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  • 1. Towards Understanding QCD Phase Diagram Lattice and RHIC Experiments Atsushi Nakamura in Collaboration with K.Nagata Lattice QCD at finite temperature and density 20 Jan. 2014 KEK 1 /38 14年1月20日月曜日
  • 2. QCD Phase Diagram Very Exciting ! Now it’s time for Lattice QCD. But how do you reveal it? Please No Sales-Talk ! 14年1月20日月曜日 From Wiki-Pedia “QCD matter” 2
  • 3. Canonical Partition Functions Experiments Lattice QCD Simulations 3 / 38 14年1月20日月曜日
  • 4. A few years ago, Z(µ) = DU det (µ)e SG Nagata and I were looking for a Reduction formula for Wilson fermions in a finite density QCD project: det ˜ = det Reduction Matrix Original Fermion Matrix Rank(det ˜ ) < Rank (det ) Nagata and Nakamura Phys. Rev. D82 094027 (arXiv:1009.2149) 4 14年1月20日月曜日
  • 5. Obtained formula has the form of the fugacity expansion Z(µ) = DU cn n e SG n = Zn n n µ/T e Fugacity Zn : Canonical Partition Function 5 14年1月20日月曜日
  • 6. Z(µ, T ) Zn (T ) Z(µ, T ) = Tr e = If n|e ˆ (H µN )/T ˆ (H µN )/T n = n|e H/T n = Zn (T ) n µn/T |n e µ/T e Fugacity 6 /38 14年1月20日月曜日 n |n
  • 7. RHIC (Relativistic Heavy Ion Collider) 7 14年1月20日月曜日
  • 8. Multiplicity Distribution of RHIC Wao, Multiplicity ! Interesting ! It is almost Zn 14年1月20日月曜日
  • 9. We assume the Fireballs created in High Energy Nuclear Collisons are described as a Statistical System. with µ (chemical Potential) and T (Temperature) Z(µ, T ) Grand Canonical partition function All QCD Phase Information is in Z(µ, T ) 14年1月20日月曜日
  • 10. Experiment Partition Function is Sum of the Probabilities with n ... If I know , then I have Zn. number 10 /38 14年1月20日月曜日
  • 11. How can we extract Zn n from Pn = Zn ? Observables in Experiments Experiment unknown We require (Particle-AntiParticle Symmetry) Z+n = Z 14年1月20日月曜日 n
  • 12. From Experiment = 1.0 Z Zn n = 1.4 Z = 1.2 = 1.5 Zn Z n n Zn n n 12 /38 14年1月20日月曜日 Zn n 07:12-08:05 n n Z Final Value = 1.534
  • 13. Experiment Demand Z+n = Z n sqrt(s)=62.4 1 Fit 0.1 0.01 0.001 P-n sqrt(s)=62.4 1 Pn P-n Pn 0.01 Zn Z-n Z-n Zn 0.0001 0.0001 1e-06 1e-05 1e-08 1e-06 1e-07 -20 14年1月20日月曜日 -15 -10 -5 n 0 5 10 15 20 1e-10 -20 -15 -10 -5 0 5 10 Net proton number 15 20
  • 14. Fitted are very consistent with those by Freeze-out Analysis. 12 Chemical Freeze-Out x This work Freeze-out 10 ξ 8 6 4 2 0 µ/T e 14年1月20日月曜日 0 50 100 1/2 sNN 150 200 J.Cleymans, H.Oeschler, K.Redlich and S.Wheaton Phys. Rev. C73, 034905 (2006)
