Fluctuations at finite temperature and density

Transcription

1 Fluctuations at finite temperature and density [ , , , , ] Szabolcs Borsanyi Wuppertal-Budapest collaboration: R. Bellwied, Z. Fodor, S. Dürr, S. D. Katz, S. Krieg, A. Pasztor, C, Ratti and K. K. Szabo Bergische Universität Wuppertal July 18, 215

2 The QCD phase diagram Quark Gluon Plasma Temperature Early Universe Hadron Gas Deconfined Quarks and Gluons Vacuum Collision experiments Nuclei Compact Stars Colour Flavour Locking Quark Density (Chemical potential)

3 Beam energy scan and freeze-out curve At RHIC a broad energy range snn = GeV has been scanned with heavy ion collisions. Last inelastic scattering: chemical freeze-out. Chiral crossover region from lattice: T c = [WB hep-lat/61114,hep-lat/6968, ,15.358] Traditional method: Hadron Resonance Gas (HRG)-based statistical fit of pion, kaon, proton, etc yields. Fit result at s NN = 13GeV µ B = 38(12) MeV and T ch = 165(5) MeV. [Andronic et al nucl-th/51171] [HotQCD ]

4 Do conserved charges fluctuate? [Cartoon by V. Koch] There is an acceptance cut in rapidity (as well as tranverse momentum), so that only particles from the fireball should be counted. The cuts also implement a sub-volume counting: a sufficiently small sub-volume can behave as a grand-canonical ensemble.

5 Fluctuations in a grand canonical ensemble The expectation value of a conserved charge is a derivative with respect to the chemical potential. N q = T log Z(T, V, {µ q}) µ q The response of the system to the thermodynamic force µ q is proportional to the fluctuation of the conserved charge: N i µ j = T 2 log Z(T, V, {µ q }) µ j µ i = 1 T ( N in j N i N j ) The higher derivatives are the generalized quark number susceptibilities: with ˆµ q = µ q /T. χ u,d,s,c i+j+k+l (p/t 4 ) i,j,k,l = ( ˆµ u ) i ( ˆµ d ) j ( ˆµ s ) k ( ˆµ c ) l

6 Fluctuations on the lattice The partition function of the lattice gauge theory with staggered fermions is Z = DU e Sg (det M u(µ u)) 1/4 (det M d (µ d )) 1/4 (det M s(µ s)) 1/4 = DU e S eff where S g is the gauge action. First derivative of the free energy density: i log Z = 1 DU i e S eff = A i, A i = 1 d log det M i (µ i ) = 1 Z 4 dµ i 4 TrM M 1 When we make further derivatives, the following chain rule applies: j X = X ( j log Z) + X j e S eff + j X = XA j X A j + j X. With these rules the 2nd derivative reads Isospin and baryon directions: i j log Z = A i A j A i A j + δ ij da i /dµ ( u d ) 2 log Z = 2 da/dµ, ( u + d ) 2 log Z = 4[ AA A A ] + 2 da/dµ

7 The sign problem The fermion determinant detm = detm e iθ has a fluctuating phase at µ > : [Allton hep-lat/2413] θ = 1 [ ] 4 N ln detm f Im µ + µ3 ln detm +... µ 3! µ The fluctuation of A = ln detm/ µ gives at LO for the phase: θ 2 = 1 9 µ2 BL 3 TN 2 f χ ud 11 ( 1 at µ B 1 MeV, with T = T c and LT = 3.) χ ud 11 N t =8 N t =1 N t =12 N t =16 N t =2 N t =24 cont -.2 HRG (212) HRG (214) T=135 MeV histogram of the estimator of dlog det M / dmu 64 3 x16 lattice 48 3 x16 lattice 64 3 x24 lattice 32 3 x8 lattice Im η + M -1 M η + /4 / ( Nx 3 /N t ) 1/2 / [ χ ud 11 ]1/2 [Wuppertal-Budapest ]

8 Translating from quark numbers to B,Q and S In terms of physical derivatives we have d dµ B = 1 3 u d s, d dµ Q = 2 3 u 1 3 d 1 3 s, d dµ S = s χ B 4 = 1 81 χ Q 4 = 1 81 χ B 2 = χ Q 2 = χ I 2 = 1 1 ] [2 u 2 + s u s + 2 u d log Z, VT ] [5 u 2 + s 2 2 u s 4 u d log Z, VT ] [ u 2 u d log Z. VT 2 [ ] 2χ u 4 + χ s 4 + 6χ ud χ us χ us χ us χ ud χ uds χ uds 112, [ ] 17χ u 4 + χ s χ ud χ us 22 4χ us 13 28χ us 31 4χ ud χ uds χ uds 112

