We report on a continuum extrapolated result (arXiv:1309.5258) for the
equation of state (EoS) of QCD with $N_f=2+1$ dynamical quark flavors and
discuss preliminary results obtained with an additional dynamical charm quark
($N_f=2+1+1$). For all our final results, the systematics are controlled, quark
masses are set to their physical values, and the continuum limit is taken using
at least three lattice spacings corresponding to temporal extents up to
$N_t=16$.Comment: Conference proceedings: The 32nd International Symposium on Lattice
  Field Theory - Lattice 2014, June 23-28, 2014, Columbia University, New York,
  New Yor

Borsanyi, Szabolcs

Fodor, Zoltan

Hoelbling, Christian

Katz, Sandor D.

Krieg, Stefan

Ratti, Claudia

Szabo, Kalman K.

English

arXiv

We report on a continuum extrapolated result [1] for the equation of state (EoS) of QCD with N f = 2+ 1 dynamical quark flavors and discuss preliminary results obtained with an additional dynamical charm quark (N f = 2+1+1). For all our final results, the systematics are controlled, quark masses are set to their physical values, and the continuum limit is taken using at least three lattice spacings corresponding to temporal extents up to Nt = 16

Szabolcs Borsányia

Zoltán Fodora

Christian Hoelblinga

Sándor D. Katzc

Stefan Krieg

Claudia Rattie

Kálmán K. Szabóa

CiteSeerX

Recent results on the Equation of State of QCD

We report on a continuum extrapolated result for the equation of state (EoS) of QCD with Nf = 2 + 1 dynamical quark flavors and discuss preliminary results obtained with an additional dynamical charm quark (Nf = 2 + 1 + 1). For all our final results, the systematics are controlled, quark masses are set to their physical values, and the continuum limit is taken using at least three lattice spacings corresponding to temporal extents up to Nt = 16

Borsányi, Szabolcs

Fodor, Zoltán

Hoelbling, C.

Katz, Sándor

Krieg, S.

Ratti, C.

Szabó, K. K.

Repository of the Academy's Library

Recent results on the equation of state of QCD

We report on a continuum extrapolated result (arXiv:1309.5258) for the equation of state (EoS) of QCD with $N_f=2+1$ dynamical quark flavors and discuss preliminary results obtained with an additional dynamical charm quark ($N_f=2+1+1$). For all our final results, the systematics are controlled, quark masses are set to their physical values, and the continuum limit is taken using at least three lattice spacings corresponding to temporal extents up to $N_t=16$

