Freeze-out parameters: lattice meets experiment

Transcription

1 Freeze-out parameters: lattice meets experiment Claudia Ratti Università degli Studi di Torino and INFN, Sezione di Torino In collaboration with R. Bellwied, S. Borsanyi, Z. Fodor, S. Katz, S. Krieg, K. Szabo Claudia Ratti 1

2 Motivation We live in a very exciting era to understand the fundamental constituents of matter and the evolution of the Universe We can create the deconfined phase of QCD in the laboratory Lattice QCD simulations: unprecedented levels of accuracy physical quark masses several lattice spacings continuum limit The joint information between theory and experiment can help us to shed light on the QCD phase diagram Claudia Ratti 2

3 Fluctuations of conserved charges The deconfined phase of QCD can be reached in the laboratory Need for unambiguous observables to identify the phase transition fluctuations of conserved charges (baryon number, electric charge, strangeness) S. Jeon and V. Koch (2000), M. Asakawa, U. Heinz, B. Müller (2000) A rapid change of these observables in the vicinity oft c provides an unambiguous signal for deconfinement They can be calculated on the lattice as combinations of quark number susceptibilities They can be directly compared to experimental measurements Claudia Ratti 3

4 The observables under study Fluctuations (susceptibilities) are defined as follows: The chemical potentials are related: χ BSQ lmn = l+m+n p/t 4 (µ B /T) l (µ S /T) m (µ Q /T) n. µ u = 1 3 µ B µ Q; µ d = 1 3 µ B 1 3 µ Q; µ s = 1 3 µ B 1 3 µ Q µ S. Quadratic fluctuations: χ X 2 = 1 VT 3 N2 X Correlators between different charges: χ XY 11 = 1 VT 3 N XN Y. Claudia Ratti 4

5 Physical meaning Diagonal susceptibilities measure the response of quark densities to an infinitesimal change in the chemical potential χ X 2 = 2 p/t 4 (µ X /T) 2 = (µ X /T) ( nx /T 3) A rapid increase of these observables in a certain temperature range signals a phase transition Non-diagonal susceptibilities measure the correlation between different quark flavors χ XY 11 = 2 p/t 4 (µ X /T) (µ Y /T) = (µ Y /T) ( nx /T 3) They can provide information about bound-state survival above the phase transition Claudia Ratti 5

6 Results: susceptibilities of baryon number and electric charge χ B 2 = 1 9 ( ) 2χ u 2 +χ s 2 +2χ ud 11 +4χ us 11 ; χ Q 2 = 1 9 ( ) 5χ u 2 +χ s 2 4χ ud 11 2χ us SB limit SB limit χ 2 B N t =6 N t =8 N t =10 N t =12 N t =16 cont. HRG χ 2 Q N t =6 N t =8 N t =10 N t =12 N t =16 cont. HRG T [MeV] T [MeV] rapid rise aroundt c They reach 90% of ideal gas value at large temperatures WB collaboration, JHEP (2012) Claudia Ratti 6

7 Comparison between light and strange quark susceptibilities 1 SB limit χ 2 s /χ2 u cont. HRG T [MeV] strange quark susceptibilities have their rapid rise at larger temperatures compared to the light quark ones they rise more slowly as a function oft There is a difference of MeV between the inflection points of the two curves WB collaboration, JHEP (2012) Claudia Ratti 7

8 Are there bound states in the QGP? Comparison of lattice to PNJL (C.R., R. Bellwied, M. Cristoforetti, M. Barbaro, PRD (2012)) χ 11 us SB limit N t =8 N t =10 N t =12 N t =16 cont. PNJL MF PNJL PL PNJL MC PNJL MF: pure mean field calculation PNJL PL: mean field plus Polyakov loop fluctuations PNJL MC: full Monte Carlo result with all fluctuations taken into account the red curve falls on the blue forv T/T c Even the inclusion of fluctuations is not enough to describe lattice data abovet c There is space for a contribution from bound states C BS SB limit stout cont. PNJL MF PNJL PL T/T c Claudia Ratti 8

9 Freeze-out temperature from experiment Fit to yields of identified particles: Statistical Hadronization Model (SHM) Model-dependent. Parameters: freeze-out temperature and chemical potential Claudia Ratti 9

10 Caveats Lattice results for susceptibilities are first-principle calculations However,T c cannot be univocally defined The experimental results are fitted by means of the Statistical Hadronization Model It would be nice to have a direct comparison between first-principle calculations and experimental results Claudia Ratti 10

11 Higher order susceptibilities and ratios susceptibilities are defined as follows: χ BSQ lmn = l+m+n p/t 4 (µ B /T) l (µ S /T) m (µ Q /T) n. we are now interested in higher order susceptibilities and in ratios Ratios have a very peculiar shape which allows to unambiguously spot the transition They can be directly related to an experimental measurement: no need for model interpretation! Claudia Ratti 11