  • 15. Zn from RHIC data s = 19.6GeV s = 27GeV s = 39GeV 1 0.01 1 'Zn_27' 'Zn_19.6' 0.0001 0.01 0.0001 1e-06 1e-06 1e-08 1e-08 1e-10 1e-10 1e-12 'Zn_39' 0.01 0.0001 1e-06 Experiment 1e-12 1e-08 1e-10 1e-12 1e-14 1e-16 1e-18 -25 -20 -15 -10 -5 0 5 10 15 20 25 1e-14 -25 -20 -15 -10 -5 0 5 10 15 25 1e-14 -25 -20 -15 -10 -5 0 5 s = 200GeV s = 62.4GeV 0.1 20 1 'Zn_62.4' 'Zn_200' 0.01 0.1 0.001 0.01 0.0001 0.001 1e-05 1e-06 0.0001 1e-07 1e-05 1e-08 1e-06 1e-09 1e-10 -20 -15 -10 -5 0 5 10 15 20 15 /38 14年1月20日月曜日 1e-07 -15 -10 -5 0 5 10 15 10 15 20 25
  • 16. Now we have Zn of RHIC data (sqrt(s)= 10.5,19.6, 27, 39, 62.4, 200 GeV) Wao ! We can calculate at any density ! This includes all QCD Phase information ! T µ/T 14年1月20日月曜日 ( µ/T e )
  • 17. Do not forget that your n is finite ! I need cooling down 14年1月20日月曜日
  • 18. Moments k 14年1月20日月曜日 ⌘ ✓ @ T @µ ◆k k log Z
  • 19. What happens ? 0.6 Number Susceptibility p if we increase these points 15% if we drop these points s = 27GeV Number Susceptibility Usually we =27 GeV consider (only) here. 0.55 0.5 freeze-out point 1 4 0.8 2 0.6 0.4 0.45 =19.6 GeV 0.2 0.4 0.35 1.2 freeze-out point 0.7 14年1月20日月曜日 0.8 0.9 1 1.1 /T 1.2 1.3 0 1.4 1.5 -0.2 1 1.1 1.2 1.3 /T 1.4 1.5 1.6
  • 20. Susceptivility Number Susceptibility, sNN1/2=19.6 0.6 Number Susceptibility, sNN1/2=27 0.6 freeze-out point 0.55 0.55 0.5 0.5 0.45 0.45 0.4 0.35 freeze-out point 0.4 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 0.35 µ/T 0.7 0.8 0.9 Number Susceptibility, sNN1/2=39 0.65 freeze-out point 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.5 0.6 0.7 0.8 0.9 1 1.1 µ/T 0.65 1 1.1 1.2 1.3 1.4 1.5 µ/T Number Susceptibility, sNN1/2=62.4 0.6 freeze-out point 0.58 0.56 0.54 0.52 0.5 0.48 0.46 0.44 0.42 0.4 0.38 1/2 =200 1.2 Number Susceptibility, sNN 0.3 0.4 0.5 0.6 0.7 1.3 1.4 µ/T freeze-out point 0.6 0.55 0.5 0.45 0.4 14年1月20日月曜日 0.1 0.2 0.3 0.4 0.5 0.6 µ/T 0.7 0.8 0.9 1 0.8 0.9 1 1.1
  • 21. Kurtosis R42, sNN1/2=19.6 R42, sNN1/2=27 1.5 freeze-out point 1 freeze-out point 1.4 1.3 0.5 1.2 1.1 0 1 Usually we consider (only) here. -0.5 -1 0.9 1 1.05 1.1 1.15 1.2 µ/T 1.25 1.3 0.8 0.7 1.35 1.4 0.6 R42, sNN1/2=39 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.86 0.88 0.9 0.92 0.94 µ/T 0.96 0.98 1 R42, sNN1/2=62.4 1.1 freeze-out point R42, sNN1/2=200 1.2 freeze-out point 1 1 0.9 freeze-out point 0.8 0.8 0.6 0.7 0.4 0.6 0.2 0.5 0.6 0.7 0.8 µ/T 14年1月20日月曜日 0.9 1 0.5 0.35 0.4 0.45 0.5 µ/T 0.55 0.6 0.65 0.1 0.15 0.2 0.25 0.3 µ/T 0.35 0.4 0.45
