Lecture 5-6: Chiral and Deconfinement Transition in QCD and the Properties of Matter at Low and High Temperatures

Lecture 5-6: Chiral and deconfinement transition in QCD and the properties of matter at low and high temperatures

Effective field theory approach

Effective theories at low temperatures (chiral perturbation theory)

Chiral transition and axial symmetry restoration

Center symmetry, deconfinement transition, color screening

High temperature QCD: perturbation theory, magnetic screening, dimensional reduction, hard-thermal loop effective theory

Virial expansion and hadron resonance (HRG) model

Taylor expansion of the pressure and fluctuations of conserved charges

Effective field theories Try to describe the physics of low momentum modes ( p < Λ ) in terms of the relevant dof only:

= 0(k<⇤)+˜(k>⇤)

eff S[;g ] S[ ,˜;g ] Seff [0;g ] Z = e i = ˜e 0 i = e j D D 0 D D 0 Z Z Z Z S [ ;geff ] e eff 0 j How to calculate the effective action ?

1) Perform the function integral in perturbation theory (partial path integration) difficult ! 2) Matching: write down the most general form of the effective Lagrangian as sum of eff d 4 different operators g j ( ⇤ ; g i ) O j / ⇤ consistent with the symmetries of the original theory

(n) (n) eff G (k = ⇤; gi)=Geff (k = ⇤; gj ) matching condition

eff eff gj = gj (⇤; gi) Linear sigma model and chiral perturbation theory

Goal to describe physics at scale < pion mass << ΛQCD (for sufficiently small quark mass) taking into account spontaneous breaking of chiral symmetry Linear sigma model 1 1 L = (@ )2 + (@ ~⇡ )2 + (2 + ~⇡ 2 v)2 + h 2 µ 2 µ 24 Invariant under chiral rotations for h~m =0: +~b ~⇡ , ~⇡ ⇡ ~b q ! · ! ¯ , ~⇡ ¯i~⌧ , = ~b ~⌧ /2 $ $ 5 · 5 Useful for the discussion at T>0 and restoration of the chiral symmetry

Problem: mσ ≈ 400 MeV ~ ΛQCD no well-defined power counting in a small parameter Only two parameters λ and v, cannot describe the experimental data => additional operators ?? Chiral perturbation theory (non-linear signa model ) : U =exp(i~⌧ ~⇡ /F ), Use only pion fields and chiral symmetry · U G† UG ,G SU(2) ! R L R,L ⇢ R,L 2 2 2 Expansions in powers p /F , (mπ/F) , F ~ ΛQCD (2) (4) (6) Gasser, Leutwyler, ‘84 Leff = L + L + L + .... (2) 1 2 2 2 2 L = F Tr @ U@U † M (U + U †) ,M=(m + m )B m 4 µ µ u d ' ⇡ Kinetic term is invariant under the chiral rotation and the mass term transforms like the quark mass term in the QCD Lagrangian TrU + h.c. Tr(G G† U)+h.c. ! L R At leading order in 1/F2 expansion we recover the Lagrangian for massive pions, but interactions are proportional to p2 ( expand U in 1/F ) but operators with d>4 => the effective theory is non-renormalizable, but no IR divergences ~p2/F 2 Next-to-leading ~p4/F4 order:

(4) 1 2 1 L = l Tr@ U †@ U l Tr(@ U †@ U)Tr(@ U †@ U)+ 1 4 µ µ 4 2 µ ⌫ µ ⌫ 1 2 1 4 2 l M Tr @ U †@ U Tr(U + U †) (l + l )M Tr(U + U †) 8 4 µ µ 16 3 4 P. Gerber, H. Leutwyler / Hadrons below the chiral phase transition 391 ⇡2T 4 T 4 ⇤ p(T )= 1+ ln p , [] 30 36F 4 T ✓ ◆

Gerber,2 Leutwyler4a , ‘894b be bb bc ⇤ 270 MeV p ' for massless pions Hadronic interatcions are weak at low T, T

Using this relation, one obtains

z6a = 3M2(G )2/8F2 (2.6)

Zsa = - 25MZ(G1)3/48F 4. (2.7)

The three-loop graph 8b involves an integral over the torus T defined by -ill2 <~ X 4 ~ j~/2, c2 = fTddx It(x)] 2. (2.8)

This integral can be expressed in terms of the derivative of the propagator with respect to the mass dG 1 G 2 dM 2 . (2.9)

In terms of this quantity, we have

z,b = M2( G1)2(8Ga - 3MZG2)/16F 4. (2.10) The chiral condensate and chiral susceptibility 2 T @ ln Z @ ¯ T @ ln Z ¯ m,q = = = disc + conn = = @m h i V @m2 h i V @mq q q 2 2 2 Need renormalization < ( ¯ ) > < ¯ > TrD q

