New results in QCD at finite µ

Transcription

1 New results in QCD at finite µ Rajiv Gavai and Sourendu Gupta ILGTI: TIFR XQCD 28, Duke University July 23, 28 sg (ILGTI: TIFR) New results at finite µ XQCD 8 1 / 37

2 Outline 1 The finite temperature transition 2 Quark Number Susceptibilities 3 Linkage 4 The Critical End Point 5 Series sums and Padé resummations 6 Summary sg (ILGTI: TIFR) New results at finite µ XQCD 8 2 / 37

3 Outline The finite temperature transition 1 The finite temperature transition 2 Quark Number Susceptibilities 3 Linkage 4 The Critical End Point 5 Series sums and Padé resummations 6 Summary sg (ILGTI: TIFR) New results at finite µ XQCD 8 3 / 37

4 The finite temperature transition Crawling towards the continuum Before this year: state of the art lattice computations of physics at finite chemical potential used lattice cutoff Λ = 4T 8 MeV near T c. Our earlier computation used m π 23 MeV and spatial sizes with LT = 2, 3, 4 and 6. This enabled extrapolation to the thermodynamic limit, i.e., L. Now: new computations with Λ = 6T 12 MeV near T c. m π remains unchanged (23 MeV), but spatial volumes are somewhat smaller (LT = 2, 3 and 4). No extrapolation to L yet. 2 5 configurations at each coupling; stochastic determination of traces with 5 random vectors on each configuration. (Gavai and SG, Phys. Rev. D 68, 23, 3456.) sg (ILGTI: TIFR) New results at finite µ XQCD 8 4 / 37

5 The finite temperature transition Locating the finite temperature cross over Cross-over coupling monitored using Polyakov Loop susceptibility: χ L, an operator which enters fourth-order QNS: (T/V) O 22 c, and an operator which enters eighth-order QNS: (T/V) O 44 c. Measures consistent with each other within the precision of this work. For m/t c =.1, we find β c = 5.425(5). Previous results bracketed this: for m/t c =.15 one had β c = 5.438(4) (Gottlieb et al, PRL 59, 1987, 1513) and for m/t c =.75 it was found that β c = ( Bernard et al, PR D 45, 1992, 3854). Transition is not first order. Computations at larger volumes are required to distinguish cross over from second order transition. sg (ILGTI: TIFR) New results at finite µ XQCD 8 5 / 37

6 The finite temperature transition Polyakov loop susceptibility.5.4 Ns=12 Ns=18 Ns=24.3 χ L β sg (ILGTI: TIFR) New results at finite µ XQCD 8 6 / 37

7 Not first order The finite temperature transition.3.2 χ max /N L 3 s N s sg (ILGTI: TIFR) New results at finite µ XQCD 8 7 / 37

8 Outline Quark Number Susceptibilities 1 The finite temperature transition 2 Quark Number Susceptibilities 3 Linkage 4 The Critical End Point 5 Series sums and Padé resummations 6 Summary sg (ILGTI: TIFR) New results at finite µ XQCD 8 8 / 37

9 What is a QNS? Quark Number Susceptibilities Taylor coefficient of the pressure in N f = 2 QCD is P(T, µ u, µ d ) = n u,n d 1 n u!n d! χ n u,n d (T)µ nu u µ n d d, and, since the two quark flavours are degenerate, χ nu,n d = χ nd,n u. Diagonal QNS have either n u = or n d =. In two flavour QCD trade µ u,d for µ B,Q. Then χ B = 2 9 (χ 2 + χ 11 ) = 2χ BQ Transforming to µ B,I3, one has χ Q = 1 9 (5χ 2 4χ 11 ). χ BI3 =, χ I3 = 1 2 (χ 2 χ 11 ). sg (ILGTI: TIFR) New results at finite µ XQCD 8 9 / 37

10 Off-diagonal QNS Quark Number Susceptibilities -.2 Nt=6 Nt=4 2 χ 11 /Τ T/Tc Sees only O 11. No evidence for lattice spacing effects. sg (ILGTI: TIFR) New results at finite µ XQCD 8 1 / 37

11 Diagonal QNS Quark Number Susceptibilities 2 χ 2 /Τ Nt=4 Nt= T/Tc Sees O 11 and O 2. Second expectation value is cutoff dependent. Also, has a cross over. We look at its susceptibility O 22 c to identify T c. sg (ILGTI: TIFR) New results at finite µ XQCD 8 11 / 37

12 Quark Number Susceptibilities Susceptibility of QNS: O 22 c 4th order QNS Nt=4 (T/V)<O 22 > Nt= T/Tc Peak at the same coupling as peak of χ L. Within the 1% precision of T/T c, the two quantities peak at the same coupling. See Gavai and Gupta, PR D72 (25) 546. sg (ILGTI: TIFR) New results at finite µ XQCD 8 12 / 37