9 All diagonal fluctuations 1.8 χ 2 i /(χ2 i )SB i=i i=q i=u i=b i=s This result: [Wuppertal-Budapest ] see also [HotQCD ]

10 Off-diagonal susceptibilities χ BS 11 [HotQCD ] χ QS 11 [HotQCD ] χ BQ 11 /χb 2 [HotQCD ] χ BQ 11 /χb [WB preliminary] χ BQ 11 /χb 2 Wuppertal-Budapest preliminary N t =8 N t =1 N t =12 N t =16 N t =2 HRG (212) HRG (214)

11 Susceptibilities with non-staggered fermions anisotropic wilson m π = 392 MeV isotropic wilson in continuum m π = 545, 44, 285 MeV. MeV χ s /T MeV 44 MeV 285 MeV physical point (staggered) T / m Ω [FASTSUM , ] [WB , ] Overlap fermions (N f = 2 m π = 35 MeV) [WB ].

12 At high temperature: lattice vs Hard Thermal Loops 1 χ HTLpt partial 4loop DR HRG U HISQ (BNL-Bielefeld): χ 2 4stout (W-B): χ 2 U 4stout (W-B): 3χ 2 B χ U 4 HTLpt HTLpt (N f =4) HRG 4stout (W-B) χ B 4.2 χ 2, HTLpt partial 4loop DR HRG 4stout HTLpt partial 4loop DR u,s HRG (χ 2,2) u,d 4stout χ 2,2 u,s 4stout χ 2, HTL results: [Haque et al , ] Dimensional reduction: [ ] Lattice results: [WB: ] See also: [Swansea-Bielefeld: hep-lat/513] [BNL-Bielefeld ] [WB: ]

13 The Hadron Resonance Gas model Hadrons are free particles in a heat bath, their interactions are introduced by adding all their resonances to the heat bath, as free particles. [Dashen, Ma, Bernstein 1969] p HRG T 4 = 1 VT 3 + i mesons log Z M/B m i = Vd i 2π 2 log Z M (T, V, m i, {µ}) + i baryons ) dkk 2 log (1 z i e m i 2+k2 /T = VT 3 2π 2 d m 2 i i T 2 (±) k+1 zk i k 2 K 2(km i /T ) k=1 log Z B (T, V, m i, {µ}) with the fugaciy factor z i = exp(b i ˆµ B + Q i ˆµ Q + S i ˆµ S ) and ˆµ = µ/t. For m i T (e.g. baryons) only the k = 1 term is relevant (Boltzmann approximation). The contribution of a particle and antiparticle combined: log Zm M/B i 2 VT 3 2π d mi 2 2 i T K2(m i/t ) cosh(b 2 i ˆµ B + Q i ˆµ Q + S i ˆµ S )

14 In this approximation and µ Q = the pressure can be simply written as as sum: p HRG (ˆµ B, ˆµ S ) = p mesons S= + p baryons S= cosh(ˆµ B ) +p mesons S =1 cosh(ˆµ S ) + p baryons S =1 cosh(ˆµ B ˆµ S ) +p baryons S =2 cosh(ˆµ B 2ˆµ S ) + p baryons S =3 cosh(ˆµ B 3ˆµ S ) And so one can form identities independently of the resonance list: χ B 4 χ B 2 = v 1 = χ BS 31 χ BS 11 = v 2 = 1 3 (χs 2 χ S 4 ) 2χ BS 13 4χ BS 22 2χ BS 31 = [BNL-Bielefleld ] For the light flavors, see: [WB ] For the charm sector, see: [BNL-Bielefeld ]

15 Equation of state at finite density p(µ Taylor approach: B ) T = p() 4 T + χ B µ 2 4 B 2 2!T + χ B µ 4 2 B 4 4!T + χ B µ 6 4 B 6 6!T MILC and BNL-Bielefeld: N t = 6 3rd order [ , hep-lat/5124] Wuppertal-Budapest: (2stout staggered) continuum, leading order [ ] This plot shows a comparison to dimensional reduction and HTL HTLpt 3-loop truncated WB HTLpt 1-loop exact DR Μ B 3MeV Μ B 2MeV [Mogliacci et al ]. Μ B 1MeV BNL-Bielefeld-CCNU: N t = 6, 8 HISQ, next-to-leading order, plus NNLO for the error estimate [ ] In both works: µ B extrapolation with strangeness neutrality n s =.