  • 22. Multiplicity tells us Not only Freeze -out points Information of wider regions Nmax →large Wider 14年1月20日月曜日 22
  • 23. Lee-Yang Zeros 23 14年1月20日月曜日
  • 24. Lee-Yang Zeros (1952) Zeros of Z( ) in Complex Fugacity Plane. Z(↵k ) = 0 Great Idea to investigate a Statistical System x x x x x x 24 /38 14年1月20日月曜日 Phase Transition
  • 25. Lee-Yang Zeros Non-trivial to obtain. But once they are got, it is easy to figure out the Free-energy Z( , T ) = e F/T Lee-Yang zeros 2-d Electro-Magnetic F: Free-energy F: Potential k :Point charge k : zeros 25 14年1月20日月曜日
  • 26. ’out’ 20 15 10 5 0 -5 -10 1 -1 0.5 -0.5 0 0 0.5 F( ) = 1 -1 log( k 26 14年1月20日月曜日 -0.5 k)
  • 27. cut Baum-Kuchen (cBK) Algorithm and 1 1 i 50 - 100 number of significant digits 14年1月20日月曜日 27/38 (  Number of Zeros in Contour C ) A Coutour is cut into four pieces if there are zeros inside.
  • 28. Is this my Original ? I donot think so. Let us wait until someone claims. 28 14年1月20日月曜日
  • 29. Is this my Original ? I donot think so. Let us wait until someone claims. 28 14年1月20日月曜日 It’s me ! Riemann (1826 - 1866)
  • 30. Experiment Lee-Yang Zeros: RHIC Experiments cos(t), cos(t),sin(t) ’plotXY11.5’ 1 1/2 =19.6 0 -1 -0.5 0 sNN 1 0.5 1/2 -0.5 -1 -1 -1 1 -1 =39 -0.5 0 Re[ξB] sNN 1 0.5 1/2 1 -1 =62.4 0 sNN 1/2 0.5 1 =200 0 -0.5 -0.5 -0.5 -1 -1 -1 -1 14年1月20日月曜日 0 Re[ξB] 0.5 Im[ξB] 0 -0.5 1 0.5 Im[ξB] 0.5 =27 0 -0.5 -0.5 1/2 0.5 Im[ξB] Im[ξB] 0 sNN 1 0.5 0.5 Im[ξB] sNN 1 -0.5 0 Re[ξB] 0.5 1 -1 -0.5 29/40 0 Re[ξB] 0.5 1 -1 -0.5 0 Re[ξB] 0.5 1
  • 31. sNN 1 1/2 =200 Im[ξB] 0.5 0 -0.5 -1 -1 14年1月20日月曜日 -0.5 0 Re[ξB] 30 0.5 1
  • 32. Lee-Yang Zeros Lattice QCD Z( ) = 1 Z3m m T/Tc=0.99 Zn = 0 Im[ξ] 0.5 0 Periodicity -0.5 -1 -1 -0.5 0 Re[ξ] 0.5 1 = 1.85 T /Tc 0.99 31 /38 14年1月20日月曜日 3m unless 2 = 3 ( n = 3m = ei )
  • 33. T/Tc=0.99 1 Im[ξ] 0.5 0 = 1.85 0.99 = 1.87 T /Tc 1.01 = 1.89 -0.5 T /Tc T /Tc 1.04 -1 -1 -0.5 0 Re[ξ] 0.5 1 './plotXY_B1870' cos(t), sin(t) 1 0.5 0 -0.5 -1 -1 -0.5 0 0.5 1 './plotXY_B1890' cos(t), sin(t) 1 0.5 0 -0.5 -1 -1 14年1月20日月曜日 -0.5 0 0.5 1 32 /38
  • 34. Lee-Yang Zeros Lattice QCD 1 T/Tc=1.20 ⇠=e Im[ξ] 0.5 ⇠ Imaginary µ The Unit Cirle in 0 -0.5 -1 -1 -0.5 0 Re[ξ] T /Tc 0.5 Roberge-Weise Transition ! 1 1.20 ( 33 /38 14年1月20日月曜日 µ/T µ/T e )
  • 35. q 1 B = T/T =1.08 c B q q 3 Im[ q] 0.5 0 -0.5 3 -1 10 4 3 8 4 -1 14年1月20日月曜日 3 -0.5 0 Re[ q] 0.5 1 10 4 -1 34 /38 -0.5 0 Re[ B] 0.5 1
  • 36. A Message to Experimentalists In the Lee-Yang Diagram constructed from your multiplicity, Zeros here No Roberge-Weise Transition 14年1月20日月曜日 q B Your Temperature 35 /38 T TRW 1.2Tc
