Role of fluctuations in detecting the QCD phase transition

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1 Role of fluctuations in detecting the QCD phase transition Fluctuations of the Polyakov loop and deconfinement in a pure SU(N) gauge theory and in QCD Fluctuations of conserved charges as probe for the chiral phase transition and deconfinement Probability distribution and O(4) criticality theoretical expectation and STAR data Krzysztof Redlich University of Wroclaw & EMMI

2 2 Susceptibilities of net charge and order parameters The generalized susceptibilities probing fluctuations of net -charge number in a system and its critical properties pressure: generalized susceptibilities ( i+ j+ k) ^4 ( i+ j+ k) p/ T i j i T µ x m Order parameter = : 1 ln < Oh >= V Z h particle number density quark number susceptibility 4 th order cumulant (2) q (4) q = < N >, VT 1 1 q 3 expressed by = < > < > VT q 3 ( N N ) N N N q q 1 4 ( ( ) 4 3 ( ) 3 ) q δn δn 3 = < > < > VT = and central moment δ N = N < N >

3 Polyakov loop on the lattice needs renormalization Introduce Polyakov loop: L cn L ren q L e β F ren < >= 2 π ik/ N cn = e Z N ( ) 0 deconfined T > Tc 0 confined T > Tc Renormalized ultraviolet divergence Usually one takes as an order parameter 1/ V 3

4 To probe deconfinement : consider fluctuations Pok Man Lo, B. Friman, O. Kaczmarek, C. Sasaki & K.R. Fluctuations of modulus of the Polyakov loop SU(3) pure gauge: LGT data However, the Polyakov loop Thus, one can consider fluctuations of the real part I L = L +il R R and the imaginary of the Polyakov loop. I

5 Fluctuations of the real and imaginary part of the renormalized Polyakov loop Real part fluctuations Imaginary part fluctuations Pok Man Lo, B. Friman, O. Kaczmarek, C. Sasaki & K.R., PRD (2013)

6 Ratios of the Polyakov loop fluctuations as an excellent probe for deconfinement Pok Man Lo, B. Friman, O. Kaczmarek, C. Sasaki & K.R., PRD (2013) In the deconfined phase RA 1 R A A = R Expand modulus and find: Indeed, in the real sector of Z(3) L L + δ L with L =< L > R 0 R 0 R I I I 0 + δ I with 0 = L L L L < > < > < ( δ ) > L L L R 0, thus = V < ( δl ) >, = V < ( δl ) > L 2 L 2 R R I I Expand the modulus, δl ( δl ) R I L = LR + LI L0 (1 + + ) 2 L0 2L0 get in leading order thus A R

7 Ratios of the Polyakov loop fluctuations as an excellent probe for deconfinement Pok Man Lo, B. Friman, O. Kaczmarek, C. Sasaki & K.R., PRD (2013) In the confined phase RA 0.43 R A = A R Expand modulus and find: Indeed, in the Z(3) symmetric phase, the probability distribution is to a first approximation Gaussian with the partition function Z = dl dle Thus A 3 2αT α R + I VT [ ( T )( L L )] R I 1 1 R =, 3 I = 3 and 2αT 2αT 1 π = (2 ), consequently π SU (3) R A = (2 ) = SU (2) 2 In the SU(2) case R A π in agreement with MC results = (2 ) = 0.363

8 Ratio Imaginary/Real of Polyakov loop fluctuations Pok Man Lo, B. Friman, O. Kaczmarek, C. Sasaki & K.R., PRD (2013) L I L R In the confined phase for any symmetry breaking operator its average vanishes, thus and = thus R = I LL =< L > < L> = LL R I In deconfined phase the ratio of / 0 I R and its value is model dependent

9 Effective chiral models and gluon potential β = 1/ T 3 i nt + * S = dτ d x iq( γµ µ Aµ δ µ 4) q V ( qq, ) + µ qqq ULL 0 V * ULL (, ) the [ (, )] Z(3) invariant Polyakov loop potential V int ( q, q) the SU(2)xSU(2) through: Nambu-Jona-Lasinio model invariant quark interactions described PNJL chiral model K. Fukushima; C. Ratti & W. Weise; B. Friman, C. Sasaki.,. coupling with meson fileds FRG thermodynamics of PQM model: PQM chiral model B.-J. Schaefer, J.M. Pawlowski & J. Wambach; B. Friman, V. Skokov,... B. Friman, V. Skokov, B. Stokic & K.R.