13 Quark Number Susceptibilities Diagonal fourth-order QNS 8 7 Nt=4 χ Nt= T/Tc Non-zero for T > T c. Has contribution from O 4, which has non-vanishing value for the ideal gas. sg (ILGTI: TIFR) New results at finite µ XQCD 8 13 / 37

14 Quark Number Susceptibilities The operator O Nt=4 (T/V)<O 4 > Nt= T/Tc Rapid cross over from a small value in the hadronic phase to a non-vanishing value for the ideal gas. sg (ILGTI: TIFR) New results at finite µ XQCD 8 14 / 37

15 Quark Number Susceptibilities Susceptibility of O 4 : O 44 c 8th order QNS (T/V) <O 44 > T/Tc This quantity peaks at the same coupling as χ L and O 22 c. Within the precision of our measurement there is no dependence of the cross over coupling on these observables. sg (ILGTI: TIFR) New results at finite µ XQCD 8 15 / 37

16 Quark Number Susceptibilities The operator O 6 6th order QNS 2. (T/V)<O 6 > Nt=4.4 Nt= T/Tc The operator expectation value O 6 has structure below T c and hence its susceptibility cannot be used to probe the cross over coupling. Similar observation for O 8. sg (ILGTI: TIFR) New results at finite µ XQCD 8 16 / 37

17 Outline Linkage 1 The finite temperature transition 2 Quark Number Susceptibilities 3 Linkage 4 The Critical End Point 5 Series sums and Padé resummations 6 Summary sg (ILGTI: TIFR) New results at finite µ XQCD 8 17 / 37

18 Linkage Linkage between quantum numbers Linkage between quantum numbers F and G is C (FG) F = χ FG χ F = C (GF) F. Measures amount of G excited per unit fluctuation in F. The quantities C (FG) F and C (FG) G not necessarily equal. C (ud) u = 2/3 at low temperature, since the pion is the lightest excitation, and at high temperature it vanishes. For N f = 2 one has C (ud) u = C (ud) d. C (BQ) Q = at low temperatures (pion is the lightest particle) and 1/5 at high temperature. C (BQ) B = 1/2 at all temperatures by definition. sg (ILGTI: TIFR) New results at finite µ XQCD 8 18 / 37

19 Results Linkage.2 (ud) u C C (BQ) Q T/Tc T/Tc Quick cross over from hadron gas behaviour to quark gas behaviour. Rounding seen close to T c. Finite-size effects need to be investigated: LT = 4. Earlier computation with N t = 4 saw less rounding with LT = 6. sg (ILGTI: TIFR) New results at finite µ XQCD 8 19 / 37

20 Outline The Critical End Point 1 The finite temperature transition 2 Quark Number Susceptibilities 3 Linkage 4 The Critical End Point 5 Series sums and Padé resummations 6 Summary sg (ILGTI: TIFR) New results at finite µ XQCD 8 2 / 37

21 The Critical End Point Finite size effects At critical point correlation length becomes infinite, appropriate susceptibilities diverge and free energy becomes singular... in the infinite volume limit (van Hove s theorem). No numerical computation ever performed on infinite volumes. Deduce the existence of a critical point through extrapolations: finite size scaling (FSS) well developed for direct simulations. Example: peak of susceptibility scales as power of volume. Smaller effect: position of peak shifts from its infinite volume position by a different power of volume χ max (L) L p, T c (L) = T c a/l q, (p, q > ). FSS not well developed for series expansions; some aspects are known. sg (ILGTI: TIFR) New results at finite µ XQCD 8 21 / 37

22 Series expansions The Critical End Point ln c n n (L ) * 3 n (L ) * 2 L > L > L n (L 1) * n For a divergent quantity: χ(t, µ B ) = n c n(t)µ n B, the leading finite volume effects in the series coefficients. sg (ILGTI: TIFR) New results at finite µ XQCD 8 22 / 37

23 N t = 4 The Critical End Point 6 m π N s /T µ B Ns At fixed T/T c.95. Circles: ratio of order and 2; boxes: ratio of order 2 and 4. Gavai and SG, Phys. Rev. D 71, 25, sg (ILGTI: TIFR) New results at finite µ XQCD 8 23 / 37

24 The Critical End Point N t = 6: Radius of convergence µ/(3τ) LT=4; T/Tc= n Filled symbols: (χ () /χ (n) ) 1/n. Open symbols: χ (n 1) /χ (n+1). sg (ILGTI: TIFR) New results at finite µ XQCD 8 24 / 37

25 The Critical End Point N t = 6: Radius of convergence µ/(3τ) LT=4; T/Tc= n Filled symbols: (χ () /χ (n) ) 1/n. Open symbols: χ (n 1) /χ (n+1). sg (ILGTI: TIFR) New results at finite µ XQCD 8 25 / 37