R 0.025 l HISQ, ml=ms/20 0.02 HISQ, ml=ms/27 stout, ml=ms/27 0.015

0.01

0.005

-0.005 T [MeV] 120 130 140 150 160 170 180 190 200 210

HotQCD : Phys. Rev. D85 (2012) 054503; Bazavov, PP, PRD 87(2013) 094505

How to define the chiral transition temperature ? O(N) scaling and the chiral transition temperature

For sufficiently small ml and in the vicinity of the transition temperature:

governed by universal O(4) scaling

0 Tc is critical temperature in the mass-less limit, h0 and t0 are scale parameters Pseudo-critical temperatures for non-zero quark mass are defined as peaks in the response functions ( susceptibilities) :

0 = = = Tc in the zero quark mass limit

universal scaling function has a peak at z=zp Caveat : staggered fermions O(2) ml →0, a > 0, proper limit a →0, before ml → 0 O(N) scaling and the transition temperature The notion of the transition temperature is only useful if it can be related to the critical temperature in the chiral limit : fit the lattice data on the chiral condensate with scaling form + simple Ansatz for the regular part

0 6 parameter fit : Tc , t0, h0, a1, a2, b1 Domain wall Fermions and UA(1) symmetry restoration Domain Wall Fermions, Bazavov et al (HotQCD), PRD86 (2012) 094503

chiral: axial:

DWF DSDR 20 Asqtad Nt=8 Asqtad Nt=12 HISQ Nt=6 HISQ N =8 15 t

2 HISQ Nt=12 /T

l, disc 10 MS

0 130 140 150 160 170 180 190 200 210 220 T (MeV) Peak position roughly agrees with previous staggered results axial symmetry is effectively restored T>200 MeV ! Center symmetry and deconfinement transition

Above the phase transition temperature Z(3) (center) symmetry of SU(3) gauge theory is broken Quarks transform non-trivially under Z(3) symmetry group => static charges in fundamental representations can be screened by gluons !

Lattice set-up:

The free energy of static quark is infinite in the Continuum limit due to linear 1/a divergence => needs renormalization How to determine the deconfinement transition temperature ?

Boyd et al., Nucl. Phys. B496 (1996) 167 Necco, Nucl. Phys. B683 (2004) 167

• Use different volumes and Ferrenberg-Swedsen re-weighting to combine information collected at different gauge couplings Finite volume behavior can tell the order of the phase transition, e.g. for 1st order transition the peak height scales as spatial volume ! Continuum limit for L ? Dumitru et al, hep-th/0311223

needs renormalization ! Free energy of static quark anti-quark pair and other correlators

McLerran, Svetitsky, PRD 24 (81) 450 analog of the Wilson loop at T=0 F (r,T )/T 1 8 F (r,T )/T e = G (r, T )+ G (r, T ),G (r, T )=e 1,8 9 1 9 8 1,8 2 G(r ,T) L(r)L†(0) r = L = G1(r ,T) !1 ⌘h i !1 |h i| !1 F (r ,T)=F (T )=2FQ !1 1 At short distance the quark anti- quark interactions should not depend on T: F (r, T )+T ln 9 =

= F1(r, T )=V (r)+C, rT 1 ⌧ ⇡ V (r)= + r 12r

Kaczmarek, Karsch, P.P., Zantow, hep-lat/0309121

Jahn, Philipsen, PRD 70 (04) 0074504 The renormalized Polyakov loop in pure glue theory

Kaczmarek et al, PLB 543 (02) 41, PRD 70 (04) 074505, hep-lat/0309121 Correlation length near the transition

Kaczmarek, Phys.Rev.D62 (00) 034021

small inverse correlation length => weak 1st order phase transition QCD is far from the large N-limit ! Singlet free energy and Polyakov loop in 2+1 flavor QCD

Pure glue ≠ QCD ! The free energy of static quark anti-quark pair is screened already Deconfinement transition happens at low temperature (even at T=0) => at lower temperature but the string breaking Polyakov loop behaves smoothly around Tc , Z(3) symmetry plays apparent role Polyakov and gas of static-light hadrons

Energies of static-light mesons:

Free energy of an isolated static quark:

Megias, Arriola, Salcedo, 600 PRL 109 (12) 151601 F (T) [MeV] HISQ Q stout Bazavov, PP, PRD 87 (2013) 094505 500