16 Canonical approach: fugacity expansion Taylor method: log Z is expanded in µ Fugacity expansion: Z is expanded in exp(µ/t ) (Laurent series) Z(µ) = DUe S G [U] det[d(µ)] 2 Since only the determinant depends on µ we expand it for each gauge configuration: det[d(µ)] = q e µq/t D (q) With imaginary µ q this is a Fourier transformation: D (q) = 1 2π π π Signal-to-noise ratio in the canonical determinants: D 2 q D q 2 e q(m N /3 m π/2)/t dϕe iqϕ det[d(µ/t = iϕ] [Gattringer et al , , ]

17 Quark number susceptibility from fugacity expansion For small lattices (up to 123 6) σ 2 /M can be calculated at µb > left: [Gattringer ] right [Suzuki Thu] Comparison to direct reweighting: [Fukuda ] [Nagata ]

18 Physics at imaginary µ B Many exploratory studies: [de Forcrand & Philipsen hep-lat/2516] [Philipsen ] [Philipsen ] [Cea et al hep-lat/61218, ,122.57] At imaginary µ B there is no sign problem. The observables, including the crossover line, are analytical functions of µ 2 B. 24 Polyakov loop orientation Ising critical point chiral crossover 1st order π 2π 3π 4π 5π 6π µ B I /T = 3µq I /T The phase diagram is perodic µ B µ B + i2πt, with simultaneous rotation between the Z(3) sectors.

19 Curvature of the phase diagram The T c(µ B ) can be expanded around µ B = (Taylor method) or found through analytical continuation with Im µ B >. T c(µ B ) T c ( ) 2 µb = 1 κ +... T c(µ) [Kaczmarek et al ] [Endrodi et al ] [Cea et al ] [Bonati et al , ] This conference: µ s =, imaginary µ B.825πT S =, imaginary µ B.625πT χ S 2 (T) µ B = <S>=, extrapolted from µ B = <S>= direct simulaton µ s =µ u =µ d extrapolated from µ B = µ s =µ u =µ d extrapolated from <S>=, µ B > magnetic equation of state expanded in µ B x1 lattice µ I B /T=5π/ [Bonati et al ; M. Mesiti Tue] [Wuppertal-Budapest prelim. J. Günther Tue] [BNL-Bielefeld-CCNU preliminary; P. Hegde Tue]

20 Curvature of the phase diagram The T c(µ B ) can be expandedd around µ B = (Taylor method) or found through analytical continuation with Im µ B >. This conference: T c(µ B ) T c ( ) 2 µb = 1 κ +... T c(µ) Lattice Curvature of the phase diagram Hegde HISQ Nt=6 Taylor Τ [MeV] Cosmai HISQ Nt=6,8,1,12 Analytical Mesiti 2stout Nt=6,8,1,12 Analytical [ ] Guenther 4stout Nt=1,12,16 Analytical analytical continuation from lattice Dyson-Schwinger [C. Fischer et al.] 6 freeze-out [Becattini, Cleymans 25] 4 modified statistical fit [Becattini 212] 2 freeze-out from fluctuations [Alba et al 214] µ B [MeV] [J. Günther Tue]

