Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Probability distribution of conserved charges and the QCD phase transition K. Redlich HIC crossover CP or Triple Point ? Qarkyonic Matter ? QCD phase boundary, its O(4) „scaling” & relation to freezeout in HIC Moments and probablility distributions of conserved charges as probes of the O(4) crossover criticality STAR data & expectiations with: P. Braun-Munzinger, B. Friman F. Karsch, V. Skokov Chemical freezeout and QCD crtitical line O(4) etuniversality A. Andronic al., Nucl.Phys.A837:65-86,2010. Relating freezeout and QCD critical line at small baryon chemical potential Particle yields and their ratios as well as LGT thermodynamics at T Tc are well described by the HRG Partition Function QCD crossover line remnant of the O(4) criticality LGT and phenomenological HRG model C. Allton et al., S. Ejiri, F. Karsch & K.R. n 1 P q dn 4 T ( / T )n q , I (T ) The HRG partition function provides and excellent approximation of the QCD thermodynamics at T Tc See also: C. Ratti et al., P. Huovinen et al., B. Muller et al.,……. 3 Kurtosis as an excellent probe of deconfinement S. Ejiri, F. Karsch & K.R. R4,2 B 1 c4 9 c2 HRG factorization of pressure: P (T , q ) F (T ) cosh(3q / T ) Kurtosis B F. Karsch, Ch. Schmidt consequently: c4 / c2 9 in HRG 2 In QGP, SB 6 / Kurtosis=Ratio of cumulants c4q / c2q B The R4,2 measures the quark content of particles carrying baryon number ( N q )4 ( N q )2 3 ( N q )2 excellent probe of deconfinement O(4) scaling and critical behavior Near Tc critical properties obtained from the singular part of the free energy density F Freg FS with T Tc with t Tc Tc 2 FS (t , h) b d F (b1/ t , b / h) Phase transition encoded in the “equation of state” Fs pseudo-critical line h 1/ F ( z ) , z th h h1/ h1/ O(4) scaling of net-baryon number fluctuations The fluctuations are quantified by susceptibilities B (n) n (P / T 4 ) : cn n ( B / T ) P Preg PSingular From free energy and scaling function one gets c h (n) B (n) r 2 n /2 c h ( n) B with ( n) r 2 n f ( n /2) ( n) ( z ) for 0 and n even f ( z) for 0 Resulting in singular structures in n-th order moments which appear for n 6 at 0 and for n 3 at 0 since 0.2 in O(4) univ. class Ratio of cumulants at finite density PQM -RG See talk of Vladimir Skokov HRG R4,2 c4 / c2 HRG Deviations of the ratios of odd and even order cumulants from their asymptotic, low T-value, c4 / c2 c3 / c1 9 are increasing with / T and the cumulant order Properties essential in HIC to discriminate the phase change by measuring baryon number fluctuations ! Ratio of higher order cumulants at finite density See talk of Vladimir Skokov Deviations of the ratios from their asymptotic, low T-value, are increasing with the order of the cumulant order and with increasing chemical potential Properties essential in HIC to discriminate the phase change by measuring baryon number fluctuations ! Properties of fluctuations in HRG B( n ) n ( P(T , B ) / T 4 ) (B / T )n Calculate generalized susceptibilities: from Hadron Resonance Gas (HRG) partition function: P HRG Pmesons Pbaryons and Pbaryons T 4 F (m / T ) cosh(B / T ) baryons B( 4) 1 (2 ) B B(3) 1 (1) B ( 4) resulting in: B B2 B( 2 ) B B(2) B(3) coth( B / T ) and tanh( B / T ) (1) (2) B B B2 B(2) (1) M B B S B B 2 B(3) ( 2) B Compare these HRG model predictions with STAR data at RHIC: Coparison of the Hadron Resonance Gas Model with STAR data Frithjof Karsch & K.R. B(2) coth( B / T ) (1) B RHIC data follow generic properties expected within HRG model for (4) B different ratios of 1 (2) B the first four moments of baryon number fluctuations B(3) tanh( B / T ) (2) B Error Estimation for Moments Analysis: Xiaofeng Luo arXiv:1109.0593v Mean, variance, skewness and kurtosis obtained by STAR and rescaled HRG STAR Au-Au s 200 M p p 8.5 STAR Au-Au s 200 M p p 1.8 these data, due to transverse momentum cut. Account effectively for the above in