  • 37. Lee-Yang Zeros: RHIC Experiments sNN 1 1/2 =19.6 1/2 =27 0 =39 0 -0.5 -0.5 -0.5 -1 -1 -1 -1 -0.5 0 Re[ξB] 0.5 1 -1 sNN 1 1/2 -0.5 0 Re[ξB] =62.4 1 -1 sNN 1/2 -0.5 =200 0.5 Im[ξB] Im[ξB] 0.5 1 0.5 0 0 -0.5 -0.5 -1 -1 -1 14年1月20日月曜日 1/2 0.5 Im[ξB] 0 sNN 1 0.5 Im[ξB] 0.5 Im[ξB] sNN 1 -0.5 0 Re[ξB] 0.5 1 36/38 -1 -0.5 0 Re[ξB] 0.5 1 0 Re[ξB] 0.5 1
  • 38. Effects of Nmax Kim’s Model In Confinement Z(µq ) = I0 + 3 (⇠q + ⇠q 3 6 +(⇠q + ⇠q 6 )I2 + · · · )I1 Ik :Modified Bessel Lesson from the Model Nmax Large Lee Yang Zeros Large |µ| regions It should be so! 14年1月20日月曜日 37
  • 39. Summary Grand-Partition functions, Z(µ, T ) , provide us the QCD phase information, which can be constructed from Zn . Lattice QCD can calculate Zn But we need much more works to obtain reliable Experiments provide us the multiplicities We can calculate Zn from them. Present data are those of net-proton, which are not conserved quantities. Either correction, or ask experimentalists to measure net-baryon Charge multiplicity is a conserved quantity, and another probe. Large Nmax are wanted, but even finite Nmax data give us the lower bound. Lee-Yang zeros provide us a new tool of the QCD phase study. They are sensitive to the data, i.e., they teach us which regions are important. 38 14年1月20日月曜日
  • 40. Backup Slide 39 14年1月20日月曜日
  • 41. BES(Beem Energy Scan) 14年1月20日月曜日
  • 42. Hunting the QCD Phase Transition Regions Find Rooms where No Criminal. The Target is in other Room. Not here ! Then, .. 14年1月20日月曜日 Lower Bound
  • 43. 0.17 Temperature (GeV) 200 62.4 39 27 0.16 0.150 14年1月20日月曜日 19.6 Freeze-out Point Lower bound determined by susceptivility Lower bound determined by negative Kurtosis Phase Transition Regions estimated by Lee-Yang Zero distribution 0.1 0.2 0.3 Chemical Potential 0.4 (GeV) 0.5
  • 44. Other Messages Net proton multiplicity is Not a conserved quantity. Baryon multiplicity is perfect Can you estimate Baryon multiplicity from that of Proton ? Another conserved quantity is the Charge multiplicity. It should work as well. 43 14年1月20日月曜日
  • 45. You have a Big Chance to find QCD phase Transition ! 44 /40 14年1月20日月曜日
  • 46. Canonical Partition Functions is a Bridge between Two Approaches to Study QCD Phase Lattice QCD Simulaitions Experiments 45 /40 14年1月20日月曜日
