QCD Thermodynamics at Intermediate Coupling. Nan Su. Frankfurt Institute for Advanced Studies

Transcription

1 Nan Su p. 1 QCD Thermodynamics at Intermediate Coupling Nan Su Frankfurt Institute for Advanced Studies Collaborators: Jens O. Andersen & Lars E. Leganger (NTNU), Michael Strickland (Gettysburg) Phys. Rev. Lett. 104, (2010) & JHEP 1008, 113 (2010) & Phys. Lett. B 696, (2011) & arxiv: [hep-ph] Quarks, Gluons, and Hadronic Matter under Extreme Conditions St. Goar, 17/03/2011 FIAS Frankfurt Institute for Advanced Studies

2 Nan Su p. 2 Introduction: Heavy ion collisions QGP or QGL? RHIC: T MeV 2T c. LHC: T MeV 4-6T c. Quark Gluon Plasma (QGP) or Quark Gluon Liquid (QGL) at LHC? Running coupling expected is g s 2 or α s 0.3. Neither infinitesimally small, nor infinitely large: intermediate coupling. Can pqcd methods reproduce lattice data for thermodynamic functions at such intermediate couplings (g s 2)? More importantly the resulting machinery should be able to address dynamics as well.

3 Nan Su p. 3 Intro: Nonconvergence of canonical thermal QCD The weak-coupling expansion of the QCD free energy, F, is known to order α 3 s logα s. 1,2,3,4,5,6 At temperatures expected at RHIC energies, T 0.35 GeV, the running coupling constant α s is approximately 0.3, or g s 2. The successive terms contributing to F can strictly only form a decreasing series if α s < 1/20 which corresponds to T 10 5 GeV. 1 Shuryak, Toimela, Arnold and Zhai, 94/95. Perturbative QCD free energy with N c = 3 and N f = 3 vs temperature. (πt µ 4πT ) (α s = g 2 s /4π) 4 Kastening and Zhai, Braaten and Nieto, Kajantie, Laine, Rummukainen and Schröder, 02.

4 Nan Su p. 4 Anharmonic Oscillator Consider the perturbation series for the ground state energy, E, of a simple anharmonic oscillator with potential V(x) = 1 2 ω2 x 2 + g 4 x4 (ω 2,g > 0) Weak-coupling expansion of the ground state energy E(g) is known to all orders (Bender and Wu, 69/73) ( g ) n E(g) = ω c BW n 4ω 3 c BW n n=0 rational coeff.: lim n c BW n = ( 1) n+1 6 π 3 3 n (n 1 2 )! Because of the factorial growth, the expansion is an asymptotic series with zero radius of convergence!

5 Anharmonic Oscillator Nan Su p. 5

6 Nan Su p. 6 Variational Perturbation Theory (VPT) (Janke and Kleinert, 95/97) Split the harmonic term into two pieces ( and treat the second as part of the interaction (with r 2 g ω 2 Ω 2) ) ω 2 Ω 2 + ( ω 2 Ω 2) N ( g ) n = E N (g,r) = Ω c n (r) 4Ω 3 n=0 c n not arbitrary: c n (c BW ) Fix Ω N : E N Ω Ω=ΩN = 0

7 Nan Su p. 7 Hard-Thermal-Loop Perturbation Theory (HTLpt) HTLpt is a reorganization of the perturbative series for QCD in spirit to VPT (Andersen, Braaten and Strickland, 99) L HTLpt = (L QCD +L HTL δl HTL ) g δg The Hard-Thermal-Loop (HTL) effective action reads ( L HTL = 1 y α 2 m2 D Tr y β ) G µα (y D) 2 G µ β δ counts the number of HTL dressed loops. Adding L HTL shifts the expansion to an ideal gas of massive quasiparticles, which are the appropriate d.o.f. at high T. Interested in T > 2 3T c. y