12 Relating lattice results to experimental measurement the first four cumulants are: χ 1 = (δx) χ 2 = (δx) 2 χ 3 = (δx) 3 χ 4 = (δx) 4 3 (δx) 4 2 we can relate them to higher moments of multiplicity distributions: mean : M = χ 1 variance : σ 2 = χ 2 skewness : S = χ 3 /χ 3/2 2 kurtosis : κ = χ 4 /χ 2 2 Sσ = χ 3 /χ 2 κσ 2 = χ 4 /χ 2 F. Karsch (2012) M/σ 2 = χ 1 /χ 2 Sσ 3 /M = χ 3 /χ 1 Claudia Ratti 12

13 Thermometer and Baryometer R Q 31 : thermometer R Q 31 (T,µ B) = χq 3 (T,µ B) χ Q 1 (T,µ B) = χ QB 31 (T,0)+χQ 4 (T,0)q 1(T)+χ QS 31 (T,0)s 1(T) χ QB 11 (T,0)+χQ 2 (T,0)q 1(T)+χ QS 11 (T,0)s 1(T) +O(µ2 B ) Expand numerator and denominator aroundµ B = 0: ratio is independent ofµ B R Q 12 : baryometer R Q 12 (T,µ B) = χq 1 (T,µ B) χ Q 2 (T,µ B) = χ QB 11 (T,0)+χQ 2 (T,0)q 1(T)+χ QS 11 (T,0)s 1(T) χ Q 2 (T,0) µ B T +O(µ3 B ). Expand numerator and denominator aroundµ B = 0: ratio is proportional toµ B Claudia Ratti 13

14 Experimental measurement I Star Collaboration: arxiv Claudia Ratti 14

15 Experimental measurement II Star Collaboration: arxiv S Q σ Q = χ Q 3 /χq 2. Average of two most central measurements over s = 27, 39, Claudia Ratti 15

16 Extracting freeze-out parameters S Q σ Q 3 / MQ lattice continuum [WB, this work] N t =8 N t =10 N t =12 N t =16 BNL-Bielefeld N t =8 [ ] STAR preliminary [ ] HRG T [MeV] WB Collaboration: PRL (2013) M Q /σ Q 2 T=145 MeV T=150 MeV T=155 MeV T=160 MeV STAR, 62.4 GeV STAR, 27 GeV STAR, 39 GeV µ B [MeV] Upper limit: T f 157±4MeV s[gev] µ f B [MeV] (5) 39 75(7) 27 95(9) Claudia Ratti 16

17 Alternative: use baryon number S B σ B 3 / MB Nt =6 N t =8 N t =10 N t =12 WB continuum limit M B /σ B T=145 MeV T=150 MeV T=155 MeV T=160 MeV T [MeV] µ B [MeV] WB Collaboration: PRL (2013) Independently extractt f andµ Bf Check thermodynamic consistency Experimental measurement still missing Claudia Ratti 17

18 Strange vs light thermometer R. Bellwied et al.: PRL (2013) Flavor-specific kurtosis confirms separation between light and strange quarks w f = χ Bf 13 χbf 11 more sensitive to flavor content: in the hadronic phase it only receives contribution from hadrons with more than one quark of flavor f Claudia Ratti 18

19 Charm quark number susceptibilities χ c 2 = T V 2 lnz µ c µ c µi =0 charm susceptibilities rise at much larger temperatures compared to the light quark ones their rise with temperature is much slower Claudia Ratti 19

20 Conclusions High precision (continuum limit) lattice QCD results allow to extract the freeze-out parameters of heavy-ion collisions there are indications of a flavor separation in the transition temperatures this could lead to a short mixed phase of degrees of freedom in which strange particle formation is dominant this should lead to measurable effects in the strange hadron yields (evidence from ALICE) new model-independent comparison between theory and experiment: χ 4 /χ 2 work in progress: define a meaningful experimental measurement ultimate test: hadronic spectral functions at finite-temperature Claudia Ratti 20

21 Backup slides Claudia Ratti 21

22 Relations between chemical potentials µ B,µ S andµ Q are NOT independent: n S = 0 n Q = Z A n B Z A = 0.4 By expandingn B,n S andn Q up toµ 3 B we get: 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 +... Claudia Ratti 22

23 Taylor coefficients: results q 1 (T) q 3 (T)/q 1 (T) N t =8 N t =10 N t =12 N t =16 extrapolation BNL-Bielefeld [ ] HRG s 1 (T) s 3 (T)/s 1 (T) N t =8 N t =10 N t =12 N t =16 extrapolation BNL-Bielefeld [ ] HRG T [MeV] T [MeV] WB Collaboration: PRL (2013) µ Q turns out to be very small Agreement between WB and BNL-Bielefeld collaborations Claudia Ratti 23

24 Defining the experimental measurement Problem: we need to go from theb,q,s basis to the basis ofu, d, s quark flavors In principle we need to measure all light and strange quark final states Experimentally impossible, most resonances are too rare and cannot be reconstructed Most resonances decay one to one to their ground state Strange weak decays need to be reconstructed Ongoing project with P. Alba, W. M. Alberico, R. Bellwied, M. Bluhm, D. Chinellato and M. Weber Claudia Ratti 24

25 Charm quark number susceptibilities χ c 2 = T V 2 lnz µ c µ c µi =0 charm susceptibilities rise at much larger temperatures compared to the light quark ones their rise with temperature is much slower Claudia Ratti 25