Szabo, Kalman

Juelich Shared Electronic Resources

Recent results on the Equation of State of QCDSzabolcs Borsányia, Zoltán Fodora,b,c, Christian Hoelblinga, Sándor D. Katzc,d ,Stefan Krieg∗,a,b, Claudia Rattie, Kálmán K. Szabóa,baDepartment of Physics, University of Wuppertal,Gaußstr. 20, D-42119, GermanybForschungszentrum Jülich, Jülich, D-52425, GermanycInstitute for Theoretical Physics, Eötvös University, Pázmány 1, H-1117 Budapest, HungarydMTA-ELTE Lendület Lattice Gauge Theory Research GroupeUniversità di Torino and INFN, Sezione di Torino via Giuria 1, I-10125 Torino, ItalyWe report on a continuum extrapolated result [1] for the equation of state (EoS) of QCD withN f = 2+ 1 dynamical quark flavors and discuss preliminary results obtained with an additionaldynamical charm quark (N f = 2+1+1). For all our final results, the systematics are controlled,quark masses are set to their physical values, and the continuum limit is taken using at least threelattice spacings corresponding to temporal extents up to Nt = 16.The 32nd International Symposium on Lattice Field Theory,23-28 June, 2014Columbia University New York, NY∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/arXiv:1410.7917v1  [hep-lat]  29 Oct 2014EoS of QCD Stefan Krieg1. Full result for the N f = 2+1 equation of stateThe rapid transition from the quark-gluon-plasma ’phase’1 to the hadronic phase in the earlyuniverse and the QCD phase diagram are subjects of intense study in present heavy-ion experiments(LHC@CERN, RHIC@BNL, and the upcoming FAIR@GSI). This transition can be studied in asystematic way in Lattice QCD (for recent reviews see, e.g., [3, 4, 5, 6]). The equation of state(EoS) of QCD, (i.e, the pressure p, energy density ε , trace anomaly I = ε −3p, entropy s = (ε +p)/T , and the speed of sound c2s = dp/dε as functions of the temperature) has been determined byseveral lattice groups, however, a full result was, until recently, still lacking. Ref. [7] constitutes afull result at three characteristic temperatures, which we have now extended to the full temperaturerange in [1] and made available electronically [8].Our calculation is based on a tree-level Symanzik improved gauge action with 2-step stout-link improved staggered fermions. The precise definition of the action can be found in ref. [9], itsadvantageous scaling properties are studied in ref. [10, 11, 12]. In particular, while it approachesthe continuum value of the Stefan-Boltzmann limit in the infinite temperature limit T → ∞ slowerthan actions with p4 or Naik terms (the latter is an additional fermionic term in the asqtad and hisqactions), it behaves monotonous and reaches the asymptotic a2 behavior quite early. Extrapolationsfrom moderate temporal extents, e.g., using Nt ≥ 8, allow for a smooth continuum extrapolationand provide an accuracy on the percent level, the typical accuracy one aims to reach. Additionally,applying simple tree-level improvement factors for the bulk thermodynamic observables brings theindividual data points for the different Nt very close to the continuum limit.This points to an important advancement of the calculation described here over our previousresults of [7]: we now include a large range of Nt = 12 data points and one Nt = 16 data pointlocated at the peak position. Previously, we only had Nt = 12 results at three characteristic tem-perature values available. The T → 0 limit is difficult due to taste-breaking effects, but is crucialsince the renormalization is done at zero temperature, i.e. p(T=0)=0. A mismatch at T=0 leads toa shift in the whole EoS. Previously, we calculated the difference in the pressure between the phys-ical theory and its counterpart with 720 MeV heavy pions at a selected temperature (100 MeV),where the latter theory has practically zero pressure, and we, therefore, get p(T = 100 MeV) inthe physical theory with the desired normalization. The difference of this result and the predictionby the Hadron Resonance Gas model (HRG) was then included in the systematical error. With ourincreased range of temporal extents, we now can use five lattice spacings to fix the additive term inthe pressure, integrating in the quark mass down to the physical values (see Figure 1), arriving at acomplete agreement with the hadron resonance gas model at low