the HRG model by rescaling the volume parameter by the factor 1.8/8.5 Moments obtained from probability distributions Moments obtained form the probability distribution N k N k P( N ) N Probability quantified by all moments 1 P( N ) 2 2 dy exp[iyN (iy)] 0 k ( y ) V [ p ( T , y ) p ( T , ) ] y Moments generating function: k k In statistical physics N ZC ( N ) T P( N ) e Z GC i S i S H [S , H ] 0 e He conservation on the average Z (T , S ,V ) Tr [e GC Z GC S S e S S / T ( H S S ) Z C S exact conservation ] Z (T ,V ) TrS [e C S 1 Z S (T ,V ) 2 2 d e 0 iS Z GC (T , S T H ] i ) Probability quantified by Sn , Sn :mean numbers of charged 1,2 and 3 particles & their antiparticles P(N) in the Hadron Resonance Gas for netproton number The probability distribution for net baryon number N is governed in HRG by the Skellam distribution P( N ) The probability distribution for net proton number N is entirely given in terms of (measurable) mean number of protons b and anti-protons b Probability distribution obtained form measured particle spectra: P( N ) f ( p , p ) pt pt STAR data P ( N ) broken lines Mean proton and anti-proton calculated in the HRG with (T , B ) taken at freezeut curve and Volume obtained form the net-mean proton number Comparing HRG Model with Preliminary STAR data: efficiency uncorrected Data presented at QM’11, H.G. Ritter, GSI Colloquium s 7.7 GeV s 11 GeV Data consistent with Skellam distribution: No sign for criticality HRG energy dependence versus STAR data arXiv:1106.2926 Xiaofeng Luo for the STAR Collaboration Data consitent with Skellam distribution Data presented at QM’11, H.G. Ritter, GSI Colloquium HRG shows broader distribution Observed schrinking of the probability distribution already expected due to deconfinement properties of QCD In the first approximation the B quantifies the width of P(N) P( N ) exp( N 2 / 2VT 3 B ) HRG LGT hotQCD Coll. Similar: Budapest-Wuppertal Coll Data versus UrQMD V. Skokov et al.. UrQMD provide much too narower probability distribution of net proton number Centrality dependence of O(4) criticality? Weak dependence of chemcial Freezeout on centrality Decrease of chemical potentil In the chiral limit decrease of volume reduce criticlity J. Engels Decrease of centrality shifts the chemical line out of chiral O(4) cross over Influence of volume on criticality requires detaile knowlege of Finite Size Scaling Function for free energy in O(4) universality class Centrality dependence at Preliminary STAR data H.G. Ritter, GSI Colloquium s NN 62 and 200 GeV Published STAR data-efficiency uncorected HRG shows increasing broadening of distributions with centrality Centrality dependence of HRG model deviations from data: Thanks to Xu Nu Criticality increases with centrality: seen in the probability distributions and moments compared to data O(4) criticality schematic diagram Hadronic freezout Based on the relation between HRG model and presently aveliable data we could conclude that: O(4) criticality HRG freezeout The separation between chemical freezeout line and the O(4) chiral line might appear in HIC at collision energies larger than 39 GeV?? Probability distributions and their energy dependence for 0.4 GeV p 0.8 GeV t The maxima of P ( N ) have very similar values for all sNN thus N p N p const. for all 7 GeV sNN 2.8 TeV Conclusions: Probability distributions and higher order cumulants are excellent probes of O(4) criticality in HIC Hadron resonance gas provides reference for O(4) critical behavior in HIC and LGT results In the HRG the P(N) is uniqually determined by the measure multiplicities of charged particles. This avoids ambiguity in determining freezeout parameters and volume of the system Deviation of P(N) form HRG sets in at s 39GeV and increases with centrality: This might indicated remnants of O(4) criticality in STAR data on net proton probability distribution in central Au-Au collisions at s 39GeV