21 Fluctuations from experiment s window is reprenucleons tion, obtion [31]. illions):.7, 11.5, At RHIC STAR has measured the mean, variance, skewness and kurtosis of the event-by-event net charge4 distribution at various energies and centralities. Mean (M) 4 a) Net-Charge Au+Au 3 η <.5 2 ve (N ) 1.5 and 5 and antilate net- c).8 d) 2 GeV ge distrises. The.6 39 GeV 62.4 GeV.2 27 GeV ariations 19.6 GeV duce adts GeV prod- 7.7 GeV.1.2 CLT entrality weighted here the e of the ach such N part N part y remove centrales of the corrected (a) mean, (b) standard deviation, (c) skewness and [STAR: FIG. 1: ] (Color online) The efficiency and centrality bin width The effiobtained (d) kurtosis of the net-charge multiplicity distributions as a function of number of participating nucleons ( Npart ) for erage effor Au+Au collisions. The dotted lines represent calculations most from the central limit theorem. The error bars are statistical Skewness (S) Std. Deviation (σ) Kurtosis (κ) 2 b) 15 1 σ 2 M Sσ κσ a).6 b) 5 c) Net-charge Au+Au η <.5-5% 7-8% -5% NBD 7-8% NBD -5% Poisson 7-8% Poisson s NN (GeV) FIG. 4: (Color online) Beam-energy dependence of (a) σ 2 /M, (b) Sσ, and(c)κσ 2,afterallcorrections,formostcentral(- 5%) and peripheral (7-8%) bins. The error bars are statistical and the caps represent systematic errors. Results from the Poisson and the NBD baselines are superimposed. The values of κσ 2 for Poisson baseline are always unity. also be us tion on che measured h lattice QCD functions a in heavy-io rameters b been obtai ally, by usi ment of th be extracte els [24, 42]. ses [44] usin for central tracted fre MeV and µ that this is sured highe cal freeze-o The freezemoments a tional meth higher sens which warr In summ charge mul tion of cent energies fro These data QCD phas point. We tonically w

22 Caveats Effects due to volume variation because of finite centrality bin width Experimentally corrected by centrality-bin-width correction method Finite reconstruction efficiency Experimentally corrected based on binomial distribution [A. Bzdak, V. Koch, PRC (212)] Spallation protons Experimentally removed with proper cuts in p T Canonical vs Grand Canonical ensemble Experimental cuts in the kinematics and acceptance [V. Koch, S. Jeon, PRL (2)] Proton multiplicity distributions vs baryon number fluctuations Numerically very similar once protons are properly treated [M. Asakawa and M. Kitazawa], [PRC (212), M. Nahrgang et al., ] Final-state interactions in the hadronic phase [ J.Steinheimer et al., PRL (213)] Consistency between different charges = fundamental test

23 Thermometer from the skewness A possible thermometer [BNL-Bielefeld ] T ch is found through Sσ 3 /M experiment (beam energy) = Sσ 3 /M lattice (T ch ) Comparing Wuppertal-Budapest lattice results with STAR data: Net electric charge: Net baryon number: S Q σ Q 3 / MQ lattice continuum [WB, this work] N t =8 N t =1 N t =12 N t =16 BNL-Bielefeld N t =8 [ ] STAR preliminary [ ] HRG R B 31 =S B σ B 3 / M B STAR N t =6 N t =8 N t =1 N t =12 WB continuum limit [Wuppertal-Budapest ], [STAR ]. Conclusion T ch 157MeV [Wuppertal-Budapest ], [STAR ] (protons). Conclusion T ch 151MeV

24 Baryometers from proton and charge fluctuations R B 12 =M B / 2 B STAR, 27 GeV STAR, 62.4 GeV STAR, 39 GeV.3 T=14 MeV.2 T=145 MeV STAR, 2 GeV T=15 MeV.1 µ B [MeV] R Q 12 =M Q / 2 Q T=14 MeV T=145 MeV T=15 MeV STAR, 62.4 GeV STAR, 39 GeV STAR, 27 GeV.2 STAR, 2 GeV µ B [MeV] The second order fluctuations are accurate both on the lattice and in experiment. The temperature range 14 MeV < T < 15MeV is motivated by the skewness data. Are these chemical potentials consistent between baryon and charge? [Wuppertal-Budapest ]

25 Baryometers from proton and charge fluctuations s[gev ] µ f B [MeV] (from B) µ f B [MeV] (from Q) ± ± ± ± ±11 11± ±13.8 The chemical potentials are now consistent and they also agree with the statistical model s parametrization [Andronic ] [Wuppertal-Budapest ] µ B [MeV] from B from Q SHM model 2 s [GeV]

26 A new thermometer Our thermometer, the skewness, is difficult to work with both on lattice and in experiment. If we assume that the consistency between proton and charge freeze-out holds exactly (same temperature and chemical potential), then a ratio of ratios can be defined..2 [M Q / Q 2 ] / [MB / B 2 ] Our conclusion: STAR, 39 GeV µ B = 1 MeV STAR, 62.4 GeV µ B = 65 MeV STAR, 2 GeV µ B = 25 MeV µ B = T f = 144 ± 6MeV (please remember the caveats)