temperatures. We also improvedthe precision on our line of constant physics (LCP, see ref. [1]), and used two different methods toset the scale (based on the w0 scale [13] or on fk) in order to control the systematical error relatedto scale setting.These two different scale setting procedures entered into our ’histogram’ method [14] used toestimate systematical errors, along with a range of other fit methods, each of which is an in prin-ciple completely valid approach. We then calculated the goodness of fit Q and weights based onthe Akaike information criterion AICc [15, 16] and looked at the unweighted or weighted (based1Since this transition is a cross-over [2], this use of the term ’phase’ is somewhat abusive, and indicates only thedominant degrees of freedom.2EoS of QCD Stefan Krieg200 300 400 500T[MeV]012345(ε-3p)/T4continuum limitNt4 stout crosscheck=8   10   12   160 1 2 3 4 5 6log(mq/mphys)-0.200.20.40.60.8-Nt4� lnZ� ln(ma)Nt=12 Nt=16lightstrangeFigure 1: Left: The trace anomaly as a function of the temperature. The continuum extrapolated resultwith total errors is given by the shaded band. Also shown is a cross-check point computed in the continuumlimit with our new and different lattice action at T = 214 MeV, indicated by a smaller filled red point, whichserves as a crosscheck on the peak’s hight. Right: Setting the overall scale of the pressure: integration fromthe infinitely large mass region down to the physical point using a range of dedicated ensembles and timeextents up to Nt = 16; the sum of the areas under the curves gives p/T 4.on Q or AICc) distribution of the results. The median is the central value, whereas the centralregion containing 68% of all the possible methods gives an estimate on the systematic uncertain-ties. This procedure provides very conservative errors. Here, we had four basic types of continuumextrapolation methods (with or without tree level improvement for the pressure and with a2 aloneor a2 and a4 discretization effects) and two continuum extrapolation ranges (including or excludingthe coarsest lattice Nt=6 in the analysis). We used seven ways to determine the subtraction termat T=0 (subtracting directly at the same gauge coupling β or interpolating between the β valueswith various orders of interpolation functions), and the aforementioned two scale procedures. Fi-nally, we had eight options to determine the final trace anomaly by choosing among various splinefunctions, giving altogether 4·2·7·2·8=896 methods. Note that using either an AICc or Q baseddistribution changed the result only by a tiny fraction of the systematic uncertainty. Furthermore,the unweighted distribution always delivered consistent results within systematical errors.The systematic error procedure clearly demonstrates the robustness of our final result. Even inthe case of applying or not applying tree level improvement, where the data points at finite latticespacing change considerably, the agreement between the continuum extrapolated results, and hencethe contribution to the systematic error, is on the few percent level.The continuum extrapolated trace anomaly is shown in Figure 1.2. Update on the N f = 2+1+1 Equation of StateSo far, the equation of state is known only in 2+1 flavor QCD. The contribution from the seacharm quarks most likely matter at least for T > 300−400 MeV (for an illustration, see Figure 2).The N f = 2+1 lattice results of the previous section agree with the HRG at low temperaturesand are correct for the small to medium temperatures, and, as is shown in Figure 2, at temperatures3EoS of QCD Stefan Krieg100 200 400 600 800 1000T / MeV0246p / T4resonance gasanalytic recipeinterpolationNf = 4Nf = 3Figure 2: Left: Laine and Schroeder’s perturbative estimate of the effect of the charm in the QCD equationof state [17]. Right: Wuppertal-Budapest [7] and perturbative (up to O(g5)) results for the equation of state. 