26 The Critical End Point N t = 6: Radius of convergence µ/(3τ) LT=4; T/Tc= n Filled symbols: (χ () /χ (n) ) 1/n. Open symbols: χ (n 1) /χ (n+1). sg (ILGTI: TIFR) New results at finite µ XQCD 8 26 / 37

27 Radius of convergence The Critical End Point 5 4 Nt=6 Nt=4 3 µ (2) * /T T/Tc Lattice spacing dependence quantifies possible systematic errors. sg (ILGTI: TIFR) New results at finite µ XQCD 8 27 / 37

28 The Critical End Point N t = 6: Finite size scaling µ/(3τ) LT=2; T/Tc= n Filled symbols: (χ () /χ (n) ) 1/n. Open symbols: χ (n 1) /χ (n+1). sg (ILGTI: TIFR) New results at finite µ XQCD 8 28 / 37

29 The Critical End Point N t = 6: Finite size scaling µ/(3τ) LT=4; T/Tc= n Filled symbols: (χ () /χ (n) ) 1/n. Open symbols: χ (n 1) /χ (n+1). sg (ILGTI: TIFR) New results at finite µ XQCD 8 29 / 37

30 The Critical End Point Critical end point Multiple criteria agree: Stability of radius of convergence with order and estimator Pinching of the radius of convergence with T. Smallest T where all the coefficients are positive. Finite size effects roughly correct; more planned for the future. This gives T E T c =.94 ±.1 and µ E B = 1.8 ±.1 TE with Nf = 2 when m π /m ρ.3 at a finite volume with LT = 4 and lattice cutoff of a = 1/6T E. For a lattice cutoff of a = 1/4T E at the same renormalized quark mass and on the same volume we had found a similar value for T E /T c and µ E B /T E = 1.3 ±.3. Extrapolation to L reduced this to 1.1 ±.1. sg (ILGTI: TIFR) New results at finite µ XQCD 8 3 / 37

31 Outline Series sums and Padé resummations 1 The finite temperature transition 2 Quark Number Susceptibilities 3 Linkage 4 The Critical End Point 5 Series sums and Padé resummations 6 Summary sg (ILGTI: TIFR) New results at finite µ XQCD 8 31 / 37

32 Series sums and Padé resummations Fluctuations of Baryon number Suggestion by Stephanov, Rajagopal, Shuryak; Asakawa, Heinz, Muller; Jeon, Koch ) P( B) = exp ( ( B)2. 2VTχ B Extrapolate χ B to finite chemical potential: peak at T c? Series coefficients for χ B T/Tc sg (ILGTI: TIFR) New results at finite µ XQCD 8 32 / 37

33 Sum the series Series sums and Padé resummations 5 4 n=4 χ Β /Τ n= µ/τ Summing the series never shows critical behaviour: sum is a polynomial and smoothly behaved. The sum peaks at T c : incorrect (see SG, SEWM 26). sg (ILGTI: TIFR) New results at finite µ XQCD 8 33 / 37

34 Critical fluctuations Series sums and Padé resummations 5 P 1 4 χ Β /Τ P µ/τ Use Padé approximants for the extrapolations: divergence at the critical end point (see Lombardo, Mumbai 25). Error propagation requires care: see arxiv: [hep-lat]. sg (ILGTI: TIFR) New results at finite µ XQCD 8 34 / 37

35 Outline Summary 1 The finite temperature transition 2 Quark Number Susceptibilities 3 Linkage 4 The Critical End Point 5 Series sums and Padé resummations 6 Summary sg (ILGTI: TIFR) New results at finite µ XQCD 8 35 / 37

36 Summary Summary 1: finite temperature 1 Simulations of N f = 2 QCD (staggered quarks, Wilson action) with renormalized quark mass m π 23 MeV with 1/a 12 MeV and LT = 2, 3 and 4. 2 Finite temperature cross over located at β c = 5.425(5), consistent with previous computations at neighbouring masses. Consistent measurements obtained with χ L, (T/V) O 22 c and (T/V) O 44 c within precision of this computation. 3 Cutoff artifacts seen in many QNS. Surprisingly, measurements are more well-behaved at smaller lattice spacing (see χ 6 and χ 8, for example). sg (ILGTI: TIFR) New results at finite µ XQCD 8 36 / 37

37 Summary Summary 2: finite chemical potential 1 Very stable estimate of the critical end point: three criteria agree. T E T c =.94 ±.1 and µ E B = 1.8 ±.1 T E with lattice cutoff of a = 1/6T E 11 MeV, compared to µ E B /T E = 1.3 ±.3 at a = 1/4T E 75 MeV on the same spatial volume. 2 Series extrapolation needs resummation: Padé approximants are one possible resummation. 3 Linkage between u and d quantum numbers disappears at T T c when µ B =. How abrupt? Requires finite size scaling. sg (ILGTI: TIFR) New results at finite µ XQCD 8 37 / 37