27 Summary Diagonal as well as off-diagonal quark number susceptibilities have been well explored with staggered fermions in the physical point Several 4th order moments have been calculated in the continuum We made comparisons to the Hadron Resance Gas model (T < T c ) and Hard Termal Loop PT (T T c ) Many extrapolatory results for fluctuations in the canonical approach Several groups have presented the leading µ B dependence of the transition temperature: all agree First contact to experiment has been made Challenges: Higher order charge fluctuations ; most sensitive to pion physics thermometer candidate for LHC Extrapolations to larger finite chemical potential, where RHIC experiments run I thank all those who sent me contributions and apologize for not including all.

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29 Implementing the conserved charges For RHIC physics we consider the grand canonical ensemble at a set of baryon, charge and strangeness chemical potentials such that: S = and Q / B = Z Au /A Au.4. In the statistical models these constraints are fulfilled exactly. In lattice one uses a Taylor expansion. [BNL-Bielefeld ] µ Q (T, µ B ) = q 1(T )µ B + q 3(T )µ 3 B +..., µ S (T, µ B ) = s 1(T )µ B + s 3(T )µ 3 B +... The coefficients are now to NLO ([BNL-Bielefeld ] [WB ]) Simulations at Im µ B > with strangeness neutrality [J Günther Monday] s 1 (T) s 3 (T)/s 1 (T) N t =8 N t =1 N t =12 N t =16 WB continuum limit BNL-Bielefeld [ ] HRG µ S I (T) /T µb I =.79 T µ B I = 1.18 T µ B I = 1.57 T µ B I = 1.96 T

30 Baryon fluctuations In the Hadron Gas phase there is a negligible contribution from B > 1 resonances. In the Boltzmann approximation χ B 4 χ B 2 = 2 VT 3 2π 2 2 VT 3 2π 2 i B4 i d i m2 i T K 2 2 (m i /T ) 1 i B2 i d i m2 i T 2 K 2 (m i /T ) Stefan-Boltzmann limit: χ B 4 /χb 2 = π2 /6/9 fugacity expansion: Derivatives at µ B : R B 42 =χb 4 /χb 2 HRG N t =6 N t =8 N t =1 N t =12 WB continuum limit [Gattringer ] See also: [BNL-Bielefeld: ] [Wuppertal Budapest: ] [Wuppertal-Budapest ]

31 Physics at imaginary µ B Many exploratory studies: [de Forcrand & Philipsen hep-lat/2516] [Philipsen ] [Philipsen ] [Cea et al hep-lat/61218, ,122.57] Polyakov loop orientation Ising critical point chiral crossover 1st order π 2π 3π 4π 5π 6π I I µ B/T = 3µq/T χ B 1 T = T c T > T RW T c < T < T RW At low T only B states contribute: χ B 1 sin(µ B /T ) Then χ B 4 /χ B 2 = 1 as we find in simulations. T< T c π/2 π 3π/2 2π Im µ /T At high T fractional charges are required to make a first order transition at µ B = πt : χ B 1 sin(µ B /T /3)

32 Melting and abundance of charm A comparison to HRG is difficult in the charm sector, only few resonances have been found so far. Strategy of the Bielefeld group [ ]: study HRG identities melting of the charm resonance list from the Quark Model Comparison of light/strange/charm sectors for melting:

33 Hierarchy in perturbation theory 1 χ u 4 Perturbatively: χ 4 6/π 2 α, χ 22 α 3/2, χ 11 α 3 log α χ 31 α 5 log α.1.1 6/π 2 -χ u 4 χ u,d 2,2 χ u,d 1,1 χ u,d 3,1 4stout lattice data at T=24 MeV [Toimela PLB (1983) B124,47;] [Blaizot hep-ph/11369] [Vuorinen hep-ph/212283] First contribution to χ 11 : We compare three flavors: light-light, light-strange, light-charm (see plot ) Lattice data: [WB ].1 2 1/N t χ 11 leading log HRG.1 ud 4stout -χ 11 us 4stout -χ 11 uc 4stout -χ T c 2T c 3T c 4T c 5T c 1