0 1 2 3 4 5 6 200  400  600  800  1000  1200T [MeV](ε-3p)/T4 HTL Nf=4continuum limit Nf=2+1Nf=2+1+1 @ 214 MeVNf=2+1+1 Nt=6Nf=2+1+1 Nt=8Nf=2+1+1 Nt=10HRG 0 1 2 3 4 5 6 200  400  600  800  1000  1200T [MeV]p/T4lattice continuum limitHRGNf=2+1+1 flavor Nt=6Nf=2+1+1 flavor Nt=8Nf=2+1+1 flavor Nt=10Figure 3: Left: Preliminary results for the charmed EoS. For comparison, we show the HRG result, theN f = 2+ 1 band, and, at high Temperatures, the HTL result [18], where the central line marks the HTLexpectation for the EoS with the band resulting from (large) variations of the renormalization scale. Right:Preliminary result for the pressure, errors indicate the Stefan-Boltzmann value. All errors are statistical only.of about 1 GeV perturbative results become sufficiently precise. Therefore, we need to calculate theEoS with a dynamical charm only for the remaining temperatures in the region of approximately300 MeV < T < 1000 MeV.We are using a new lattice action for these calculations. More precisely, where our N f = 2+1calculation used an action with 2 levels of stout gauge link averaging in the coupling of the fermionsto the gauge fields, we increased this to 4 levels with a smearing parameter of ρ = 0.125 (forfurther details see [1]). The crosscheck point shown in Figure 1 was computed using this newaction. Since it perfectly agrees with the N f = 2+1 results, even though it was computed using adynamical charm, we can be certain that at temperatures at and below T = 214 MeV, we can relyon the N f = 2+1 results.4EoS of QCD Stefan KriegOur preliminary results are shown in Figure 3, all errors are statistical only. Our results spana region of temperatures from T = 214 MeV up to T = 1.2 GeV. At the low end we make contactto the N f = 2+1 equation of state, and at large temperatures to the HTL result. Thereby, we coverthe full region of temperatures, from low temperatures, where the HRG gives reliable results, tohigh temperatures, where we make contact with perturbation theory. Our present set of data pointswill be extended to Nt = 12, in order to make a controlled continuum extrapolation possible.2.1 Line of constant physicsWith the switch to a new lattice action comes the need to (re-) compute the LCP. Since wewould like to span the temperature range from approximately 300 MeV < T < 1000 MeV, we haveto compute the LCP for a large range of couplings or lattice spacings. We split this range up inthree overlapping (since we have to make sure that the derivative is smooth) regions according tothe applicable simulation strategies.At medium to coarse lattice spacings (region I) one can afford to use spectroscopy to tunethe parameters. This is shown in Figure 4. Here, we bracketed the physical point defined throughMpi/ fpi and (2MK−Mpi)/ fpi and, through interpolation, tune the light and strange quark masses toper-mill precision.Using the parameters computed in this way, we then performed simulations in the SU(3) flavor-symmetrical point [19], extrapolating the results to our target couplings. There, we tuned theparameters to reproduce the extrapolated results. Since the quark masses are larger than physical,such simulations are considerably less costly than using spectroscopy as for region I, and we arethus able to compute a precise LCP down to fine lattice spacings of a= 0.05 fm (region II), wherethe HMC starts being affected by the freezing of topology.For finer lattice spacings we thus used our established step scaling procedure [1] based on thew0 scale. To this end, we computed the observableO = tddt[t2E(t)]∣∣∣∣0.01L2at three different lattice spacings (a0, a1, a2) and volumes (164, 204, 244) chosen to keep the phys-ical volume fixed, extrapolated to a3 = 24/32a2, and tuned the coupling to match the extrapolatedresult. Using this method, we extended the LCP to very fine lattice spacings with a < 0.05 fm(region III).3. ConclusionsThe precision of Lattice QCD results at finite temperature has increased significantly over thelast years. We have discussed a full result (all sources of uncertainties controlled) for the N f = 2+1EoS by the Budapest-Wuppertal collaboration [1], shown how to include a dynamical charm quarkfor the N f = 2+1+1 EoS and presented preliminary results.AcknowledgmentsComputations were performed on JUQUEEN at Forschungszentrum Jülich, HERMIT at HLRS,Stuttgart, and on the QPACE machine and on GPU clusters [20] at University of Wuppertal. We5EoS of QCD Stefan Krieg1.04 1.05 1.06 1.07 1.08 1.09 1.126.82727.227.427.627.82828.22 2Mpi  / fpi22 ]  /  f pi2 −Mp i[ 2 Mk abeta=3.55, data for i=2, j=2, k=1, l=2 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 3.5  3.55  3.6  3.65  3.7  3.75  3.8  3.85  3.9  3.95a [fm]βFigure 