400 Ground state and first excited states are from lattice QCD Michael, Shindler, Wagner, 300 arXiv1004.4235 Wagner, Wiese, JHEP 1107 016,2011 200 Higher excited state energies are estimated from potential model T [MeV] 100 Gas of static-light mesons 120 140 160 180 200 only works for T < 145 MeV Gulon self energy and color screening in perturbation theory QCD at high temperatures

Arnold, Zhai, Phys.Rev. D51 (1995) 1906, Kastening, Zhai, Phys.Rev. D52 (1995) 7232

Bosonic contribution:

Static resummation: 1 m2 A2 2 D 0 !n,0

Fermionic contribution: Arnold, Zhai, Phys.Rev. D51 (1995) 1906, Kastening, Zhai, Phys.Rev. D52 (1995) 7232 Kajantie et al, Phys. Rev. D 67 (2004) 105008

very poor convergence ! Pressure at order g6 and magnetic mass Dimensional reduction at high temperatures

F = D A D A µ⌫ µ µ ⌫ µ A 1/2A µ ! µ

EQCD

Integrate out A0 Braaten, Nieto, PRD 51 (95) 6990, PRD 53 (96) 3421 Kajantie et al, NPB 503 (97) 357, PRD 67 (03) 105008 3d YM theory MQCD 3d F = F(non-static) + T F 3d 6 F ~ g3 Spatial string tension at T>0 and dimensional reduction

EQCD :

non-perturbative Cheng et al., Phys.Rev.D78 (08) 034506

Laine, Schröder, JHEP0503(05) 067

magnetic screening Calculated perturbatively Pressure and screening mass in the 3d effective theory

Pressure in 3d theory Electric screening mass Kajantie et al, 2009 Cuchieri et al, 2001 2.8 2.0 3.0 2.6 2.5 2.4 Stefan-Boltzmann law 1.5 6 1 6 O(g ln ) + fitted O(g ) 2.2 2.0 -g

4 6 2 full EQCD + fitted O(g ) /T 4

E 1.5 1.0 4d lattice data m 1.8 p / T

Stefan-Boltzmann law 1.6(e - 3p) / T 1.0 6 6 O(g ln 1 ) + fitted O(g ) 0.5 -g 1.4 6 0.5 full EQCD + fitted O(g ) 1.2 4d lattice data 0.0 1 0.0 0 1 2 3 0 1 2 3 10 10 10 10 1 10 1010 10010 100010 10000 T / Tc T / Tc T/T 4 c Figure 6: Left: the best fit of our result to 4d lattice data for p/T at Nf =0[28].Ourresultcomes FIG. 16. The electric screening masses in unitsLine of the is the temper fit toature. the 4d Shown lattice areresults. the electric masses mE for the first (filled out as a function of T/Λ ;forconvertingtoT/Tc we scanned the interval Tc/Λ =1.10 ...1.35 circles) and the secondMS (open circles) set of h.ThelinerepresentthefitforthetemperaturedependenceofMS the electric mass (dark band). For comparison, with the light band we show theDifferent outcome for symbolsF R ≡ 0[29].Right:the correspond to MS from 4d simulations. The open triangles4 are4 the valuesdifferent of thechoiceselectric of the mass 3d for massh values obtained from perturbative reduction corresponding trace anomaly, (e − 3p)/T = T d(p/T )/dT . ∼ ∼ in the metastable region (the last column in Tableparameter III). Some data points at the temperature T 70Tc and T 9000Tc have been shifted in the temperature scale for better visibility. physical stripe of Fig. 1. Comparing with the lower order terms in Eq. (A.1), Eq. (4.2) also has a reasonable magnitude, and indicates that perturbationtheorywithinEQCDisinfacta As one can see from the figure the agreement between the masses obtained from 4d and 3d simulation is rather useful tool at all parameter values corresponding to the 4d theory. In principle, the numerical good.result could The also electric be compared mass shows with improved some dependence resummation meton hodsh.Forrelativelylowtemperatures( defined within EQCD T<50Tc)thebestagreement with(see, e.g., the refs. 4d [31]–[33]). data for the electric mass is obtained for values of h corresponding to 2-loop dimensional reduction and lyingAt the in same the metastable time, from the region. phenomenological This fact point motivated of view, ou ourrresultissomewhatofa choice of the parameters of the effective theory at T =2Tc in sectiondisappointment: III. For as higher shown in temperatures, Fig. 6, the match however, to 4d lattice practicall data in theynodistinctioncanbemadebetweenthethreechoicesof range T/Tc ≥ 3.0is h. notBefore particularly closing smooth. the In discussion fact, as shown on in the the choice figure, inclof theuding parameter the newly determinedsoftheeff≥ective5- 3d theory let us compare our procedure ofloop fixing remainder the decreases parameters the quality of the of the effective fit significantly theory.Thisimpliesthat,unfortunately, with that proposed in [9]. In Ref. [9] the gauge coupling was fixed bywe are matching not in a position the data to realize on Polyakov our original loop goal of correlators offering consolidated determine crosschecksdinlatticesimulationtothecorrespondingvalue for calculatedthe Nf $=0QCDpressureintheinterestingtemperaturerange in lattice perturabtion theory. The resultingT/Tc g∼auge1.5 ... couplings3.0. turned out to be considerably smaller than the 2 2 correspondingOn the other hand, ones our used study in raises our the analysis, theoretical e.g. question for T of=2 whatTc kindit gave of effgects=1 could.43 while our procedure gives g (2Tc)=2.89. Thebe responsible scalar couplings for the mismatch were betweenfixed according our results to and 1-loop that of perturba 4d latticetion simulations. theory On [9]. Using this procedure the authors of Ref. [9]the obtained mundane side, a good one possibility description would of be the a substantial Polyakov con looptribution correla fromtor the in condensate terms of the 3d effective theory, however, the spatial ˆ2 2 Wilsondenoted by loop∂xF whoseMS = &(Tr large [A0 distance]) 'a − ... behavior.Unfortunatelyitssystematicinclusionwould determines the spatial string tension was not well described in the reduced 3drequire theory. a significant The reason amount forof new this analytic is the and fact numerical that the work; value moreover, of the asPolyakov order-ofloop correlator is cut-off dependent and thereforemagnitude estimates it is not and very previous useful preliminary for extracting simulations thesuggest, renormali it appearszed coupling. unlikely that this condensate could significantly change the qualitative behaviour that we have observed. On a more adventurous note, let us point out that, qualitatively, the reason for the mismatch is that the condensate in Eq. (4.1) is too large,andgrowsrapidlyasy (or T )de- 0 3 creases (note that for Nf =0,therangeT/Tc =10...10 corresponds to y ( 0.3...1.2, or

