Kenji Morita 24 June, XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” Probing the QCD Phase Boundary with Fluctuations of Conserved Charges Kenji Morita (FIAS)

24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” Search for Phase Transition in QCD Fukushima-Hatsuda, Rep.Prog.Phys.74 ‘11 Towards understanding the origin of the matter Chiral transition Confinement Theoretical Expection (from lattice QCD and models) For the statistical system specified by ( T, m ) Heavy Ion Collisions : Evidence for high energy density matter (QGP) Phase Transition ? K

Kenji Morita 24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” Multiplicity Fluctuations in Heavy Ion Collisions O (10 7 ) Counting # of particles within a given circumstance (centrality, acceptance etc) Averaging over events STAR, net-proton, PRL112 (’14)

Kenji Morita 24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” Fluctuations Indicate Phase Transition n -th order phase transition n -th order phase transition Singular behavior in Singular behavior in n=1 : Entropy, order parameter, charge density n=1 : Entropy, order parameter, charge density n=2 : specific heat n=2 : specific heat chiral susceptibility chiral susceptibility fluctuation of conserved charge fluctuation of conserved charge Observable through multiplicity distribution P(N) Higher order – more sensitive to criticality Not observable… (Stephanov ’09) of Conserved Charges

Kenji Morita 24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” Quantifying Fluctuations Prob. Distribution ~ Canonical Partition Function Shape : Cumulants ∝ Volume Note: Need Cancellation ! Higher order c n contains → Need more information on the tail ! M : Mean, s 2 : Variance, S : Skewness, k : Kurtosis No statistically meaningful measurement of c 6 yet…

Kenji Morita 24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” Grand Canonical Ensemble Description Chemical freeze-out : Equilibrated hadron gas Particle number (1 st moment) : OK T fo ～ phase boundary Expectation : measured fluctuations = those of GCE

Kenji Morita 24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” Net Baryon Number Fluctuation in HRG =Skellam Distribution # of baryons （ Poisson) # of antibaryon (Poisson)   Statistical mechanics : Boltzmann distribution   2 parameters Expectation : c 1 and c 2 are well described by HRG Deviation from Skellam in higher order c n should reflect the phase transition Ratio cancels Volume dependence Karsch-Redlich ‘11

Kenji Morita 24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” QCD ≒ Hadron Resonance Gas ？ Equation of state Baryon number fluctuations Wuppertal- Budapest BNL- Bielefeld T<155MeV : Good approx. Substantial deviation at Higher order

Kenji Morita 24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” Net-Proton Measurements by STAR Correction by bin width and reconstruction efficiency STAR, PRL114 Substantial Deviation from the HRG expectation Caveat : Net-proton≠Net-baryon Q : Is this deviation consistent with phase transition? Is there any other explanations? What is the underlying probability distribution?

Kenji Morita 24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” Th. Expectations for the Cumulants Chiral Limit T m TCP? mqmq 2 nd Order, 3d O(4) Phys. quark mass m phys Crossover – Feel O(4)? CP? c 6 changes the sign across crossover General property from O(4) scaling function (Engels and Karsch ’11, Friman et al., ’11) c 2 divergence c 3,4 change the sign Hatta-Ikeda ’03, Asakawa et al.,’09, Skokov et al., ’11, Stephanov, ‘11 Pisarski-Wilczek ‘84

Kenji Morita 24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” Probability Distribution P(N) and Cumulants c n at nonzero m From QCD : Sign Problem ! (No MC simulation reliable) Use of chiral effective model (L s M, NJL, etc) Extract qualitative feature relevant to QCD Z(2) CP : Still uncertain Remnant of O(4) in Crossover : Our suggestion

Kenji Morita 24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” P(N) w/ Chiral Quark-Meson Model (N f = 2 ) s, p, q (also below T c ) s, p, q (also below T c ) Crossover at m =0 : T pc =214 MeV Crossover at m =0 : T pc =214 MeV Critical m Common fluctuation property with QCD near phase boundary Reminder : Criticality appears higher order cumulants Solving the model : proper treatment of the critical flucuations KM et al., PRC88 ’13 for detail

Kenji Morita 24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” QM model w/ FRG approach Effective potential is obtained by solving the exact flow equation (Wetterich eq.) with approximations preserving correct critical exponents - Full propagators with k < q < L qq Integrating from k = L to k = 0 gives a full quantum effective potential Put obtained W k =0 ( s min )into the integral formula for P(N) (Stokic-Friman-Redlich ’10 ) G L = S classical

Kenji Morita 24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” Reference for the Critical Behavior Higher order cumulants need tail of P(N) Skellam distribution w/ same s 2 Skellam distribution w/ same s 2 Estimate # of data points to get correct c 6 in Skellam, then rescale Estimate # of data points to get correct c 6 in Skellam, then rescale Removing different VT 3 effect in various P(N) data Removing different VT 3 effect in various P(N) data P(N) N6N6 P(N) reproducing c 6