4: Region of the LCP, for coarse to medium lattice spacing (a > 0.08 fm). Here, dedicated sim-ulations bracketing the physical point archive a sub-percent accuracy for the LCP. Left: Bracketing of thephysical point defined through Mpi/ fpi and (2MK −Mpi)/ fpi . The strange quark mass is tuned (ms/ml is notfixed) and the ratio of the charm to strange quark mass is set at mc/ms = 11.85. Right: LCP computedthrough spectroscopy.Figure 5: Using the LCP computed from spectroscopy for coarse to medium lattice spacings (regionI), dedicated simulations in the SU(3) flavor-symmetrical point [19] using these parameters are extrapolatedtowards the continuum. At the target coupling, the parameters are tuned until they reproduce the extrapolatedvalue. In this way the LCP is extended to medium to small lattice spacings of 0.08 > a > 0.05 fm (regionII).acknowledge PRACE for awarding us resources on JUQUEEN at Forschungszentrum Jülich. Thiswork was partially supported by the DFG Grant SFB/TRR 55 and ERC no. 208740.References[1] S. Borsanyi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, et al., “Full result for the QCD equation ofstate with 2+1 flavors,” Phys.Lett. B730 (2014) 99–104, arXiv:1309.5258 [hep-lat].[2] Y. Aoki, G. Endrodi, Z. Fodor, S. Katz, and K. Szabo, “The Order of the quantum chromodynamics6EoS of QCD Stefan Kriegtransition predicted by the standard model of particle physics,” Nature 443 (2006) 675–678,arXiv:hep-lat/0611014 [hep-lat].[3] Z. Fodor and S. Katz, “The Phase diagram of quantum chromodynamics,” arXiv:0908.3341[hep-ph].[4] O. Philipsen, “The QCD equation of state from the lattice,” Prog.Part.Nucl.Phys. 70 (2013) 55–107,arXiv:1207.5999 [hep-lat].[5] P. Petreczky, “Review of recent highlights in lattice calculations at finite temperature and finitedensity,” PoS ConfinementX (2012) 028, arXiv:1301.6188 [hep-lat].[6] C. Hoelbling, “Lattice QCD: concepts, techniques and some results,” arXiv:1410.3403[hep-lat].[7] S. Borsanyi, G. Endrodi, Z. Fodor, A. Jakovac, S. D. Katz, et al., “The QCD equation of state withdynamical quarks,” JHEP 1011 (2010) 077, arXiv:1007.2580 [hep-lat].[8] The results are available as an ancillary file to [1], where they are freely downloadable. The filecontains tabulated data for the trace anomaly, pressure, entropy, energy density, and speed of soundfor our temperature range.[9] Y. Aoki, Z. Fodor, S. Katz, and K. Szabo, “The Equation of state in lattice QCD: With physical quarkmasses towards the continuum limit,” JHEP 0601 (2006) 089, arXiv:hep-lat/0510084[hep-lat].[10] Y. Aoki, Z. Fodor, S. Katz, and K. Szabo, “The QCD transition temperature: Results with physicalmasses in the continuum limit,” Phys.Lett. B643 (2006) 46–54, arXiv:hep-lat/0609068[hep-lat].[11] Y. Aoki, S. Borsanyi, S. Durr, Z. Fodor, S. D. Katz, et al., “The QCD transition temperature: resultswith physical masses in the continuum limit II.,” JHEP 0906 (2009) 088, arXiv:0903.4155[hep-lat].[12] S. Borsanyi et al., “Is there still any Tc mystery in lattice QCD? Results with physical masses in thecontinuum limit III,” JHEP 1009 (2010) 073, arXiv:1005.3508 [hep-lat].[13] S. Borsanyi, S. Durr, Z. Fodor, C. Hoelbling, S. D. Katz, et al., “High-precision scale setting in latticeQCD,” JHEP 1209 (2012) 010, arXiv:1203.4469 [hep-lat].[14] S. Durr, Z. Fodor, J. Frison, C. Hoelbling, R. Hoffmann, et al., “Ab-Initio Determination of LightHadron Masses,” Science 322 (2008) 1224–1227, arXiv:0906.3599 [hep-lat].[15] H. Akaike IEEE Transactions on Automatic Control 19 (1974) 716–723.[16] C. Hurvich and C.-L. Tsai Biometrica 76 (1989) 297–307.[17] M. Laine and Y. Schroder, “Quark mass thresholds in QCD thermodynamics,” Phys.Rev. D73 (2006)085009, arXiv:hep-ph/0603048 [hep-ph].[18] J. O. Andersen, L. E. Leganger, M. Strickland, and N. Su, “Three-loop HTL QCD thermodynamics,”JHEP 1108 (2011) 053, arXiv:1103.2528 [hep-ph].[19] W. Bietenholz, V. Bornyakov, M. Gockeler, R. Horsley, W. Lockhart, et al., “Flavour blindness andpatterns of flavour symmetry breaking in lattice simulations of up, down and strange quarks,”Phys.Rev. D84 (2011) 054509, arXiv:1102.5300 [hep-lat].[20] G. I. Egri, Z. Fodor, C. Hoelbling, S. D. Katz, D. Nogradi, et al., “Lattice QCD as a video game,”Comput.Phys.Commun. 177 (2007) 631–639, arXiv:hep-lat/0611022 [hep-lat].7