10 Polyakov loop parameters, fixed from a pure glue Lattice Thermodynamics b ( T) k U( L, L ) = b ( T) LL b ( T) ln[ M( L, L)] * * * 2 3 fixed to reproduce pure SU(3) lattice results K. Fukushima, C. Ratti & W. Weise First order deconfinement phase transition at T0 270 MeV L

11 Effective Polyakov loop Potential from Y-M Lagrangian Chihiro Sasaki & K.R. (Gross, Pisarski & Yaffe).

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13 Asymptotic properties of the gluon potential Expanding the log under integral one recovers all potential used so far, in this, the strong-coupling expansion effective lattice action to the next-to-leading order

14 The minimal potential needed to incorporate Polyakov loop fluctuatiions fluctuations Pok Man Lo, B. Friman, O. Kaczmarek, C. Sasaki & K.R.

15 The influence of fermions on ratios of the Polyakov loop susceptibilities Z(3) symmetry broken, however ratios still showing the transition Pok Man Lo, B. Friman, O. Kaczmarek, C. Sasaki & K.R. Change of the slopes at fixed T

16 Probing deconfinement in QCD lattice with p4 fermion action Pok Man Lo, B. Friman, (013) O. Kaczmarek, C. Sasaki & K.R. B S. Ejiri, F. Karsch & K.R. (06) P ( T, µ ) = F( T)cosh(3 µ / T) S. Ejiri, F. Karsch & K.R. (06) m π 0.77GeV, µ B =0 HRG factorization of pressure: q Kurtosis measures the squared of the baryon number carried by leading particles in a medium S. Ejiri, F. Karsch & K.R. (06) q 2 κσ ( 4) PC B 2 = B = (2) B T > TPC T < T

17 Probing deconfinement in QCD lattice with p4 fermion action Pok Man Lo, B. Friman, (013) O. Kaczmarek, C. Sasaki & K.R. S. Ejiri, F. Karsch & K.R. (06) m π 0.77GeV, µ B =0 The change of the slope of the ratio of the Polyakov L L loop susceptibilities A / R appears at the same T where the kurtosis drops from its HRG asymptotic value In the presence of quarks there is remnant of Z(N) L L symmetry in the / ratio, indicating deconfinement of quarks A R

18 Probing deconfinement in QCD Pok Man Lo, B. Friman, O. Kaczmarek, C. Sasaki & K.R. B 4 B 2 L A L / R S. Ejiri, F. Karsch & K.R. (06) Ch. Schmidt m π 0.77GeV, µ B =0 Pok Man Lo T[MeV] The change of the slope of the ratio of the Polyakov L L loop susceptibilities A / R appears at the same T where the kurtosis drops from its HRG asymptotic value In the presence of quarks there is remnant of Z(N) L L symmetry in the / ratio, indicating deconfinement of quarks A R

19 Remnant of the O(4) chiral phase transtion in QCD Critical region CP Pisarki & Wilczek conjecture Asakawa-Yazaki Stephanov et al., Hatta & Ikeda The QCD crossover line can appear in the O(4) critical region! This has been indeed shown in LQCD calculations by: Fig. from Ch. Schmidt At the CP: Divergence of Fluctuations, Correlation length and specific heat BNL-Bielefeld group Phys. Rev. D83, (2011) Phys. Rev. D80, (2009)