26 Possible interpretations u χ 2 /T 2 cont. s χ 2 /T 2 cont. c χ 2 /T 2, Nt =12 HRG T [MeV] survival of open charm hadrons up to T 2T c? HRG results agree with the lattice up to the inflection point in the data Claudia Ratti 26

27 Freeze-out parameters Possible interpretations u χ 2 /T 2 cont. s χ 2 /T 2 cont. c χ 2 /T 2, Nt =12 HRG u χ 2 /T 2 cont. s χ 2 /T 2 cont. c χ 2 /T 2, Nt =12 charm quark gas or T [MeV] T [MeV] survival of open charm hadrons up to T 2T c? HRG results agree with the lattice up to the inflection point in the data thermal excitation of charm quarks takes place at larger temperatures ideal gas of charm quarks agrees with lattice need for non-diagonal quark number susceptibilities Claudia Ratti 27

28 Simple experimental verification Yields of strange particles should be enhanced relative to yields of non-strange particles yields of strange particles should result in a higher temperature than yields of non-strange particles when fitted with a statistical hadronization model (SHM) Claudia Ratti 28

29 Simultaneous fit givest = 152 MeV SHM yield fit L. Milano for ALICE, QM 2012 However, protons are overestimated and strange baryons are underestimated Claudia Ratti 29

30 SHM yield fit Fitting them separately, a hierarchy is evident R. Bellwied (2012) Claudia Ratti 30

31 Caveats Lattice results for susceptibilities are first-principle calculations However,T c cannot be univocally defined The experimental results are fitted by means of the Statistical Hadronization Model It would be nice to have a direct comparison between first-principle calculations and experimental results Claudia Ratti 31

32 Higher order susceptibilities and ratios susceptibilities are defined as follows: χ BSQ lmn = l+m+n p/t 4 (µ B /T) l (µ S /T) m (µ Q /T) n. we are now interested in fourth order susceptibilities (χ 4 ) and in particular in ratiosχ 4 /χ 2 Ratios have a very peculiar shape which allows to unambiguously spot the transition They can be directly related to an experimental measurement: no need for model interpretation! Claudia Ratti 32

33 Relating lattice results to experimental measurement the first four cumulants are: χ 1 = (δx) χ 2 = (δx) 2 χ 3 = (δx) 3 χ 4 = (δx) 4 3 (δx) 4 2 we can relate them to higher moments of multiplicity distributions: variance : σ 2 = χ 2 standarddeviation : σ = χ 2 skewness : S = χ 3 /χ 3/2 2 kurtosis : κ = χ 4 /χ 2 2 Sσ = χ 3 /χ 2 κσ 2 = χ 4 /χ 2 and therefore: F. Karsch (2012) Claudia Ratti 33

34 Preliminary lattice results light and strange quarkχ 4 /χ 2 still exhibit 20 MeV difference in the transition temperature transition is easy to spot: non-monotonic behavior WB collaboration, preliminary Claudia Ratti 34

35 Defining the experimental measurement Problem: we need to go from theb,q,s basis to the basis ofu, d, s quark flavors In principle we need to measure all light and strange quark final states Experimentally impossible, most resonances are too rare and cannot be reconstructed Most resonances decay one to one to their ground state Strange weak decays need to be reconstructed Ongoing project with W. M. Alberico, R. Bellwied, M. Bluhm, D. Chinellato and M. Weber Claudia Ratti 35

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38 Comparison with previous lattice data χ u 2 = T V 2 lnz µ u µ u µi =0 χ 2 u /T SB limit stout N t =6 stout N t =8 stout N t =10 stout N t =12, N s =32 stout N t =12, N s =36 asqtad N t =6 asqtad N t =8 p4 N t =6 p4 N t = T/T c physical quark massesm s /m u,d 28, before it wasm s /m u,d 10 finer lattice spacings and continuum limit the phase transition turns out to be much smoother Claudia Ratti 38

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42 All path approach Our goal: determine the equation of state for several pion masses reduce the uncertainty related to the choice ofβ 0 conventional path: A, though B, C or any other paths are possible generalize: take all paths into account Claudia Ratti 42

43 Finite volume and discretization effects finitev : N s /N t = 3 and 6 (8 times larger volume): no sizable difference finite a: improvement program of lattice QCD (action observables) tree-level improvement for p (thermodynamic relations fix the others) trace anomaly for threet -s: hight, transitiont, lowt continuum limitn t = 6,8,10,12: same with or without improvement improvement strongly reduces cutoff effects: slope 0 (1 2σ level) Claudia Ratti 43

44 Pseudo-scalar mesons in staggered formulation Staggered formulation: four degenerate quark flavors ( tastes ) in the continuum limit Rooting procedure: replace fermion determinant in the partition function by its fourth root At finite lattice spacing the four tastes are not degenerate each pion is split into 16 the sixteen pseudo-scalar mesons have unequal masses only one of them has vanishing mass in the chiral limit m i 2 GeV stout a 2 fm 2 Scaling starts forn t 8. mi GeV Claudia Ratti 44