Katz, Sandor D

Szabo, Kalman K

ELTE Digital Institutional Repository (EDIT)

Recent results on the Equation of State of QCDSzabolcs Borsányia, Zoltán Fodora,b,c, Christian Hoelblinga, Sándor D. Katzc,d ,Stefan Krieg∗,a,b, Claudia Rattie, Kálmán K. Szabóa,baDepartment of Physics, University of Wuppertal,Gaußstr. 20, D-42119, GermanybForschungszentrum Jülich, Jülich, D-52425, GermanycInstitute for Theoretical Physics, Eötvös University, Pázmány 1, H-1117 Budapest, HungarydMTA-ELTE Lendület Lattice Gauge Theory Research GroupeUniversità di Torino and INFN, Sezione di Torino via Giuria 1, I-10125 Torino, ItalyWe report on a continuum extrapolated result [1] for the equation of state (EoS) of QCD withN f = 2+ 1 dynamical quark flavors and discuss preliminary results obtained with an additionaldynamical charm quark (N f = 2+1+1). For all our final results, the systematics are controlled,quark masses are set to their physical values, and the continuum limit is taken using at least threelattice spacings corresponding to temporal extents up to Nt = 16.The 32nd International Symposium on Lattice Field Theory,23-28 June, 2014Columbia University New York, NY∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/arXiv:1410.7917v1  [hep-lat]  29 Oct 2014EoS of QCD Stefan Krieg1. Full result for the N f = 2+1 equation of stateThe rapid transition from the quark-gluon-plasma ’phase’1 to the hadronic phase in the earlyuniverse and the QCD phase diagram are subjects of intense study in present heavy-ion experiments(LHC@CERN, RHIC@BNL, and the upcoming FAIR@GSI). This transition can be studied in asystematic way in Lattice QCD (for recent reviews see, e.g., [3, 4, 5, 6]). The equation of state(EoS) of QCD, (i.e, the pressure p, energy density ε , trace anomaly I = ε −3p, entropy s = (ε +p)/T , and the speed of sound c2s = d p/dε as functions of the temperature) has been determined byseveral lattice groups, however, a full result was, until recently, still lacking. Ref. [7] constitutes afull result at three characteristic temperatures, which we have now extended to the full temperaturerange in [1] and made available electronically [8].Our calculation is based on a tree-level Symanzik improved gauge action with 2-step stout-link improved staggered fermions. The precise definition of the action can be found in ref. [9], itsadvantageous scaling properties are studied in ref. [10, 11, 12]. In particular, while it approachesthe continuum value of the Stefan-Boltzmann limit in the infinite temperature limit T → ∞ slowerthan actions with p4 or Naik terms (the latter is an additional fermionic term in the asqtad and hisqactions), it behaves monotonous and reaches the asymptotic a2 behavior quite early. Extrapolationsfrom moderate temporal extents, e.g., using Nt ≥ 8, allow for a smooth continuum extrapolationand provide an accuracy on the percent level, the typical accuracy one aims to reach. Additionally,applying simple tree-level improvement factors for the bulk thermodynamic observables brings theindividual data points for the different Nt very close to the continuum limit.This points to an important advancement of the calculation described here over our previousresults of [7]: we now include a large range of Nt = 12 data points and one Nt = 16 data pointlocated at the peak position. Previously, we only had Nt = 12 results at three characteristic tem-perature values available. The T → 0 limit is difficult due to taste-breaking effects, but is crucialsince the renormalization is done at zero temperature, i.e. p(T =0)=0. A mismatch at T=0 leads toa shift in the whole EoS. Previously, we calculated the difference in the pressure between the phys-ical theory and its counterpart with 720 MeV heavy pions at a selected temperature (100 MeV),where the latter theory has practically zero pressure, and we, therefore, get p(T = 100 MeV) inthe physical theory with the desired normalization. The difference of this result and the predictionby the Hadron Resonance Gas model (HRG) was then included in the systematical error. With ourincreased range of temporal extents, we now can use five lattice spacings to fix the additive term inthe pressure, integrating in the quark mass down to the physical values (see Figure 1), arriving at acomplete agreement with the hadron resonance gas model at low temperatures. We also improvedthe precision on our line of constant physics (LCP, see ref. [1]), and used two different methods toset the scale (based on the w0 scale [13] or on fk) in order to control the systematical error relatedto scale setting.These two different scale setting procedures entered into our ’histogram’ method [14] used toestimate systematical errors, along with a range of other fit methods, each of which is an in prin-ciple completely valid approach. We then calculated the goodness of fit Q and weights based onthe Akaike information