Kenji Morita 16 Nov Baryon Number Probability Distribution in Quark-Meson Model Based on Functional Renormalization Group Approach.

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Presentation transcript:

Kenji Morita 16 Nov Baryon Number Probability Distribution in Quark-Meson Model Based on Functional Renormalization Group Approach Kenji Morita (Yukawa Institute for Theoretical Physics, Kyoto University) In collaboration with B. Friman (GSI), K. Redlich (Wroclaw), V. Skokov (BNL) Special Thanks to : Xiaofeng Luo and Nu Xu Refs) KM, Skokov, Friman, Redlich, arXiv:1211.xxxx KM, Friman, Redlich, Skokov, in preparation

Kenji Morita 16 Nov /17 Fluctuations of conserved charges GC ensemble : specified by ( T, m ) V Counting # of charges in a subvolume V Characteristic behavior in fluctuations near phase transition

Kenji Morita 16 Nov /17 Chart GC partition function Cumulants Pressure Probability Distribution What makes c n Critical? This Work

Kenji Morita 16 Nov /17 Probing chiral transition w/ net baryon number fluctuations T r = N B =V (Schematic / oversimplified) Crossover (m phys ) [2 nd order O(4) (chiral limit)] Critical point (m phys ) [Tricritical point (chiral limit)] 1 st order 1-10 r 0 High energy : RHIC/LHC Signature : Diverging c 2 Signature : Diverging c 6 ( m =0) and c 3 ( m ≠0) Remnant : Negative c 6 Higher order cumulants are sensitive to chiral transition Stephanov, Hatta,… Karsch, Redlich, Friman, Skokov,…

Kenji Morita 16 Nov /17 How to compute P(N) Needed : Canonical Partition Function Z Coefficients of Laurent Expansion Special case : C contains | l |= 1 (Analytic in imaginary m ) Thermodynamic potential given by a model

Kenji Morita 16 Nov /17 Relation to Analytic Structure m -mc-mc mcmc -ipT-ipT ipTipT Phase transition : singularity in complex m plane Singularity - Narrower P(N) (←Two saddle points m s and m s *) Landau Theory Stephanov ’06, Skokov-KM-Friman ‘10

Kenji Morita 16 Nov /17 Quark-meson model w/ FRG approach Effective potential is obtained by solving the exact flow equation (Wetterich eq.) with approximations giving 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 16 Nov /17 Reference P(N) : Skellam distribution # of baryon (Poisson)# of anti-baryon (Poisson) No criticality! [Note : c 6 < 0 for O(4) transition ] Reference #1 : “Constituent quark gas” Reference #2 : “Same variance” as the FRG result (comparing the shape of P(N))

Kenji Morita 16 Nov /17 P(N) in the QM model Parameters : m p =135 MeV, m s =640 MeV, f p =93 MeV Crossover at m =0 w/ T pc =214 MeV vs Skellam #1 at various T Wider at higher T Always narrower than Skellam #1

Kenji Morita 16 Nov /17 Cumlants in the QM model Reconstructing c 2, c 4, and c 6 from P(N) using central moments Deviation : Decreasing quark mass and quantum statistics effect c 6 /c ! c 6 from P(N) well reproduces “exact” results – What is characteristics of P(N)?

Kenji Morita 16 Nov /17 P(N) in the QM model vs Skellam #2 (same s ) at various T Note: Points and Lines have the same c 2 But different c 4 and c 6 ! P(N) cannot be the same!

Kenji Morita 16 Nov /17 P(N) ratio in the QM model Taking ratio to Skellam #2 at various T Ratio < 1 at large |N| for c 6 /c 2 < 1 T

Kenji Morita 16 Nov /17 Finite chemical potential FRG : e m N/T Skellam : b > b Asymmetric P(N) (c odd ≠0) Same variance=freeze-out line Large N at large m is beyond present numerical precision

Kenji Morita 16 Nov /17 Ratio at Finite chemical potential FRG : e m N/T Skellam : b > b Asymmetric P(N) (c odd ≠0) Ratio > 1 for N- << 0 Same variance=freeze-out line

Kenji Morita 16 Nov /17 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 P(N)/P(N) Skellam < 1 at large |N| → negative c 6

Kenji Morita 16 Nov /17 Experimental data Data : Au+Au by STAR (thanks to Xiaofeng Luo and Nu Xu) The tail part shows systematic variation with centrality 0-5% data shows ratio < 1 ! Centrality dependence Large N part shows ratio < 1 Ratio > 1 at negative N Not seen in 5-10% Energy dependence

Kenji Morita 16 Nov /17 Concluding Remarks Probability Distribution P(N) of net baryon # Calculating canonical partition function Z(T,V,N) in O(4) chiral model w/ FRG method Singularity of W at complex m leads to narrow P(N) Comparison w/ Skellam distributions Narrower than the “quark gas” case Same variance case : systematic change close to T pc Seen in STAR data for centrality dependence at 200GeV Critical behavior (c 6 <0) m =0 : P(N)/P(N) Skellam < 1 at positive large N Finite m : P(N)/P(N) Skellam > 1 at negative N- Seen in STAR data for energy dependence at the most central event

Kenji Morita 16 Nov /17 Backup

Kenji Morita 16 Nov /17 Illustration by Landau Theory (up to s 4 ) Thermodynamic Potential (below T c ) Periodic in m I / T : responsible for quantized baryon number Parameters: T c =0.15 GeV d = p 4 /30 (massless gas) T / T c = 0.98 m c =20.8 MeV ( m = m c ) = 11.4 for V =30 fm 3 Phase Diagram

Kenji Morita 16 Nov /17 Illustration by Landau Theory Fluctuations a = 0 : Mean Field a = : 3d O(4) Large R Small R N P(N) c 3 and c 4 (and higher) diverge in the critical case (n>2)

Kenji Morita 16 Nov /17 Probability Distribution P(N) Fugacity Shift of peak Enhance large N N0 P(N) m =0 N0 m ≫ 0 P(N) Tail of P(N) may become significant at finite m !

Kenji Morita 16 Nov /17 Fluctuations in QM model Crossover Crossover T m