Reorganizing the QCD pressure at intermediate coupling

Transcription

1 Reorganizing the QCD pressure at intermediate coupling Michael Strickland Gettysburg College and Frankfurt Institute for Advanced Studies (FIAS) Collaborators: Nan Su (FIAS) and Jens Andersen (NTNU) Reference: arxiv: and forthcoming University of Virginia, Charlottesville, VA 27 Oct 2009

2 Introduction - Heavy Ion Collisions QGP or QGL? RHIC has made extensive studies of the matter generated during heavy ion collisions, T MeV 2T c. LHC will continue this investigation at even higher temperatures, T MeV 4 5T c. Early RHIC data hinted that the perturbative approach was insufficient to explain observations, and that a strongly-coupled nearly perfect liquid may be more appropriate. LHC? Should we be surprised since at RHIC and LHC the running coupling expected is g s 2 or α s 0.3? Strong coupling limit has some very nice features, but g s. Can perturbative QCD results reproduce lattice data thermodynamic functions at such intermediate couplings (g s 2)?

3 Introduction - Perturbative Thermodynamics The weak-coupling expansion of the QCD free energy, F, has been calculated to order α 3 s log α s. 1,2,3,4 At temperatures expected at RHIC energies, T 0.3 GeV, the running coupling constant α s (2πT) is approximately 1/3, 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 Arnold and Zhai, 94/95. Perturbative QCD free energy vs temperature. (πt µ 4πT ) QCD with N c = 3 and N f = 2. 4-d lattice results from Karsch et al, 03. (Here α s = g 2 s /4π) 2 Kastening and Zhai, Braaten and Nieto, Kajantie, Laine, Rummukainen and Schröder, 02.

4 Introduction - NLO HTLpt result Hard-thermal-loop (HTL) perturbation theory 4,5 is a systematic, self-consistent and gauge-invariant reorganization of thermal quantum fields. Hard-thermal-loop perturbation theory is formulated in Minkowski space, therefore it is in principle possible to carry out real time calculations. Interested in T > 2 3 Tc. 4 Andersen, Braaten, Strickland, 99/99/99. 5 Andersen, Braaten, Petitgirard, Strickland, 02; LO and NLO HTLpt free energy of QCD with N c = 3 and N f = 2 Andersen, Petitgirard, Strickland, 03. together with the perturbative prediction accurate to g 5.

5 But there is still work to do! Problems remain: g 4 and g 5 terms can t be fully fixed at NLO. For example, when the NLO HTLpt is expanded in a truncated series in g, it is found that the g 5 term has approximately the right magnitude, but the wrong sign when comparing to the known weak-coupling expansion. Running coupling doesn t enter at NLO. At this order, running coupling needs to be put in by hand. Can be fixed by going to NNLO. Time to roll up your sleeves...

6 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, c BW 1 4ω 3 n = 2, 3 4, 21 8, 333 } 16, ,... n=0 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!

7 Anharmonic Oscillator

8 Variational Perturbation Theory (Janke and Kleinert 95/97) Split the harmonic term into two pieces and treat the second as part of the interaction ω 2 Ω 2 + ( ω 2 Ω 2) N ( g ) n = E N (g,r) = Ω c n (r) 4Ω 3 n=0 where r 2 g ( ω 2 Ω 2) The new coefficients c n can be obtained by ( ) n c n (r) = c BW (1 3j)/2 j (2rΩ) n j n j j=0 Fix Ω N by requiring that at each order N E N Ω = 0 Ω=ΩN

9 Variational Perturbation Theory

10 Finite Temperature QED/QCD Primer Long-wavelength chromoelectric fields with momentum k λ 1 gt are screened by an induced mass called the Debye mass m D. At high temperatures particles massive quasiparticles k gt defines the soft scale, k T defines the hard scale. The inverse Debye mass is called the Debye screening length, ie r D = 1/m D. V Coloumb V Debye e m Dr r e r/r D r r D - + -

11 Hard Thermal Loops: Propagator Resummation

12 Finite Temperature QED/QCD Primer At leading order in the coupling constant m 2 D = g2 T 2 for QCD and m 2 D = e2 T 2 /3 for QED; however, this is not the end of the story. Since QCD and QED are gauge theories, there are relationships between the n-point functions which must be maintained in order to preserve gauge invariance. These are called Ward-Takahashi or Slavnov-Taylor identities, e.g. p µ Γ µ (p,q,r) = S 1 (q) S 1 (r) must be obeyed by the fermion-gauge field vertex function Γ µ and propagator S. All n-point functions of the theory must be consistently derived. L HTL = 1 2 m2 DTr ( G µα y α y β (y D) 2 y G µ β )

13 Finite Temperature QED/QCD Primer L YM + L HTL = 1 4 Gµν G µν 1 2 m2 DTr ( G µα y α y β (y D) 2 y G µ β ) G µν = µ A ν ν A µ + ig[a µ,a ν ] D µ = µ + iga µ Expanding to quadratic order in A gives propagator (2-point function) Expanding to cubic order in A gives dressed gluon three-vertex Expanding to quartic order in A gives gressed gluon four-vertex Contains an infinite number of higher order vertices