8 Nan Su p. 8 HTLpt: LO free energy for pure-glue QCD Separation into hard and soft contributions (d = 3 2ǫ) F g = 1 2 P { (d 1)ln[ HTL T (P)]+ln[ HTL L (P)] } P Hard momenta (ω,p T) F (h) g = d 1 2 P 2 ln(p 2 )+ 1 2 m2 D P 1 p 2 P 2 2d 1 p 4T P P 2 1 D 4(d 1) m4 P 1 p 2 P 2T P +d 1 p 4(T P) 2 ] [ 1 (P 2 ) 2 +O(m 6 D ) Soft momenta (ω,p gt) F (s) g = 1 2 T p ln(p 2 +m 2 D )

9 Nan Su p. 9 HTLpt: QCD diagrams through NNLO Π = Gluon self-energy insertion Dressed propagators Σ = Quark self-energy insertion Dressed vertex Γ = Vertex insertion F g 3a F g 3b F g 3c F g 3d g Π g = + + One-loop pure gauge contribution to gluon self-energy g F g 3e F g 3f F g 3g F g 3h Π Π Γ Γ F g 3i F g 3l Π F g 3j F g 3k F g 1a F g 1b F f 1b Π F g 3m F f 3a F f 3b F f 3c Π Σ Π Π Σ F g 2a F g 2b F g 2c F g 2d F f 3f F f 3g F f 3d F f 3e Σ Σ Γ Γ F f 2a F f 2b F f 2d Σ F f 3j F f 3k F f 3l F f 3i g g F f 3m F f 3n F f 3o

10 Nan Su p. 10 HTLpt: NNLO thermodynamic potential for QCD For QCD with general N c and N f (F ideal (N2 c 1)π2 T 4 45, ˆx D x Ω NNLO F ideal = [ d F da 15 4 ˆm3 D +c A α s 15 3π ˆm D ˆm2 D s F α s π +( ca αs 3π [ ζ ( 3) ζ( 3) +( ca αs 3π ( sf αs π 15 ) 2 [ 45 4 ) )( sf αs π ζ ( 1) ζ( 1) ( ) 2 [ ˆm D ˆm D 8 ( )[ 15 2 ζ ( 3) ζ( 3) ˆm D 12 ( ( log ˆµ γ E+2log2 ) ( log ˆµ γ E ˆm 3 D 90ˆm2 q ˆm D log ˆµ log ˆm D γ E ] )ˆm D log ˆµ γ E+log2 π ˆm D 16 ) ( log ˆµ γ E+2log2 ( ( ] ζ ( 1) ζ( 1) ) 2πT ) ˆm 3 D log ˆµ log ˆm D γ E log2 ) ] log ˆµ log2+γ E+ 9 ˆm 14 D +90 ˆm2 q ˆm D log ˆµ γ E log2+4 5 ) PURELY ANALYTIC!!! ζ ( 1) ζ( 1) 2 5 ζ ( 3) ζ( 3) ˆm D +30 ˆm2 q ˆm D ] +s 2F ( α s π ) 2 [ (35 32log2) 45 2 ˆm D ) ] ]

11 Nan Su p. 11 HTLpt: Mass prescriptions NNLO variational gap equations give complex m D and m q = 0. Weak-coupling expansion of Debye mass involves the nonperturbative magnetic scale g 2 T (Linde, 80; Gross, Pisarski and Yaffe, 81) and is IR divergent (Rebhan, 94; Arnold and Yaffe, 95). Use the gauge-invariant NLO electric mass from dimensional reduction: hard contribution (from the scale T ) to Debye mass and well defined to all orders (Braaten and Nieto, 96) m 2 D = 4πα { s 3 T2 c A +s F + c2 A α ( s 5 3π γ E log ˆµ ) 2 + c ( As F α s 3 π log γ E log ˆµ ) 2 + s2 F α ( s 1 π log2 2 3 γ E 2 3 log ˆµ ) 3 } s 2F α s 2 2 π

12 Nan Su p. 12 HTLpt: Pure-glue and full QCD free energies Pure-glue QCD QCD with N f = 3 Andersen, Strickland and Su, PRL 104, (2010) & JHEP 1008, 113 (2010) Andersen, Leganger, Strickland and Su, PLB 696, (2011) & arxiv: [hep-ph]