20 Chemical freezeout and the QCD chiral crossover LHC Chiral O(4) universality crossover µ / T = f( s) µ Tc µ A. Andronic et al., Nucl.Phys.A837:65-86,2010. = Tc T LHC J. Stachel et. al. 2 Is there a memory that the system has passed through a region of QCD O(4)-chiral crossover transition? HotQCD Collaboration HRG model Chiral crosover Temperature from LGT T = 154 ± 9 MeV c HotQCD Coll. (QM 12) Chemical Freezeout LHC (ALICE) T = 156 ± 2 MeV f see talk of J. Stachel

21 Quark fluctuations and O(4) universality class Due to expected O(4) scaling in QCD the free energy: F = F T + b b t b 1 (2 α) βδ/ ν R (, µ q, µ I) FS (, h) 1 Consider generalized susceptibilities of net-quark number c n 4 ( n) ( P/ T ) B n ( µ B / T) = = T < T pc c ( n) R ( n ) cr + c S with Since for, are well described by the ( n) search for deviations (in particular for larger n) from HRG ( n) to quantify the contributions of, i.e. the O(4) criticality c S c ( n) s (2 α n)/ βδ ( n = dh f ) ( z) HRG F. Karsch & K. R. Phys.Lett. B695 (2011) 136 B. Friman, et al.. Phys.Lett. B708 (2012) 179, Nucl.Phys. A880 (2012) ±

22 Quark-meson model w/ FRG approach Effective potential is obtained by solving the exact flow equation (Wetterich eq.) with the approximations resulting in the O(4) critical exponents B.J. Schaefer & J. Wambach,; B. Stokic, B. Friman & K.R. q - q Full propagators with k < q < L G L =S classical Integrating from k=l to k=0 gives a full quantum effective potential Put W k=0 (s min ) into the integral formula for P(N)

23 B. Stokic, B. Friman & K.R. Fluctuations & susceptibilities t ( h) max ( h) max + γ 1.6 1/( γ β) 0.49 O β (4) 0.38 MF γ = 1 Two type of susceptibility related with order parameter 1. longitudinal 2. transverse = = σ / h l t σ = = σ / h π Scaling properties at t=0 and h 0 gδ = = σ π Bh δ 1/ ( tm ax h γ+ β) σ ( t ) t 1/ 1 max mx a γ

24 Ratios of cumulants at finite density in PQM model with FRG B. Friman, F. Karsch, V. Skokov &K.R. Eur.Phys.J. C71 (2011) B 4 B 2 HRG value 18 9 B. Friman, V. Skokov &K.R. Phys.Rev. C83 (2011) B 6 B 2 HRG value Deviations from low -T HRG values are increasing with and the cumulant order. Negative fluctuations near chiral crossover. µ / T

25 Fluctuations of 6th and 8th order moments exhibit strong variations from HRG predictions: Their negative values near chiral transition to be seen in heavy ion collisions at LHC & RHIC The range of negative fluctuations near chiral cross-over: PNJL model results with quantum fluctuations being included : These properties are due to O(4) scaling, thus should be also there in QCD. 25

26 STAR data on the first four moments of net baryon number Deviations from the HRG S σ = (3) B, (2) B 2 κ σ = ( 4) B (2) B Np Np S σ HRG =, κσ HRG = 1 N + N p p HRG Data qualitatively consistent with the change of these ratios due to the contribution of the O(4) singular part to the free energy

27 Kurtosis saturates near the O(4) phase boundary B. Friman, et al. EPJC 71, (2011) The energy dependence of measured kurtosis consistent with expectations due to contribution of the O(4) criticality. Can that be also seen in the higher moments?

