6 Sep 2012 ERG2012@Aussois 1

Kenji Morita

Effect of critical fluctuations on the baryon Effect of critical fluctuations on the baryon number probability distribution near number probability distribution near

chiralchiral phase transition phase transition

Kenji MoritaKenji Morita((YukawaYukawa Institute for Theoretical Physics, Kyoto University)Institute for Theoretical Physics, Kyoto University)

In collaboration withIn collaboration withV.Skokov (BNL), V.Skokov (BNL), B. Friman (GSI), K. Redlich (Wroclaw)B. Friman (GSI), K. Redlich (Wroclaw)

1. Fluctuations and probability distribution

2. Illustration by Laudau Theory

3. Canonical partition function & Complex m

4. Probability distribution P(N)

5. Results from QM model+FRG

6 Sep 2012 2/14

Kenji Morita

ERG2012@Aussois

Fluctuations of conserved chargesFluctuations of conserved charges

GC ensemble : specified by (TTTT, m)

VVVV

Characteristic behavior in fluctuations near phase transition Characteristic behavior in fluctuations near phase transition Characteristic behavior in fluctuations near phase transition Characteristic behavior in fluctuations near phase transition

6 Sep 2012 3/14

Kenji Morita

ERG2012@Aussois

ChartChart

GC partition function

Cumulants

Pressure

Probability Distribution

What makes ccccnnnnCritical?

This WorkThis WorkThis WorkThis Work

6 Sep 2012 4/14

Kenji Morita

ERG2012@Aussois

Illustration by Landau Theory (up to Illustration by Landau Theory (up to ss44))

Thermodynamic Potential (below TThermodynamic Potential (below Tcc))

Periodic in mIIII////TTTT : responsible for

quantized baryon number

Parameters:Parameters:Parameters:Parameters:

• TTTTcccc=0.15 =0.15 =0.15 =0.15 GeVGeVGeVGeV

• dddd====p4444/30 (/30 (/30 (/30 (masslessmasslessmasslessmassless gas)gas)gas)gas)

• TTTT////TTTTcccc = 0.98 = 0.98 = 0.98 = 0.98

• mcccc=20.8 =20.8 =20.8 =20.8 MeVMeVMeVMeV

• <<<<NNNN>(>(>(>(m====mcccc) = 11.4 for ) = 11.4 for ) = 11.4 for ) = 11.4 for VVVV=30 fm=30 fm=30 fm=30 fm3333

Phase DiagramPhase DiagramPhase DiagramPhase Diagram

6 Sep 2012 5/14

Kenji Morita

ERG2012@Aussois

Illustration by Landau TheoryIllustration by Landau Theory

Fluctuations Fluctuations

• a====0000 : Mean Field

• a= = = = -0.210.210.210.21 : 3d O(4)

Large RLarge RLarge RLarge R

Small RSmall RSmall RSmall R

NNNN

P(N)P(N)P(N)P(N)

cccc3333 and cand cand cand c4444 (and higher) diverge in the critical case(and higher) diverge in the critical case(and higher) diverge in the critical case(and higher) diverge in the critical case

(n>2)(n>2)(n>2)(n>2)

6 Sep 2012 6/14

Kenji Morita

ERG2012@Aussois

Probability Distribution P(N) Probability Distribution P(N)

Fugacity Fugacity Fugacity Fugacity

Shift of peakShift of peakShift of peakShift of peak

Enhance large NEnhance large NEnhance large NEnhance large N

NNNN0000

P(N)P(N)P(N)P(N) m=0=0=0=0

NNNN0000

m ≫≫≫≫ 0000P(N)P(N)P(N)P(N)

Tail of P(N) may become significant at finite Tail of P(N) may become significant at finite Tail of P(N) may become significant at finite Tail of P(N) may become significant at finite m!!!!

6 Sep 2012 7/14

Kenji Morita

ERG2012@Aussois

How to compute P(N) How to compute P(N)

NeededNeeded : Canonical Partition Function Z: Canonical Partition Function Z

Coefficients of Coefficients of Coefficients of Coefficients of Laurent ExpansionLaurent ExpansionLaurent ExpansionLaurent Expansion

Special case : Special case : CCCCCCCC contains |contains |ll|=|=1 1 1 1 1 1 1 1 (Analytic in imaginary (Analytic in imaginary mm))

6 Sep 2012 8/14

Kenji Morita

ERG2012@Aussois

Comparison of P(N) Comparison of P(N)

MF : Analytically solvableMF : Analytically solvable

Critical : Numerical Integration Critical : Numerical Integration (Work only |N|<60)(Work only |N|<60)

Difference at large NDifference at large NDifference at large NDifference at large N Common structure btw. MF and Critical at small NCommon structure btw. MF and Critical at small NCommon structure btw. MF and Critical at small NCommon structure btw. MF and Critical at small N

Large N Shrink (Broadening) in Critical (MF)Large N Shrink (Broadening) in Critical (MF)Large N Shrink (Broadening) in Critical (MF)Large N Shrink (Broadening) in Critical (MF)

6 Sep 2012 9/14

Kenji Morita

ERG2012@Aussois

Relation to Analytic StructureRelation to Analytic Structure

m

----mcccc mcccc

----iiiipTTTT

iiiipTTTT

N < N < N < N < NcNcNcNc : N = <N>: N = <N>: N = <N>: N = <N>N > N > N > N > NcNcNcNc : N : N : N : N ≠≠≠≠ <N><N><N><N> Two saddle points beyond Two saddle points beyond Two saddle points beyond Two saddle points beyond NcNcNcNc

Complex ConjugateComplex ConjugateComplex ConjugateComplex Conjugate

→→→→OscillationOscillationOscillationOscillation

Steepest descent path indicates contribution from cutsSteepest descent path indicates contribution from cutsSteepest descent path indicates contribution from cutsSteepest descent path indicates contribution from cuts…………....