10 [1] D.J. Gross, R.D. Pisarski and L.G. Yaffe, Rev. Mod. Phys. 53 (1981) 43 [2] F. Karsch, A. Patkós and P. Petreczky, Phys. Lett B401 (1997) 69 [3] J.O. Andersen, E. Braaten and M. Strickland, Phys. Rev. Lett. 83 (1999) 2139; Phys.Rev. D61(2000) 014017; Phys.Rev. D61(2000) 074016 [4] J.-P. Blaizot, E. Iancu and A. Rebhan, Phys. Rev. Lett. 83 (1999) 2906; Phys.Lett. B470 (1999) 181 [5] R. Kobes, G. Kunstatter and A. Rebhan, Phys. Rev. Lett. 64 (1990) 2992; Nucl. Phys. B355 (1991) 1 [6] A. Rebhan, Nucl. Phys. B430 (1994) 319 [7] P. Arnold and L.G. Yaffe, Phys. Rev. D52 (1995) 7208 [8] F. Karsch, M. Oevers and P. Petreczky, Phys. Lett. B442 (1998) 291 [9] P. Lacock, D.E. Miller and T. Reisz , Nucl. Phys. B369 (1992) 501 [10] L. Kärkkäinen, P. Lacock, B. Petersson and T.Reisz Nucl.Phys. B395 (1993) 733 [11] L. Kärkkäinen P. Lacock, D.E. Miller, B. Petersson andT.Reisz,Nucl.Phys.B418 (1994) 3 [12] K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B503 (1997) 357 [13] M. Laine and O. Philipsen, Phys. Lett. B459 (1999) 259 [14] A. Hart and O. Philipsen, Nucl.Phys. B572 (2000) 243

