, we presentresults here regarding both quantities.2. Second-order behaviourFirst we discuss what kind of behaviour we expect from the susceptibility and from the Bindercumulant in the vicinity of a second-order line. Then we carry out lattice simulations for differentquark masses, and compare them with the expectations.2.1 The chiral susceptibilityThe susceptibility of the chiral condensate is defined asχψ̄ψ ≡ ∂ψ̄ψ∂m . At the transition temper-ature it is supposed to show a pronounced peak. Distinguishing between first-order, second-ordertransitions and crossovers can be achieved by finite-size scaling of some properties of this peak.Particularly, for second-order transitions, the height ofthe peak should diverge at the critical point.We can observe this behaviour in a statistical physical approach. Our order parameter of the tran-sition is the chiral condensatēψψ , the reduced temperature ist ≡ (T −TC)/TC, and the externalfield, which breaks the symmetry is the quark massm. The definitions of the critical indicesε ,γ ,andδ are:ψ̄ψ ∼ |t|ε , χψ̄ψ ∼ |t|−γ , ψ̄ψδ |t=0 ∼ m (2.1)Now let’s take the derivative of the last proportionality with respect tom, so the susceptibilitycan be expressed as a function of the quark mass. This will determine how the height of the peakgrows while reducing the mass.χψ̄ψ |t=0 ∼ m1δ −1 (2.2)From the first and third proportionality in (2.1) we can also obtain how the critical temperaturedepends on the quark mass:|t| ∼ m 1εδ (2.3)It is worth mentioning that if we start from the second proportionality in (2.1) and from (2.2), thenwe obtain|t| ∼ mδ−1γδ , which is identical to (2.3) (c.f. theγ = ε(δ −1) scaling rule).In the following we will analyze the susceptibility peak as afunction ofβ ≡ 6/g2. Since werestrict ourselves to the interval around the critical temprature, where the functionβ (T) can belinearized, this means that in the above formulae we can substit te the reduced temperature with(the reduced)β .The critical exponents in question can be looked up in the literature for the interesting univer-sality classes; these values are summarized in the next table [9]:1/εδ 1/δ −13D Ising 0.633 -0.7853D O(2) 0.598 -0.7943D O(4) 0.537 -0.7943The nature of the finite temperature QCD transition as a functio of the quark masses G. Endr̋odi2.2 The Binder cumulantThe cumulant is another useful quantity to distinguish betwe n different types of phase transi-tions. Roughly, it measures how much the distribution of theorder parameter is of Gaussian type.Its definition for the chiral condensate is as follows:Bψ̄ψ ≡〈(δψ̄ψ)4〉〈(δψ̄ψ)2〉2 (2.4)whereδ denotes the deviation from the average, soδψ̄ψ ≡ ψ̄ψ −〈ψ̄ψ〉. The actual value of thecumulant can be easily calculated for different distributions. We should analyze the distribution ofψ̄ψ in the infinite volume limit, at the critical temperature. For a first-order transition the distri-bution consists of two Dirac-deltas, for whichBψ̄ψ = 1. For a crossover we have one Dirac-delta,which is (through a series of finite volumes) approached by Gaussi n functions getting narrowerand narrower. In this caseBψ̄ψ ≈ 3. For second-order transitions the value of the cumulant dependson the universality class: for Z(2), that of the three-dimensio al Ising-model,Bψ̄ψ = 1.604 [10];for O(2),Bψ̄ψ = 1.242 [11]; while for O(4),Bψ̄ψ = 1.092 [9].3. ResultsOur results were obtained by lattice simulations with 2+1 flavours of staggered quarks. Weused Symanzik improved gauge and stout improved fermionic act on; the details concerning theaction and the simulation techniques are described in [1, 2,3]. For smaller volumes (ranging from103×4 to 163×4) up to 500-1000 configurations were generated. For larger volumes (up to 243×4andNT = 6 simulations) we had smaller statistics, about a few hundred configurations. Autocorre-lation time was measured to be around 5, so we used every fifth configuration for measurements.Measurement of the chiral condensate was carried out with 60random vectors.In order to approach the second-order line we had to carry outsimulations at very small quarkmasses. There are, however, limitations that we have to takeinto account. First of all, smallermasses increase CPU time by a factor of 1/m. Still very important is, that we have to keep thelattice sizes much larger than the characteristic length oft e system. This length is given by theinverse of the pion mass:ℓψ̄ψ = 1/mπ ≈ 1/√m, so for smaller masses we also needed largerlattices. Paying attention to these phenomena, we carried out simulations for quark masses rangingfrom 200% down to 5% of their physical values.3.1 The chiral susceptibilityFor every quark mass we used, we had to search for the susceptibility peak on theχψ̄ψ − βplane. These peaks are plotted on figure 2., for the case ofm/mphys= 0.4. . .2. The height of thepeak increases as smaller quark masses are used, which indicates the strengthening of the transition.As shown by equations (2.2) and (2.3), the height and the position of the susceptibility peakshould follow a power-like behaviour, which has a singular point (non-analytical point for the caseof the position) at the critical mass of the second-order point, denoted in the following bym0.The critical indeces for these power functions (as summarized n table 2.1.) are rather close toeach other, particularly for the case of the height. This means that it is very difficult to distinguish4The nature of the finite temperature QCD transition as a functio of the quark masses G. Endr̋odiFigure 2: The susceptibility peaks for different quark masses. Apparently the transition gets stronger forsmaller masses, as it is shown by higher peaks.between different universality classes from observing thebehaviour of these quantities. However,if we suppose that that we are dealing with a given universality c ass (namely Z(2) for our case),than we may keep the exponent of the power function fixed, and perform a fit for the critical mass.These fits are shown on figure 3.Figure 3: The position (left panel) and the height (middle panel forNT = 4, right panel forNT = 6) of thesusceptibility peak as a function of the quark mass. With solid lines shown are the power function fits, withexponents fixed at values from different universality classes (which give the same for the height).We performed these fits for different fit intervals. The results from the height of the peak canbe seen on figure 4. As we narrow the fit interval by excluding points with largest masses, theestimate form0 reaches a nice plateu, which indicates that we are already inthe dominant region ofthe second-order point at smaller masses. We can obtain an upper bound from this analysis, whichis m0 . 