Kenji Morita 24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” Characterize Critical Behavior : P(N) ratio Ratio < 1 at large |N| for c 6 / c 2 < 1 T Narrower tail as approaching T pc c6/c2c6/c2 At T=0.98Tc, c 6 < 0 is consequence of the O(4) chiral transition KM, Friman, Redlich,

Kenji Morita 24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” P(N) Ratio at Small m Along phase boundary Dropping at small d N/N 6 for larger m KM, Friman, Redlich,

Kenji Morita 24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” How Exp. Data Look Like? Most central only : Avoid volume fluctuations N ev > 100 : Avoid effects from large error Very similar behavior to FRG/Skellam i.e., remnant of O(4) KM, Friman, Redlich,

Kenji Morita 24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” Concluding Remarks Fluctuation measurements in Heavy Ion Collisions Deviation from a reference distribution may signal critical phenomena in QCD Particle yield and lower cumulants validate thermal equilibrium picture Property of P(N B ) near Chiral Phase Transition Narrowing (relative to Skellam) in the tail near T pc P(N) Ratio supplements cumulant analyses Most central data of net-proton show perfect coincidence Consistent w/ O(4) expectation Non-critical effects on P(N) ratio? Q: Does the dropping survive after correction? Unambiguous interpretation needs c 6 – higher statistics

Kenji Morita 24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects”Outlook Other conserved charges Electric charge and net strangeness (net-kaon) measurement done by STAR Electric charge and net strangeness (net-kaon) measurement done by STAR Interpretation is more difficult because of other non-critical effects Interpretation is more difficult because of other non-critical effects Theoretical Challenges Theoretical Challenges More elaborated effective models or solving the sign problem in QCD to locate the phase boundary More elaborated effective models or solving the sign problem in QCD to locate the phase boundary More complete information from Lee-Yang zero (work in progress) More complete information from Lee-Yang zero (work in progress) Understanding possible other non-critical effects Understanding possible other non-critical effects Connection to higher density regime (FAIR, NICA, etc…) Connection to higher density regime (FAIR, NICA, etc…)

Kenji Morita 24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” Backup

Kenji Morita 24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” P(N) Ratio at Small m Fix m Asymmetric structure at nonzero m Dropping ratio turns into increasing at m 0 KM, Friman, Redlich, arXiv:

Kenji Morita 24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” Can NBD/BD Reproduce Criticality? Take NBD/BD for baryon and antibaryon 4 parameters (r,p) for B and (r’,p’) for B At m =0, r=r’ and p=p’ (2 parameters) At m >0, one can fit the first four cumulants c 1 - c 4 ! At m >0, one can fit the first four cumulants c 1 - c 4 ! Whether P(N proton ) is NBD/BD or not ? – Theoretically ill- defined (Not a conserved number) Whether P(N proton ) is NBD/BD or not ? – Theoretically ill- defined (Not a conserved number) NBD clearly cannot describe critical P(N) c 4 / c 2, c 6 / c 2 > 1 c 4 / c 2, c 6 / c 2 > 1

Kenji Morita 24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” Can NBD/BD Reproduce Criticality? Fix parameters to reproduce model c 2 and c 4 BD: Possible for each cumulant ratio Impossible for simultaneous description!

Kenji Morita 24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” P(N) vs Critical exponent a Model based on Landau theory & Scaling func. Skellam dist. (Boltzman gas) Critical exponent for the specific heat a= 0.11 : 3d Z(2) 0 : Mean Field : 3d O(4) Singular part : reproduce singular cumulants at T c

Kenji Morita 24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” P(N) vs Critical exponent a Model based on Landau theory & Scaling func. Ratio to Skellam reveals differences Long tail – divergence

Kenji Morita 24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” Experimental Results

Kenji Morita 24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” Step 3 : Scaling Property Scattered

Kenji Morita 24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” Selection of Data sets Centrality > 40% : Deviation from Scaling Apparent deviation by chemical potential Insufficient Statistics Check same underlying physics : scaling property w/ s – close to HRG Too small # of events (< 100)

Kenji Morita 24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” N max dependence of c n V dep – N/V 1/2 c 2 : Exact c 4,c 6 : Approxmate

Kenji Morita 24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” Fermi gas Cumulants Narrow Broad Narrow Critical < Fermi < Skellam Broad But deviation from Skellam in the tail is as large as critical case

Kenji Morita 24 June, /19 XXX-th International Workshop on High Energy Physics “Particle and Astroparticle Physics, Gravitation and Cosmology: Predictions, Observations and New Projects” Why P(N)? - Tail of P(N) is important in c 6 Higher order cumulants need P(N) at large N Cut here N max P(N max )~ to get correct c 6