17 Pressure and trace anomaly in HTL perturbation theory

Perform resummation with LHTL

Andersen et al, 2011 Blaizot, Iancu, Rebhan, 2000 s/sSB 1 0.5 D

0.9 2 0.4 GeV 0.8 @

2 0.3 T ê 0.7 L P

3 0.2 NNLO HTLpt - 1 loop running

- NNLO HTLpt - 3 loop running RBC-Bielefeld asqtad 0.6 E H 0.1 hotQCD asqtad hotQCD p4 hotQCD HISQ 1.5 2 2.5 3 3.5 4 4.5 5 0.0 Wuppertal-Budapest 200 400 600 800 1000 T MeV 1 FIG. 12. Comparison of the HTL entropy (full lines), the NLA results for cΛ = 2 ...2 1 (dash-dotted lines) as well as cΛ = 4 ...4(graydash-dottedlines),andthefreeentropyofbosonsFIG. 3. NNLO HTLpt result for the Nf = 3 QCD trace anomaly over temperature squared with mass (5.13) such as to reproduce the correct perturbative plasmoncompared eff withect lattice (dotted data from lines), the Wuppertal-Budapest all H L [25], hotQCD [26, 27], and RBC-Bielefeld with MS renormalization scaleµ ¯ = πT...4πT ,withthelatticeresultofRef.[40]forpureSU(3)[28, 29] collaborations. For both HTLpt curves the renormalization scale was taken to be µ =2⇡T . gauge theory (dark-gray band). much lower temperatures. This discrepancy could be due to their fixed scale approach not having suciently large N at high temperatures as noted in their paper [30]. Puttingµ ¯ = cµ¯2πT in Eq. (5.14) and assuming cµ¯ 1prescribesreasonablysmallvalues⌧ ∼ 2 ˆ In Fig. 3 we present the Nf = 3 QCD NNLO HTLpt prediction for ( 3 )/T and for αs and thus form ˆ D/(2πT )andM/(πT )forallT>Tc so as to make it interesting E P to compare the above HTL and NLA expressions with nonperturbcompareative to resultslattice results from available lattice from the Wuppertal-Budapest [25] and hotQCD collaborations [26, 27]. The lattice data from the Wuppertal-Budapest collaboration uses the gauge theory. Indeed, we have found that, form ˆ D 2πT and Mˆ πT ,thedeviation from the free Stefan-Boltzmann result is small enough" to makeasemi-perturbativepicturestout action" and have been continuum estimated by averaging the trace anomaly measured minimally tenable, although it is clear that the physics of theusing phase their two transition smallest lattice itself spacings is corresponding to N⌧ =8andN⌧ =10.Thelattice data from the hotQCD collaboration are their N =8resultsusingtheasqtad,p4,and completely beyond reach. On the other hand, the strictly perturbative results up to and ⌧ HISQ actions which have not been continuum extrapolated [26, 27]. The lattice data from including the order g3 are such that entropy and pressure would be much higher than the RBC-Bielefeld collaboration is N⌧ =6andhavealsonotbeencontinuumextrapolated their Stefan-Boltzmann values, indicating a complete loss of convergence[28, 29]. of strict thermal perturbation theory. As before, we present HTLpt results using both one- and three-loop running of the In order to have some indication of the theoretical uncertainty involved, we consider, 1 7 again as done in Ref. [11], a variation of the renormalizationscalebyafactorofcµ¯ = 2 ...2. For purely gluonic QCD, the lattice results involve the leastuncertainties.InRef.[40],the thermodynamic potentials of pure SU(3) gauge theory have been calculated from plaquette 3 action densities on lattices up to 8 32 for temperatures up to about 4.5Tc and extrapolating to the continuum limit by comparing× different lattice sizes. The lattice result for the entropy density is rendered in Fig. 12 by a grey band whose thickness ismeanttogivearoughidea of the errors reported in Ref. [40]. Our result for the HTL entropy as displayed in Fig. 7 translates into a range of values bounded by the choicesµ ¯ = πT (lower full line) andµ ¯ =4πT (upper full line). This already gives a remarkably good approximation of the latticeresultforT > 2Tc,somewhat underestimating the values at higher temperatures. In all ofthistheparameterˆ∼ mD/T takes

49 Free energy of a static quark anti-quark pair at high T Static quark anti-quark free energy in 2+1f QCD

• The strong T-dependence for T<200 MeV is not necessarily related to color screening

• The free energy has much stronger T-dependence than the singlet free energy due to the octet contribution

• At high T the temperature dependence of the free energy can be entirely understood in terms of F1 and Casimir scaling F1=-8 F8 Static quark anti-quark free energy in 2+1f QCD (cont’d)

Leading order weak coupling :

Screening function: Virial expansion and Hadron Resonance Gas

Chiral perturbation theory is limited to pion gas. Other hadrons, resonances ? Relativistic virial expansion : compute thermodynamic quantities in terms as a gas of non-interacting particles and S – matrix Dashen, Ma, Bernstein, ‘69 Free gas interactions

µi /T µi /T ln Z =lnZ0 + e 1 e 2 b(i1,i2) i ,i X1 2 3 1 V d p (p2+E2)1/2/T 1 @S @S b(i ,i )= dEe AS(S S) 1 2 4⇡i (2⇡)3 @E @E Z Z finalX  Elastic scattering only (final state = initial state)

I Partial wave S(E)= 0(2l + 1)(2I + 1) exp(2il (E)) decomposition Xl,I perform the integral over the 3-momentum

