Kenji Morita Baryon Number Probability Distribution in QuarkMeson

Kenji Morita 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) Refs) KM, Skokov, Friman, Redlich, ar. Xiv: 1211. xxxx KM, Friman, Redlich, Skokov, in preparation Special Thanks to : Xiaofeng. Luo and Nu Xu 16 Nov 2012 ATHIC 2012@Pusan 1

Kenji Morita 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 16 Nov 2012 ATHIC 2012@Pusan 2

Kenji Morita Chart This Work GC partition function Probability Distribution What makes cn Critical? Pressure Cumulants 16 Nov 2012 ATHIC 2012@Pusan 3

Kenji Morita Probing chiral transition w/ net baryon number fluctuations (Schematic / oversimplified) Karsch, Redlich, Friman, Skokov, … Remnant : Negative c 6 Crossover (mphys) [2 nd order O(4) (chiral limit)] T Signature : Diverging c 6 (m=0) and c 3 (m≠ 0) Critical point (mphys) High energy : RHIC/LHC BES@RHIC [Tricritical point (chiral limit)] Signature : Diverging c 2 Stephanov, Hatta, … 1 st order 1 -10 r 0 16 Nov 2012 Higher order cumulants are sensitive to chiral transition r=NB=V ATHIC 2012@Pusan 4

Kenji Morita How to compute P(N) Needed : Canonical Partition Function Z Thermodynamic potential given by a model Coefficients of Laurent Expansion Special case : C contains |l|=1 (Analytic in imaginary m) 16 Nov 2012 ATHIC 2012@Pusan 5

Kenji Morita Relation to Analytic Structure Phase transition : singularity in complex m Stephanov ’ 06, Skokov-KM-Friman ‘ 10 plane Landau Theory m ip. T -mc mc -ip. T Singularity - Narrower P(N) (←Two saddle points ms and ms*) 16 Nov 2012 ATHIC 2012@Pusan 6

Kenji Morita Quark-meson model w/ FRG approach Effective potential is obtained by solving the exact flow equation (Wetterich eq. ) with approximations giving correct critical exponents (Stokic-Friman-Redlich ’ 10) q GL=Sclassical q - Full propagators with k < q < L Integrating from k=L to k=0 gives a full quantum effective potential Put obtained Wk=0(smin)into the integral formula for P(N) 16 Nov 2012 ATHIC 2012@Pusan 7

Kenji Morita Reference P(N) : Skellamdistribution # 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)) 16 Nov 2012 ATHIC 2012@Pusan 8

Kenji Morita P(N) in the QM model vs Skellam #1 at various T Wider at higher T Always narrower than Skellam #1 Parameters : mp=135 Me. V, ms=640 Me. V, fp=93 Me. V Crossover at m=0 w/ Tpc=214 Me. V 16 Nov 2012 ATHIC 2012@Pusan 9

Kenji Morita Cumlantsin 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 2 < 0 at T/Tpc > 0. 96 ! c 6 from P(N) well reproduces “exact” results – What is characteristics of P(N)? 16 Nov 2012 ATHIC 2012@Pusan 10

Kenji Morita 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! 16 Nov 2012 ATHIC 2012@Pusan 11

Kenji Morita 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 16 Nov 2012 ATHIC 2012@Pusan 12

Kenji Morita Finite chemical potential FRG : em. N/T Skellam : b > b Asymmetric P(N) (codd≠ 0) Large N at large m is beyond present numerical precision Same variance=freeze-out line 16 Nov 2012 ATHIC 2012@Pusan 13

Kenji Morita Ratio at Finite chemical potential FRG : em. N/T Skellam : b > b Same variance=freeze-out line Asymmetric P(N) (codd≠ 0) Ratio > 1 for N-<N> << 0 16 Nov 2012 ATHIC 2012@Pusan 14

Kenji Morita Tail of P(N) is important in 6 c Higher order cumulants need P(N) at large N Cut here Nmax P(Nmax)~ 10 -10 to get correct c 6 P(N)/P(N)Skellam < 1 at large |N| → negative c 6 16 Nov 2012 ATHIC 2012@Pusan 15

Kenji Morita Experimental data Data : Au+Au by STAR (thanks to Xiaofeng Luo and Nu Xu) Energy dependence Centrality dependence • The tail part shows systematic variation with centrality • 0 -5% data shows ratio < 1 ! 16 Nov 2012 • Large N part shows ratio < 1 • Ratio > 1 at negative N • Not seen in 5 -10% ATHIC 2012@Pusan 16

Kenji Morita 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 Tpc Seen in STAR data for centrality dependence at 200 Ge. V 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-<N> Seen in STAR data for energy dependence at the most central event 16 Nov 2012 ATHIC 2012@Pusan 17

Kenji Morita Backup 16 Nov 2012 ATHIC 2012@Pusan 18

Kenji Morita Illustration by Landau Theory (up to s 4) Thermodynamic Potential (below T c) Phase Diagram Periodic in m. I/T : responsible for quantized baryon number Parameters: • Tc=0. 15 Ge. V • d=p 4/30 (massless gas) • T/Tc = 0. 98 • mc=20. 8 Me. V • <N>(m=mc) = 11. 4 for V=30 fm 3 16 Nov 2012 ATHIC 2012@Pusan 19

Kenji Morita Illustration by Landau Theory Fluctuations • a=0 : Mean Field • a= -0. 21 : 3 d O(4) P(N) (n>2) Large R Small R N 16 Nov 2012 c 3 and c 4 (and higher) diverge in the critical case ATHIC 2012@Pusan 20

Kenji Morita Probability Distribution P(N) Fugacity P(N) m=0 m ≫ 0 Shift of peak Enhance large N 0 0 N N Tail of P(N) may become significant at finite m! 16 Nov 2012 ATHIC 2012@Pusan 21

Kenji Morita Fluctuations in QM model Crossover T m 16 Nov 2012 ATHIC 2012@Pusan 22