28 STAR DATA Presented at QM 12 Lizhu Chen for STAR Coll. V. Skokov, B. Friman & K.R., F. Karsch et al. c / c HRG 6 2 The HRG reference predicts: O(4) singular part contribution: strong deviations from HRG: negative structure already at vanishing baryon density c6 / c 2 = 1

29 Moments of the net conserved charges Obtained as susceptibilities from Pressure c = P T µ n n 4 n ( / )/ ( B / T) or since they are expressed as polynomials in the central moment δ N = N < N >

30 Moments obtained from probability distributions Moments obtained from probability distribution k k < N >= N PN ( ) N Probability quantified by all cumulants 2π 1 P( N) = dy exp[ iyn ( iy) ] 2π 0 ( y) = β V[ p( T, y+ µ ) p( T, µ )] = k y k In statistical physics Cumulants generating function: P( N) = ZC ( N) e Z GC µ N T k

31 Influence of criticality on P(N) Consider Landau model: Ω=Ω bg + t µ σ + σ 2 4 t ( T, ) µ µ Scaling properties: Mean Field α = 0 1 (, ) 2 4 t T µ µ tµ ( T) consider and α 0 α 0.2 α 0.1 n c n sin g 3 n c n sin g 2

32 Contribution of a sigular part to P(N) P( N) = Z C ( N, T, V) Z GC e µ N T Get numerically from: For MF and α > 0 broader for α < 0 narrower than Skellam

33 What is the influence of O(4) criticality on P(N)? For the net baryon number use the Skellam distribution (HRG baseline) B P( N) = IN (2 BB)exp[ ( B+ B)] B as the reference for the non-critical behavior N /2 Calculate P(N) in an effective chiral model which exhibits O(4) scaling and compare to Skellam distribtuion

34 The influence of O(4) criticality on P(N) for µ = 0 Take the ratio of P FRG ( N) which contains O(4) dynamics to Skellam distribution with the same Mean and Variance at different T T K. Morita, B. Friman &K.R. (PQM model) µ = 0 / pc Ratios less than unity near the chiral crossover, indicating the contribution of the O(4) criticality to the thermodynamic pressure

35 The influence of O(4) criticality on P(N) for µ = 0 Take the ratio of P FRG ( N) which contains O(4) dynamics to Skellam distribution with the same Mean and Variance at different T T K. Morita, B. Friman et al. µ = 0 / pc Ratio < 1 at larger N if c 6 /c 2 < 1

36 The influence of O(4) criticality on P(N) for Take the ratio of P FRG ( N) which contains O(4) dynamics to Skellam distribution with the same Mean and Variance near T pc ( µ ) K. Morita, B. Friman et al. µ = 0 µ 0 µ 0 Asymmetric P(N) Near T the ratios less pc ( µ ) than unity for N >< N > µ For sufficiently large the FRG Skellam ( N ) for P ( N)/ P > 1 N < < N >

37 Probability distribution of net proton number STAR Coll. data at RHIC Thanks to Nu Xu and Xiofeng Luo STAR data Do we also see the O(4) critical structure in these probability distributions?

38 The influence of O(4) criticality on P(N) for µ 0 In central collisions the probability behaves as being influenced by the chiral transition µ = 0 K. Morita, B. Friman & K.R. For preliminary STAR data QM 2012

39 Energy dependence for different centralities Ratios at central collisions show properties expected near O(4) chiral pseudocritical line For less central collisions the critical structure is lost

40 Conclusions: Ratios of the Polyakov loop and the net charge susceptibilities are excellent probes of deconfinement and/or the O(4) chiral crossover transition in QCD Systematics of the net-proton fluctuations and their probability distributions measured by STAR are qualitatively consistent with the expectations that thy are influenced by the O(4) criticality.

41 Kurtosis of net quark number density in PQM model V. Skokov, B. Friman &K.R. For T < T c the assymptotic value due to confinement properties 2 P ( T) cosh 3 qq Nf mq mq µ q K T Nc T T T c4 / c 2 = 9 9 c / c 4 2 For T >> T P ( T ) qq T 4 = 2 c / c = 6/ π 4 2 c µ 1 µ 7π N fnc[ ] π T 6 T 180 Smooth change with a very weak dependence on the pion mass