I T 2 @ (E) 1 0 l b2 = 2⇡3 M dEE K2(E/T ) l,I (2l + 1)(2I + 1) @E invarian mass of the pair at threshold R P 728 R, Fenugopalan. M. Prakosh / Thermal properties

m [ l

2.0

s ° ao ~ .°

- ...-" (/T4 -

1.0

.i 0,0 ~-~~~* 100 150 200 T (uev)

Fig. 3. The pressure, energy density and entropy density of interacting pions (solid curves) all scaled to dimensionless units. The dashed lines are the corresponding results for a free gas of n- and p-mesons. 726 UseR. Venugopalan,experimental M. phasePrakash shifts/Thermal to determine properties b2 , Venugopalan, Prakash ‘92 4

3 w ~K - u) 2 °a,.q q~ "0 . ~°' //, - "0 1

0 6oa -- -1 -1 - I , I i I

30 60 i ; I ' I I ° I L 20 40 6 m 260~+460a+66~'+1/ k m v 20 60 a,°÷s6o ' ~ 0 i I I I .,~ .... 11 .,, I I - o 400 800 400 800 q~m(MeV) qom(uev) Fig. I. Upper panel: ~t~t phase shifts as a function of center of mass momentum. Lower panel: After summing all the channelsFig. only 4. resonanceUpper panel: contributionsnK phase shifts survives as a function in of center-of-mass momentum. Lower panel Isospin-weighted sum of the phase shifts. 0 isospin-weighted sum of the phase shifts. @I (E) (2l + 1)(2I + 1) l @E Xl,I with the correspondingInteracting variables for hadron a free gasgas of = pions non- andnteracting p-mesons. gas The of reason hadrons for and resonances this was the nearly exact cancellation of the contributions from the S-wave attractive and repulsive channels which resulted in an effective contribution from the P-wave channel containing the p-meson resonance. The magnitudes of the S-wave phase shifts are such that their isospin-weighted sum nearly vanishes as shown in the lower panel of fig. 1. This cancellation, of dynamic origin, may be explicitly verified 20) on the basis of the low-energy phase shifts from the Weinberg chiral lagrangian 42). It is remarkable that the cancellation persists up to rather high c.m. energies.

The interacting pressure expressed as the sum of contributions "intP~) and "intp(2) from terms linear and quadratic in the scattering amplitude using eqs. (6)-(8) are shown in fig. 2. In the upper panel we show the individual contributions to the interacting pressure from the resonant J~ phase shift. Virtually the entire contribution to the pressure comes from the resonant channel, i.e from -p~2) lnt _ the term quadratic in the scattering amplitude f, since near the resonance f is almost purely imaginary. The result is very different for the non-resonant channels (see the lower panel in fig. 2). The repulsive J02 phase shift remains relatively small in magnitude with energy and therefore f is almost purely real. Hence, the dominant contribution to the interacting pressure comes from ~intP t~ - the term proportional to the real part of the forward scattering amplitude, in contrast, the magnitude of Jo° becomes increasingly |arge with energy so that both the real and the imaginary parts of f are comparable. we take these e↵ects into account (which corresponds to a distorted spectrum), the HRG curves on both ﬁgures are sensibly di↵erent from the physical ones and agree with the corresponding lattice data of hotQCD. Our continuum results on the strange susceptibility are compared to the other results, too. We observe a good agreement between our results and the “physical” HRG ones4. Notice that the agreement between lattice and HRG model results isHadron good below resonance the transition gas temperature, versus lattice while for QCD larger calculations temperatures a deviation is obviously expected. This is observed both in our results and the hotQCD ones, but the temperatures at which deviations occur are obviously di↵erent.

5 #$ 1.0 " HRG physical # 0.5 # #" HRG physical ! asqtad Nt!12 ! 0.4 asqtad!Nt!8# HRG distorted stout Nt!8 " " stout continuum $ p4 N !8 " " asqtad Nt!8 # ! ! ! ! ! ! 0.8 t # " ! 4 HRG distorted asqtad Nt 8 ! # 0.4 #" !" stout continuum p4 Nt!8 " " stout Nt!10 ! ! " 0.3 $ # "! ! ! " R 4 ! stout Nt 8 # 0.6 ! " T 3 ! 0.3 # $ asqtad Nt!8 s ! " " !