13 Nan Su p. 13 HTLpt: Pure-glue QCD energy and entropy From the free energy we can evaluate other thermodynamic variables using standard relations: P = F,E = F T df dt,s = df dt. Andersen, Strickland and Su, PRL, 104, (2010) & JHEP 1008, 113 (2010)

14 Nan Su p. 14 HTLpt: Pure-glue and full QCD trace anomalies Pure-glue QCD QCD with N f = 3 Andersen, Strickland and Su, JHEP 1008, 113 (2010) Andersen, Leganger, Strickland and Su, PLB 696, (2011) & arxiv: [hep-ph]

15 Nan Su p. 15 HTLpt: pure-glue high T pressure 1.1 Pressure/(Ideal Pressure) NLO HTLpt (Andersen, Strickland, Su 2009) NNLO HTLpt (Andersen, Strickland, Su 2009) Lattice Data (Boyd et al 1996) Lattice Data (Endrodi et al 2007) Lattice Data (Borsanyi et al 2010) T/T c

16 Nan Su p. 16 Conclusions and Outlook The poor convergence of weak-coupling expansion is generic: not just for field theory, but even for quantum mechanics. Generalized from VPT, HTLpt can improve the convergence of perturbative calculations in a gauge-invariant manner. The NNLO HTLpt results for pure-glue QCD look very good for T > 2 3 T c, and the full QCD ones are even better! Especially considering that there are no free parameters to fit. Since HTLpt is formulated in Minkowski space, it provides a systematic and self-consistent calculation scheme for both thermodynamics and dynamics. The NNLO QCD thermodynamics calculation sets the stage of generalizing HTLpt to dynamic quantities, such as viscosities, momentum diffusion, et al. Exploring the applications to other systems, e.g. cold atoms...

17 Back-up Nan Su p. 17

18 Nan Su p. 18 Weak-coupling expansion of pure-glue pressure Kastening and Zhai, 95 & Braaten and Nieto, 96

19 Nan Su p. 19 HTLpt: NNLO variational Debye mass Comparison of the real and imaginary parts of the NNLO variational Debye mass, with N c = 3 & N f = 3 Andersen, Leganger, Strickland and Su, arxiv: [hep-ph]

20 Nan Su p. 20 HTLpt: NNLO variational pressure Comparison of the real and imaginary parts of the NNLO variational pressure, with N c = 3 & N f = 3 Andersen, Leganger, Strickland and Su, arxiv: [hep-ph]

21 Nan Su p. 21 HTLpt: N c = 3 & N f = 3 variational pressure Andersen, Leganger, Strickland and Su, arxiv: [hep-ph]

22 Nan Su p. 22 HTLpt: QCD pressure with different m f Andersen, Leganger, Strickland and Su, arxiv: [hep-ph]

23 Nan Su p. 23 HTLpt: NNLO scale variation with different m f Andersen, Leganger, Strickland and Su, arxiv: [hep-ph]

24 Nan Su p. 24 HTLpt: QCD pressure at large N f Andersen, Leganger, Strickland and Su, arxiv: [hep-ph]

25 Nan Su p. 25 HTLpt: QCD pressure with different N f N f = 3 N f = 4 Andersen, Leganger, Strickland and Su, arxiv: [hep-ph]

26 Nan Su p. 26 HTLpt: QCD trace anomaly with different N f N f = 3 N f = 4 Andersen, Leganger, Strickland and Su, arxiv: [hep-ph]

27 Nan Su p. 27 Weak-coupling expansion of QED pressure Same nonconvergence pattern as the QCD case Parwani, 94; Parwani and Coriano, 95; Zhai and Kastening, 95; Andersen, 96.

28 Nan Su p. 28 HTLpt: QED pressure with different mass prescriptions Variational masses Perturbative masses Andersen, Strickland and Su, PRD 80, (2009)

29 Nan Su p. 29 Screened Perturbation Theory P P ideal loops 3 loops 4 loops (a) g(2πt) g 2 g 3 g 4 g 5 g 6 (b) 4-loop SPT pressure vs weak-couping pressure Andersen, Braaten and Strickland, 00. Andersen and Strickland, 01. Andersen and Kyllingstad, 08.