criterion AICc [15, 16] and looked at the unweighted or weighted (based1Since this transition is a cross-over [2], this use of the term ’phase’ is somewhat abusive, and indicates only thedominant degrees of freedom.2EoS of QCD Stefan Krieg200 300 400 500T[MeV]012345(ε-3p)/T4continuum limitNt4 stout crosscheck=8   10   12   160 1 2 3 4 5 6log(mq/mphys)-0.200.20.40.60.8-Nt4� lnZ� ln(ma)Nt=12 Nt=16lightstrangeFigure 1: Left: The trace anomaly as a function of the temperature. The continuum extrapolated resultwith total errors is given by the shaded band. Also shown is a cross-check point computed in the continuumlimit with our new and different lattice action at T = 214 MeV, indicated by a smaller filled red point, whichserves as a crosscheck on the peak’s hight. Right: Setting the overall scale of the pressure: integration fromthe infinitely large mass region down to the physical point using a range of dedicated ensembles and timeextents up to Nt = 16; the sum of the areas under the curves gives p/T 4.on Q or AICc) distribution of the results. The median is the central value, whereas the centralregion containing 68% of all the possible methods gives an estimate on the systematic uncertain-ties. This procedure provides very conservative errors. Here, we had four basic types of continuumextrapolation methods (with or without tree level improvement for the pressure and with a2 aloneor a2 and a4 discretization effects) and two continuum extrapolation ranges (including or excludingthe coarsest lattice Nt=6 in the analysis). We used seven ways to determine the subtraction termat T=0 (subtracting directly at the same gauge coupling β or interpolating between the β valueswith various orders of interpolation functions), and the aforementioned two scale procedures. Fi-nally, we had eight options to determine the final trace anomaly by choosing among various splinefunctions, giving altogether 4·2·7·2·8=896 methods. Note that using either an AICc or Q baseddistribution changed the result only by a tiny fraction of the systematic uncertainty. Furthermore,the unweighted distribution always delivered consistent results within systematical errors.The systematic error procedure clearly demonstrates the robustness of our final result. Even inthe case of applying or not applying tree level improvement, where the data points at finite latticespacing change considerably, the agreement between the continuum extrapolated results, and hencethe contribution to the systematic error, is on the few percent level.The continuum extrapolated trace anomaly is shown in Figure 1.2. Update on the N f = 2+1+1 Equation of StateSo far, the equation of state is known only in 2+1 flavor QCD. The contribution from the seacharm quarks most likely matter at least for T > 300−400 MeV (for an illustration, see Figure 2).The N f = 2+1 lattice results of the previous section agree with the HRG at low temperaturesand are correct for the small to medium temperatures, and, as is shown in Figure 2, at temperatures3EoS of QCD Stefan Krieg100 200 400 600 800 1000T / MeV0246p / T4resonance gasanalytic recipeinterpolationNf = 4Nf = 3Figure 2: Left: Laine and Schroeder’s perturbative estimate of the effect of the charm in the QCD equationof state [17]. Right: Wuppertal-Budapest [7] and perturbative (up to O(g5)) results for the equation of state. 0 1 2 3 4 5 6 200  400  600  800  1000  1200T [MeV](ε-3p)/T4 HTL Nf=4continuum limit Nf=2+1Nf=2+1+1 @ 214 MeVNf=2+1+1 Nt=6Nf=2+1+1 Nt=8Nf=2+1+1 Nt=10HRG 0 1 2 3 4 5 6 200  400  600  800  1000  1200T [MeV]p/T4lattice continuum limitHRGNf=2+1+1 flavor Nt=6Nf=2+1+1 flavor Nt=8Nf=2+1+1 flavor Nt=10Figure 3: Left: Preliminary results for the charmed EoS. For comparison, we show the HRG result, theN f = 2+ 1 band, and, at high Temperatures, the HTL result [18], where the central line marks the HTLexpectation for the EoS with the band resulting from (large) variations of the renormalization scale. Right:Preliminary result for the pressure, errors indicate the Stefan-Boltzmann value. All errors are statistical only.of about 1 GeV perturbative results become sufficiently precise. Therefore, we need to calculate theEoS with a dynamical charm only for the remaining temperatures in the region of approximately300 MeV < T < 1000 MeV.We are using a new lattice action for these calculations. More precisely, where our N f = 2+1calculation used an action with 2 levels of stout gauge link averaging in the coupling of the fermionsto the gauge fields, we increased this to 4 levels with a smearing parameter of ρ = 0.125 (forfurther details see [1]). The crosscheck point shown in Figure 1 was computed using this newaction. Since it perfectly agrees with the N f = 2+1 results, even though it was computed using adynamical charm, we can be certain that at temperatures at and below T = 214 MeV, we can relyon the N f = 2+1 results.4EoS