Steepest descent method Steepest descent method

6 Sep 2012 10/14

Kenji Morita

ERG2012@Aussois

FRG approach to P(N) in QM modelFRG approach to P(N) in QM model

Replace : Replace : WWCriticalCritical --> > WWQMQM--FRGFRG

Parameters : mParameters : mParameters : mParameters : mParameters : mParameters : mParameters : mParameters : mpp=135 =135 =135 =135 =135 =135 =135 =135 MeVMeVMeVMeVMeVMeVMeVMeV, m, m, m, m, m, m, m, mss=640 =640 =640 =640 =640 =640 =640 =640 MeVMeVMeVMeVMeVMeVMeVMeV, f, f, f, f, f, f, f, fpp=93 =93 =93 =93 =93 =93 =93 =93 MeVMeVMeVMeVMeVMeVMeVMeV

Explicit breaking term in Explicit breaking term in LagrangianLagrangian

Crossover at Crossover at mm=0 w/ =0 w/ TTpcpc=214 =214 MeVMeV

Solve flow equation w/ LPA and Taylor expansion Solve flow equation w/ LPA and Taylor expansion Solve flow equation w/ LPA and Taylor expansion Solve flow equation w/ LPA and Taylor expansion (cf: Stokic-Friman-Redlich ’10)

Next : Comparison with MF result

(incl. fermion vacuum term)

at T/Tpc = 0.98

6 Sep 2012 11/14

Kenji Morita

ERG2012@Aussois

P(N) in QM modelP(N) in QM model

CrossoverCrossover

MF case exhibits broader P(N)MF case exhibits broader P(N)

Different tail behavior in FRG (V=90fmDifferent tail behavior in FRG (V=90fm33))

6 Sep 2012 12/14

Kenji Morita

ERG2012@Aussois

CumulantsCumulants in QM modelin QM model

CrossoverCrossover

Cumulants reproduced until missing large N part becomes significant

6 Sep 2012 13/14

Kenji Morita

ERG2012@Aussois

P(N) in QM modelP(N) in QM model

22ndnd order (order (chiralchiral limit)limit)

Enhanced tail behavior in FRG : consistent w/ Saddle pointEnhanced tail behavior in FRG : consistent w/ Saddle point

Caveat : physical parameters are not the same as MF

6 Sep 2012 14/14

Kenji Morita

ERG2012@Aussois

Summary and Summary and OutlookOutlook

Probability Distribution P(N)Probability Distribution P(N)Calculating Calculating Calculating Calculating Calculating Calculating Calculating Calculating canonical partition functioncanonical partition functioncanonical partition functioncanonical partition functioncanonical partition functioncanonical partition functioncanonical partition functioncanonical partition function Z(T,V,N)Z(T,V,N)Z(T,V,N)Z(T,V,N)Z(T,V,N)Z(T,V,N)Z(T,V,N)Z(T,V,N)Relation to analytic structure of Relation to analytic structure of Relation to analytic structure of Relation to analytic structure of Relation to analytic structure of Relation to analytic structure of Relation to analytic structure of Relation to analytic structure of WW at at at at at at at at complex complex complex complex complex complex complex complex mm

Through the location of saddle pointsThrough the location of saddle points

Behavior of Behavior of Behavior of Behavior of Behavior of Behavior of Behavior of Behavior of cumulantscumulantscumulantscumulantscumulantscumulantscumulantscumulants : Shape of P(N): Shape of P(N): Shape of P(N): Shape of P(N): Shape of P(N): Shape of P(N): Shape of P(N): Shape of P(N)Higher order fluctuationsHigher order fluctuationsHigher order fluctuationsHigher order fluctuationsHigher order fluctuationsHigher order fluctuationsHigher order fluctuationsHigher order fluctuations –– Sensitive to Sensitive to Sensitive to Sensitive to Sensitive to Sensitive to Sensitive to Sensitive to large Nlarge Nlarge Nlarge Nlarge Nlarge Nlarge Nlarge N behaviorbehaviorbehaviorbehaviorbehaviorbehaviorbehaviorbehavior

Difference btw. MF and FRG calculationsDifference btw. MF and FRG calculations

Critical case : two saddle points lead to an Critical case : two saddle points lead to an Critical case : two saddle points lead to an Critical case : two saddle points lead to an Critical case : two saddle points lead to an Critical case : two saddle points lead to an Critical case : two saddle points lead to an Critical case : two saddle points lead to an Oscillatory factor; Oscillatory factor; Oscillatory factor; Oscillatory factor; Oscillatory factor; Oscillatory factor; Oscillatory factor; Oscillatory factor; shrinking P(N)shrinking P(N)shrinking P(N)shrinking P(N)shrinking P(N)shrinking P(N)shrinking P(N)shrinking P(N)Need mathematical techniqueNeed mathematical techniqueNeed mathematical techniqueNeed mathematical techniqueNeed mathematical techniqueNeed mathematical techniqueNeed mathematical techniqueNeed mathematical technique for wellfor wellfor wellfor wellfor wellfor wellfor wellfor well--------controlled computationcontrolled computationcontrolled computationcontrolled computationcontrolled computationcontrolled computationcontrolled computationcontrolled computation

Brute forte : Low T is very toughBrute forte : Low T is very tough……

Analytic method : asymptotic expansion method w/ cutsAnalytic method : asymptotic expansion method w/ cuts

Comparison w/ exp. Comparison w/ exp. –– incl. incl. PolyakovPolyakov looploop

Near CP?Near CP?