0.05·mphys. The same procedure was done also for the case of the positionof the peak, forwhich the fits turned out to be less stable. Nevertheless fromthis latter we hadm0 . 0.12·mphys.3.2 The Binder cumulantWe saw in section 2.2., that the values of the cumulant for thesecond-order cases and forcrossover are quite apart from each other, which makes it easier to have a more accurate estimatefor the critical mass. We measured the cumulant atTC, i.e. atβ corresponding to the position ofthe peak, which almost always coincided with the minimum of the cumulant in that temperature5The nature of the finite temperature QCD transition as a functio of the quark masses G. Endr̋odiFigure 4: The singular point of the power-function fitted to the heightof he peak, versus the upper boundof the fit interval. As largest masses are excluded from the fit, we get deeper in the dominant region, wherethe behaviour of the susceptibility is governed by the approriate critical index.interval. At larger masses the cumulant has a value consistet with the crossover behaviour, thencloser to the critical mass it starts to decrease, and at the point with smallest mass (m/mphys= 0.05for NT = 4 and 0.1 for NT = 6) it already reaches the value which represents the Z(2) universalityclass. So the behaviour of the cumulant is consistent with the assumption we posed about figure 1.If we accept this scenario, than we can have an upper bound here again form0. These results areshown on figure 5.Figure 5: The value of the Binder cumulant atTC plotted against the quark mass for 163×4 (left), 243×4(middle) and 183×6 (right). Behaviour at smaller masses indicate that the universality class of the second-order point in question is consistent with Z(2). We have upper bounds form0 indicated by the blue lines.From this analysis we can conclude for the critical mass thatm0/mphys. 0.07 forNT = 4, andm0/mphys. 0.12 for NT = 6.3.3 SummaryThe behaviour of the Binder cumulant showed that the universality class of the second-orderline is consistent with Z(2). We also obtained upper bounds for the value of the critical mass fromthe analysis of both the chiral susceptibility and the Binder cumulant. These estimates suggest6The nature of the finite temperature QCD transition as a functio of the quark masses G. Endr̋odistrongly that the critical mass is below 7% of the physical quark mass onNT = 4 and 12% onNT = 6 lattices. This means that the physical point is at least about ten times farther from the lowerleft corner of the phase diagram, than the second-order phase line. So the first-order region on figure1. is exaggarated, and looks rather like as depicted on figure6., which is our final conclusion.Figure 6: The lower left part of the QCD phase diagram, as a conclusion of ur work.References[1] Y. Aoki, G. Endr̋odi, Z. Fodor, S.D. Katz and K.K. Szabó,The order of the quantum chromodynamicsphase transition predicted by the standard model of particle physics, Nature443 (2006) 675-678[hep-lat/0611014].[2] Y. Aoki, Z. Fodor, S.D. Katz and K.K. Szabó,The QCD transition temperature: Results with physicalmasses in the continuum limit, Phys. Lett.B643 (2006) 46-54 [hep-lat/0609068].[3] Y. Aoki, Z. Fodor, S.D. Katz and K.K. Szabó,The equation of state in lattice QCD: With physicalquark masses towards the continuum limit, JHEP01 (2006) 089 [hep-lat/0510084].[4] R.D. Pisarski and F. Wilczek,Remarks on the chiral phase transition in chromodymanics, Phys. Rev.D29 (1984) 338-341.[5] P. de Forcrand and O. Philipsen,The chiral critical line of N(f) = 2+1 QCD at zero and non-zerobaryon density, JHEP01 (2007) 077 [hep-lat/0607017].[6] P. de Forcrand and O. Philipsen,The chiral critical point of Nf = 3 QCD: towards the continuum, Inproceedings ofXXV International Symposium on Lattice Field Theory(2007)PoS(LATTICE 2007)218.[7] Z. Fodor and S.D. Katz,Lattice determination of the critical point of QCD at finite Tand mu, JHEP03 (2002) 014 [hep-lat/0106002].[8] Z. Fodor and S.D. Katz,Critical point of QCD at finite T and mu, lattice results for physical quarkmasses, JHEP04 (2004) 050 [hep-lat/0402006].[9] K. Kanaya and S. Kaya,Critical exponents of a three-dimensional O(4) spin model, Phys. Rev. D51,5 (1995) 2404–2410 [hep-lat/9409001].[10] H.W.J. Blote, E. Luijten and J.R. Heringa,sing universality in three dimensions: a Monte Carlo study,Journal of Physics A Mathematical General28 (1995) 6289-6313 [cond-mat/9509016].[11] S. Holtmann, J. Engels and T. Schulze,Specific heat and energy for the three-dimensional O(2)model, Nuclear Physics B - Proceedings Supplements106 (2002) 893 [hep-lat/0109013].7

The Nature of the finite temperature QCD transition as a function of the quark masses

https://edit.elte.hu/xmlui/bitstream/handle/10831/47709/The%20nature%20of%20the%20finite%20temperature%20QCD%20transition.pdf?sequence=1&amp;isAllowed=y

The nature of the finite temperature QCD transition as a function of the
  quark masses

The nature of the finite temperature QCD transition as a function of the quark masses

Abstract

Similar works

Full text

Available Versions

ELTE Digital Institutional Repository (EDIT)