14 Hard-Thermal-Loop Perturbation Theory (HTLpt) Hard-thermal-loop perturbation theory is a reorganization of the perturbative series for QCD which is similar in spirit to variational perturbation theory L HTLpt = (L QCD + L HTL ) + L HTL (g,m 2 D(1 δ)) g δg The HTL improvement term is L HTL = 1 2 (1 δ)m2 DTr ( G µα y α y β (y D) 2 y G µ β ) where y indicates angle average

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

16 HTLpt: 1- and 2-loop diagrams for QCD 1- and 2-loop QCD diagrams contributing to HTLpt

17 HTLpt: 1- and 2-loop free energy for pure glue NLO Approximately Self-Consistent HTL Phi-Derivable (Blaizot, Iancu, Rebhan) Hard Thermal Loop Perturbation Theory (Andersen, Braaten, Petitgirard, Strickland) 4d Lattice "Pure Glue" (Boyd et al) LO and NLO HTLpt free energy of pure glue vs temperature Andersen, Braaten, Petitgirard, Strickland, 02.

18 HTLpt: naive pert. expansion of QED free energy Perturbative QED free energy

19 HTLpt: 1- and 2-loop diagrams for QED 1- and 2-loop QED diagrams contributing to HTLpt

20 HTLpt: 3-loop diagrams for QED 3-loop QED diagrams contributing to HTLpt

21 HTLpt: 3-loop diagrams for QED 3-loop HTLpt QED diagrams which can be neglected in our approach since we make a dual expansion in e and m D assuming m D e at leading order.

22 HTLpt: 3-loop thermodynamic potential for QED The NNLO thermodynamic potential reads { Ω NNLO = π2 T N f 15 4 ˆm3 D +N f α π [ N f ( α π) 2[ 15 +N 2 f( α π) 2[ ˆm D+15(log ˆµ γ+2log 2 ) ˆm 3 D 90 ˆm D ˆm 2 f 64 (35 32 log 2) 45 2 ˆm D log ˆµ γ log PURELY ANALYTIC! ] ζ ( 1) ζ( 1) 2 5 ζ ( 3) ζ( 3) ]} 1 15(log ˆµ ˆm D γ+2 log 2 ) ˆm D To eliminate the m D and m f dependence, the gap equations are imposed m D Ω(T,α,m D,m f,δ = 1) = 0 m f Ω(T,α,m D,m f,δ = 1) = 0 ]

23 HTLpt: 2- and 3-loop free energy for QED NLO and NNLO HTLpt predictions for QED free energy

24 HTLpt: comparison of different methods/schemes Comparison of three different predictions for the QED free energy at µ = 2πT 3-loop Φ-derivable result is taken from Andersen and Strickland, 05

25 NNLO HTLpt thermodynamic functions for pure glue Together with Jens Andersen (Trondheim, Norway) and my graduate student Nan Su (Frankfurt Institute for Advanced Studies, Frankfurt Germany) we have recently completed a three loop calculation of the HTLpt thermodynamic potential Ω NNLO = 1 15 F ideal 4 + N cαs 3π ˆm 3 D + N cαs 3π! 2 " 45 4 " ˆm D ˆm D 135 log ˆµ ˆµ ˆm 2 D log ˆm log ˆµ 2 + γ + log 2 π γ A ˆm D # + γ! ˆm 3 D ζ ( 1) ζ( 1) # ζ! ( 3) ζ( 3) For the Debye mass above we use the NLO expression for Π 00 (P = 0) which was derived using effective field theory methods (Braaten and Neito, 1995). For α s we use the standard 3-loop running.

26 NNLO Pressure (Andersen, Strickland, Su, forthcoming) 1 Pressure / (Ideal Pressure) Lattice Data (Boyd et al) LO HTLpt NLO HTLpt NNLO HTLpt T/T c

27 NNLO Energy (Andersen, Strickland, Su, forthcoming) (Energy Density) / (Ideal Energy Density) Lattice Data (Boyd et al) LO HTLpt NLO HTLpt NNLO HTLpt T/T c

28 NNLO Entropy (Andersen, Strickland, Su, forthcoming) 1 Entropy / (Ideal Entropy) Lattice Data (Boyd et al) LO HTLpt NLO HTLpt NNLO HTLpt T/T c

29 Conclusions and Outlook The problem of bad convergence of finite temperature weak-coupling expansion is generic. It does not just occur in gauge theories, but also in scalar theories, and even in quantum mechanics. Variational perturbation theory and hard-thermal-loop perturbation theory can improve the convergence of perturbative calculations in a gauge-invariant manner which is formulated in Minkowski space. The NNLO results for pure-glue SU(3) Yang-Mills look very good for T > 2 3T c! Especially considering that there are no free parameters to play with. Once the NNLO full QCD thermodynamics is obtained (COMING SOON!) we can start trying to use the HTLpt reorganization to calculate dynamic quantities such as momentum diffusion, viscosities, etc.