$% " , Ψ# ! $ l ! 3p # p4 N !8 $ " ! 0.2 t ! # ! #" !Ψ "! Ε# 2 ! 0.4 ! # # 0.2 ""! ! ! "!! ! " ! $ $ !"!! " 0.1 ! # stout continuum "! # 1 0.20.1 HRG physical "! ! " ! $ !" # !! # HRG distorted N !8 " !!!! " !t ! # !!!! Renormalized Polyakov loop ! " " 0.0 HRG distorted"! Nt!12 800 100 120 140 160 180 200 0.0 100 120 140 160 180 100 120140140160160 180180 200200 220220 T MeVT MeV T TMeVMeV FigureFigure 8: Left: 7: Left: Renormalized (✏ 3p)/T chiral4 as acondensate function of as the deﬁned temperature. in Eq. (4.3). Open Right: symbols Subtracted are our chiral results. condensateFull symbolsWuppertal-Budapestl,s as are deﬁned the results in eq. Collaboration for (4.4), the asqtad as a, andfunctionJHEP p4 1009 actions of the(2010) at temperature.N 073= 8 [5]. Solid Gray line: bands HRG are model the ! ! " " t ! ! " " continuumwith physical results masses. of our collaboration, Dashed lines: obtained HRG model with the with stout distorted action. spectrums. Full symbols As itare can obtained be seen, withthe the prediction asqtad and of p4 the actions HRG model [5, 11]. with In both a spectrum panels, distortion the solid line corresponding is the HRG to model the stout result action with at physicalNt = masses. 8 is already The quite error close band to corresponds the physical to one. the Theuncertainty error on in the the recent quark preliminary mass-dependence HISQ result of hadron[11] masses. is larger The than dashed the di↵ lineserence are between the HRG+ the stoutPT and model asqtad result data, with that distorted is why masses, we do not which show takethem into account here. Right: the discretization renormalized Polyakove↵ects and loop. heavier We comparequark masses our results used in with [5, 11]those for ofN thet = hotQCD 8 and Nt =Collaboration 12. (asqtad and p4 data for Nt = 8 [5]).

4 In the left panel of1.0 Fig." 7" we show the trace anomaly divided by T as a function ! ! """ " of the temperature. Our N = 8 results"" areasqtad: takenmΠ from!220 MeV Ref. [15]. Notice that, for this t ! "" " observable, we have a0.8 check-point!! at Nt = 10stout: too:mΠ the!135 results MeV ! are on top of each other. " Also shown are the results of the hotQCD! collaborationstout: mΠ!414 at MeVNt =" 8 [5] and the HRG model predictions for physical and distorted! resonance" spectrums. On the one hand, our results 0.6 ! Nt!8 s ,

l " are in good agreement with the “physical”! HRG model ones. It is important to note, # that using our mass splittings0.4 and inserting this distorted spectrum into the HRG model ! " gives a temperature dependence which lies! essentially on the physical HRG curve (at least within our accuracy).0.2 On the other hand,! a distorted spectrum based on the asqtad and ! "" p4 frameworks results in a shift of about 20! MeV to the" right. The asqtad and p4 lattice ! ! " results can be successfully described by this distorted!!!! HRG! prediction,! " too. 150 200 250 In the right panel of Fig. 7 we show the renormalized Polyakov loop (the renormalization procedure was discussed in the previousT MeV Section). The comparison with the data of

4For completeness we included in our comparisons preliminary [11] results of the hotQCD collaboration

obtained by the HISQ and asqtad actions on Nt=8 and 12, respectively. We will discuss their impact later. Figure 9: Subtracted chiral condensate l,s as a function! " of the temperature. The empty triangles are our results with physical quark masses as shown in Fig. 4. The empty rectangles are our results with an average pion mass of 587 MeV at T 135 MeV. The red curve is the result of the hotQCD ' collaboration [5]: these results are the same shown in Fig. 8: a line connects the data to lead the eye. For all sets of data we have Nt = 8. As it can– 15 be – seen, the asqtad data can be mimicked in the stout framework by using a larger quark mass. limit within the HISQ framework is still missing. This last important step (which needs quite some computational resources and also care) will hopefully eliminate the remaining minor discrepancy, too. The same two members of the hotQCD Collaboration presented

– 17 – QCD thermodynamics at non-zero chemical potential

Taylor expansion :

µS hadronic T

quark

Taylor expansion coefficients give the fluctuations and correlations of conserved charges, e.g.