of QCD Stefan KriegOur preliminary results are shown in Figure 3, all errors are statistical only. Our results spana region of temperatures from T = 214 MeV up to T = 1.2 GeV. At the low end we make contactto the N f = 2+1 equation of state, and at large temperatures to the HTL result. Thereby, we coverthe full region of temperatures, from low temperatures, where the HRG gives reliable results, tohigh temperatures, where we make contact with perturbation theory. Our present set of data pointswill be extended to Nt = 12, in order to make a controlled continuum extrapolation possible.2.1 Line of constant physicsWith the switch to a new lattice action comes the need to (re-) compute the LCP. Since wewould like to span the temperature range from approximately 300 MeV < T < 1000 MeV, we haveto compute the LCP for a large range of couplings or lattice spacings. We split this range up inthree overlapping (since we have to make sure that the derivative is smooth) regions according tothe applicable simulation strategies.At medium to coarse lattice spacings (region I) one can afford to use spectroscopy to tunethe parameters. This is shown in Figure 4. Here, we bracketed the physical point defined throughMπ/ fπ and (2MK−Mπ)/ fπ and, through interpolation, tune the light and strange quark masses toper-mill precision.Using the parameters computed in this way, we then performed simulations in the SU(3) flavor-symmetrical point [19], extrapolating the results to our target couplings. There, we tuned theparameters to reproduce the extrapolated results. Since the quark masses are larger than physical,such simulations are considerably less costly than using spectroscopy as for region I, and we arethus able to compute a precise LCP down to fine lattice spacings of a = 0.05 fm (region II), wherethe HMC starts being affected by the freezing of topology.For finer lattice spacings we thus used our established step scaling procedure [1] based on thew0 scale. To this end, we computed the observableO = tddt[t2E(t)]∣∣∣∣0.01L2at three different lattice spacings (a0, a1, a2) and volumes (164, 204, 244) chosen to keep the phys-ical volume fixed, extrapolated to a3 = 24/32a2, and tuned the coupling to match the extrapolatedresult. Using this method, we extended the LCP to very fine lattice spacings with a < 0.05 fm(region III).3. ConclusionsThe precision of Lattice QCD results at finite temperature has increased significantly over thelast years. We have discussed a full result (all sources of uncertainties controlled) for the N f = 2+1EoS by the Budapest-Wuppertal collaboration [1], shown how to include a dynamical charm quarkfor the N f = 2+1+1 EoS and presented preliminary results.AcknowledgmentsComputations were performed on JUQUEEN at Forschungszentrum Jülich, HERMIT at HLRS,Stuttgart, and on the QPACE machine and on GPU clusters [20] at University of Wuppertal. We5EoS of QCD Stefan Krieg1.04 1.05 1.06 1.07 1.08 1.09 1.126.82727.227.427.627.82828.22 2Mpi  / fpi22 ] / fpi2 −Mpi[2Mkabeta=3.55, data for i=2, j=2, k=1, l=2 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 3.5  3.55  3.6  3.65  3.7  3.75  3.8  3.85  3.9  3.95a [fm]βFigure 4: Region of the LCP, for coarse to medium lattice spacing (a > 0.08 fm). Here, dedicated sim-ulations bracketing the physical point archive a sub-percent accuracy for the LCP. Left: Bracketing of thephysical point defined through Mπ/ fπ and (2MK −Mπ)/ fπ . The strange quark mass is tuned (ms/ml is notfixed) and the ratio of the charm to strange quark mass is set at mc/ms = 11.85. Right: LCP computedthrough spectroscopy.Figure 5: Using the LCP computed from spectroscopy for coarse to medium lattice spacings (regionI), dedicated simulations in the SU(3) flavor-symmetrical point [19] using these parameters are extrapolatedtowards the continuum. At the target coupling, the parameters are tuned until they reproduce the extrapolatedvalue. In this way the LCP is extended to medium to small lattice spacings of 0.08 > a > 0.05 fm (regionII).acknowledge PRACE for awarding us resources on JUQUEEN at Forschungszentrum Jülich. Thiswork was partially supported by the DFG Grant SFB/TRR 55 and ERC no. 208740.References[1] S. Borsanyi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, et al., “Full result for the QCD equation ofstate with 2+1 flavors,” Phys.Lett. B730 (2014) 99–104, arXiv:1309.5258 [hep-lat].[2] Y. Aoki, G. Endrodi, Z. Fodor, S. Katz, and K. Szabo, “The Order of the quantum chromodynamics6EoS of QCD Stefan Kriegtransition predicted by the standard model of particle physics,” Nature 443 (2006) 675–678,arXiv:hep-lat/0611014 [hep-lat].[3] Z. Fodor and S. Katz, “The Phase diagram of quantum chromodynamics,” arXiv:0908.3341[hep-ph].[4] O. 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Recent results on the Equation of State of QCD

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