  • 47. Lattice QCD Canonical Approach Miller and Redlich Phys. Rev. D35 (1987) 2524 A.Hasenfratz and Toussaint Nucl. Phys.B371 (1992) 539 Barbour and Bell Nucl. Phys. B372 (1992) 385 Engels, Kaczmarek, Karsch and Laermann Nucl.Phys. B558 (1999) 307 deForcrand and Kratochvila Nucl. Phys. B (P.S.) 153 (2006) 62 (hep-lat/0602024) A.Li, Meng, Alexandru, K-F. Liu PoS LAT2008:032 and 178 Phys.Rev. D82(2010) 054502, D84 (2011) 071503 Danzer and Gattringer arXiv:1204-1020 Europian Journal 46 /40 14年1月20日月曜日 Lattice
  • 48. Lattice: How to Calculate Fugacity Expansion Nagata and A. Nakamura, Phys. Rev. D82 (2010) 094027 Alexandru and Wenger, Phys.Rev.D83 (2011) 034502 47 /40 14年1月20日月曜日 Lattice
  • 49. Four Excuses why not Baryon Multiplicities 1. This is a formulation. Let’s wait until Experimentalists measure Baryon multiplicities 2. After the Freeze-out, the proton number is essentially constant. 3. Expect the proton multiplicity is similar to the baryon multiplicity 4. By some event generators or models, let us calculate the proton and baryon multiplicity. From that data, we can estimate the baryon multiplicity. 48 14年1月20日月曜日
  • 50. Lattice = DU ( an = DU ( = n n )(det (0)) an Zn )(det (0)) DU an (det (0)) n 49 /40 14年1月20日月曜日 e SG Nf Nf n ZGC (µ) = n Nf e e SG SG
  • 51. Lattice Zn from lattice QCD 1e-16 'Zn1850-orig' 1e-17 'Zn1850-orig' 1 1 1e-10 1e-20 1e-18 1e-20 1e-40 1e-30 1e-60 1e-40 1e-19 1e-80 1e-50 1e-100 1e-60 1e-120 1e-20 1e-70 15 16 17 18 19 20 m=3n 1e-140 1e-80 1e-160 -60 -40 -20 0 m=3n 20 40 60 1e-90 -50 -40 -30 -20 -10 0 10 20 30 40 Im(Zn ) are used as an error 50 m=3n 'Zn1900-orig' 'Zn1950-orig' 1 1 1e-05 'Zn2000-orig' 1 1e-05 1e-10 1e-10 1e-10 1e-20 1e-15 1e-15 1e-30 1e-20 1e-25 1e-20 1e-25 1e-40 1e-30 1e-30 1e-50 1e-35 1e-40 -30 1e-35 -20 -10 0 m=3n 10 20 30 1e-60 -40 -30 -20 -10 0 10 m=3n 50 /40 14年1月20日月曜日 20 30 40 1e-40 -30 -20 -10 0 m=3n 10 20 30
  • 52. A Strange Fact There are Lee-Yang Zeros on the unit circle. Theoretically, a bit annoying. Phenomenologically, very natural 51 /40 14年1月20日月曜日
  • 53. Z(µ) = det (mf , µf )e SG f det (m, µ) is REAL if µ is pure Imaginary. On the unit circle in complex plane. ( =e µ/T 52 14年1月20日月曜日 )
  • 54. det (m, µ) is REAL and Positive, if µ is pure Imaginary and m is sufficiently large. Z(µN ) = det (Nucleon)e SG >0 Lee-Yang zeros on the unit circle tell us that Nucleon is a composite. 53 14年1月20日月曜日
  • 55. Current lattice QCD simulations assumes mu = m d Z(µN ) = 2 (det (mq , µq )) det (ms , µs ) · · · e Z(µN ) can not take zero. 54 14年1月20日月曜日 SG
  • 56. mu > md µp = 2µu + µd Pure imaginary µp does not mean µu and µd are pure imaginary. 55 14年1月20日月曜日