Computation of Taylor expansion coefficients reduces to calculating the product of inverse fermion matrix with different source vectors => can be done effectively on GPUs

Deconfinement : fluctuations of conserved charges

baryon number

electric charge

strangeness

1 Ideal gas of massless quarks : SB i/i

0.8

i=B 0.6 Q S

0.4 filled : HISQ, N=6, 8 conserved charges carried open : stout continuum by light quarks 0.2

T [MeV] HotQCD: PRD86 (2012) 034509 0 150 200 250 300 350 BW: JHEP 1201 (2012) 138, conserved charges are carried by massive hadrons Deconfinement : fluctuations of conserved charges

baryon number

electric charge

strangeness

7 Ideal gas of massless quarks : i i,SB i=B 4 /4 Q 6 S

3 filled : HISQ, N =6, 8 conserved charges carried open : stout, N=12 2 by light quarks

1 BNL-Bielefeld : talk by C. Schmidt T [MeV] BW: talk by Borsanyi 0 150 200 250 300 350 @ Confinement X conference conserved charges are carried by massive hadrons Correlations of conserved charges

P.P. J.Phys. G39 (2012) 093002 • Correlations between strange and light quarks at low T are due to the fact that strange hadrons contain both strange and light quarks but very small at high T (>250 MeV) => weakly interacting quark gas

• For baryon-strangeness correlations HISQ results are close to the physical HRG result, at T>250 MeV these correlations are very close to the ideal gas value

• The transition region where degrees of freedom change from hadronic to quark-like is broad ~ (100-150) MeV Deconfinement of strangeness Partial pressure of strange hadrons in uncorrelated hadron gas:

should vanish !

0.30 non-int. quarks

• v1 and v2 do vanish within errors 0.25 at low T 0.20 • v and v rapidly increase above B B 1 2 0.15 χ2-χ4 the transition region, eventually v1 reaching non-interacting quark 0.10 v gas values 2 0.05

uncorr. BNL-Bielefeld, arXiv:1304.7220 0.00 hadrons T [MeV]

140 180 220 260 300 340 Deconfinement of strangeness (cont’d) Using the six Taylor expansion coefficients related to strangeness

it is possible to construct combinations that give

up to terms

1.0 0.10 B3(0,0) B (0,1/6) 0.5 3 B3(1/3,1/6) HRG 0.0 P|S|=3,B 0.05

-0.5

M(0,0) -1.0 M(0,-3) 0.00 M(-6,-3) -1.5 HRG P|S|=1,M T [MeV] T [MeV] -2.0 -0.05 140 180 220 260 300 340 140 180 220 260 300 340

Hadron resonance gas descriptions breaks down for all strangeness sectors above Tc BNL-Bielefeld, arXiv:1304.7220 Quark number fluctuations at high T At high temperatures quark number fluctuations can be described by weak coupling approach due to asymptotic freedom of QCD

2nd order quark number fluctuations 4th order quark number fluctuations 1.0 1.0

0.8 0.8

0.6 0.6 c u4 c u2

c u4, c u2, SB 0.4 SB 0.4 BNL-Bielefeld, Nt=8 BNL-Bielefeld, Nt=8 WB WB 0.2 0.2 HTLpt HTLpt DR DR 0.0 0.0 200 300 400 500 600 200 300 400 500 600 T MeV T MeV Andersen, Mogliacci, Su, Vuorinen, PRD87 (2013) 074003 • Lattice results converge as the continuum limit is approachedH L • Good agreementH betweenL lattice and the weak coupling approach for 2nd order quark number fluctuations • For 4th order the weak coupling results are in reasonable agreement with lattice Staggered versus Wilson and Overlap Fermioms

Comparison with Wilson Fermion calculations, mπ ≈ 500 MeV, Borsányi et al, arXiv:1205.0440 Chiral condensate Strangeness susceptibility Polyakov loop MeV MeV MeV 125 150 175 200 225 250 275 125 150 175 200 225 250 275 150 175 200 225 250 275 0.005 2 staggered continuum 1 staggered continuum staggered continuum 0 Wilson continuum Wilson continuum Wilson continuum 1.5 -0.005 0.8 4 -0.01 2 / m

0.6 R 1 R L /T

-0.015 s R – -0.02 0.4 R 0.5 m -0.025 0.2 -0.03 0 0 -0.035 0.08 0.1 0.12 0.14 0.16 0.08 0.1 0.12 0.14 0.16 0.08 0.1 0.12 0.14 0.16 T/m T/m T/m

Comparison with overlap Fermion calculations, mπ ≈ 350 MeV Borsányi et al, PLB713 (12) 342 T [MeV] T [MeV] T [MeV] 120 140 160 180 200 220 240 260 120 140 160 180 200 220 240 260 0.01 120 140 160 180 200 220 240 260 3 3 2.5 6×12 6×12 3 0 3 1 3 6×12 8×16 8×16 3 staggered 8×16 -0.01 staggered 2 staggered 0.8 4

-0.02

/m -0.03 1.5 SB R 0.6 R

/ L

– -0.04 I R 1 m -0.05 0.4 -0.06 0.5 -0.07 0.2

-0.08 0 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 Tw0 Tw Tw0 0