QCD Equation of state in perturbation theory (and beyond... )

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1 QCD Equation of state in perturbation theory (and beyond... ) Kari Rummukainen CERN Theory & University of Oulu, Finland Physics of high baryon density K. Rummukainen (Oulu & CERN) EoS in pqcd Trento 06 / 22

2 QCD is perturbative if T, µ Λ QCD Weakly interacting quarkgluon plasma, with g 2 True: Asymptotic coupling constant expansion: T quark gluon plasma p/t 4 = g 0 +g 2 +g 3 +g nuclear matter superconductivity? False: Coefficients of the expansion are not calculable in loop expansion beyond some low order, due to infrared singularities. For pressure, this happens at order g 6 : 3-dim confinement! [Linde 980] Perturbation theory and lattice nicely complement each other µ K. Rummukainen (Oulu & CERN) EoS in pqcd Trento 06 2 / 22

3 Status (pressure = free energy): p/t 4 can be calculated only up to O(g 6 ln /g) in P.T. p(t, µ = 0) fully known [Kajantie, Laine, K.R., Schröder 02] p(t, µ i ) also known fully (T > gµ?) [Vuorinen 03] p(t = 0, µ i ) known to order O(g 4 ) [Freedman, McLerran 77] p(t, µ i ) for all T known to order O(g 4 ) [Ipp, Kajantie, Rebhan, Vuorinen 06] To go beyond O(g 6 ln /g) requires non-perturbative input (3-dim lattice, partly done... ) Bad convergence for T < 00 T c! Far from ideal gas (Stefan-Bolzmann) Still missing: all of the above is with m q = 0! K. Rummukainen (Oulu & CERN) EoS in pqcd Trento 06 3 / 22

4 Resummed perturbation theory Bad convergence in std. expansion reshuffle to improve Screened P.T. (not suitable for QCD) [Karsch, Patkos, Petreczky] Hard Thermal Loop P.T. [Andersen, Braaten, Petitgirard, Strickland] Φ-derivable approach [Blaizot, Iancu, Rebhan; Peshier] Aim: improved convergence at larger couplings (low T ) Typically include quasiparticles, Debye screening, damping etc. from the outset: more physical thermal vacuum to expand around? Reproduce std. perturbative expansion to some (low) order, depending on truncations, but include contributions from all higher orders. Non-perturbative magnetostatic sector (g 6, 3d confinement), is not included. If this is important these approaches fail (as does normal P.T.) K. Rummukainen (Oulu & CERN) EoS in pqcd Trento 06 4 / 22

5 Why precise equation of state at T > T c is desirable? Cosmology: precision measurements require < % level accuracy in EoS HIC: input for hydro; susceptibilities CBM: boundary condition for models(?) plain theoretical interest K. Rummukainen (Oulu & CERN) EoS in pqcd Trento 06 5 / 22

6 Lattice QCD Standard lattice QCD works very well when T < 5 0T c : 5 4 p/t 4 p SB /T flavour 2+ flavour 2 flavour pure gauge T [MeV] [Karsch 200] Pure gauge N f = 0: p(t T c ) 0 since glueballs heavy N f = 2, 3: at T < T c gas of (light) pions p (T c ) discontinuous genuine st order phase transition ( for pure gauge QCD) Lattice QCD runs out of steam at T 5 0T c! Hierarchy of scales Λ QCD T /a. Relation to perturbation theory? K. Rummukainen (Oulu & CERN) EoS in pqcd Trento 06 6 / 22

7 Pressure in perturbation theory The perturbative expression for the pressure is fully known (m q = 0): p/p SB = Stefan-Boltzmann +g 2 2-loop [Shuryak 78] +g 3 resummed 2-loop [Kapusta 79] +g 4 ln /g resummed 2-loop [Toimela 83] +g 4 resum 3-loop [Arnold, Zhai 94] +g 5 resum 3-loop [Kastening, Zhai 95] +g 6 ln /g resum 4-loop [Laine, Kajantie, K.R., Schroeder +g 6 not computable! [Linde 80] +g 7 + g 8 computable +g 9 + g not computable p SB = π2 T 4 45 (8 2N f 4 ) K. Rummukainen (Oulu & CERN) EoS in pqcd Trento 06 7 / 22

8 P.T. alone does not provide the answer:.5.0 p/p SB Generally bad convergence T/Λ_ MS g 2 g 3 g 4 g 5 g 6 (ln(/g)+0.7) 4d lattice [Laine, Kajantie, K.R, Schroeder 02] K. Rummukainen (Oulu & CERN) EoS in pqcd Trento 06 8 / 22

9 Dependence on g 6 -coefficient:.5.0 p/p SB 0.5 g 6 (ln(/g)+.5) g 6 (ln(/g)+.0) g 6 (ln(/g)+0.5) g 6 (ln(/g)+0.0) g 6 (ln(/g) 0.5) 4d lattice T/Λ_ MS Non-perturbative effects [parametrized here with g 6 (const.)] can be very significant! K. Rummukainen (Oulu & CERN) EoS in pqcd Trento 06 9 / 22

10 Loops at finite T Finite T ensemble: euclidean with imaginary time extent /T, with (anti)periodic b.c for bosons (fermions) p 2 { 2nπT n Z Bosons p 2 + ωn 2, ω n = (2n + )πt n Z Fermions d 4 p (2π) 4 T d 3 p (2π) 3 n Thus, all n 0 Bosonic modes and all Fermionic modes acquire a mass πt. Clearly, only bosonic n = 0 modes are infrared sensitive. /T K. Rummukainen (Oulu & CERN) EoS in pqcd Trento 06 0 / 22

11 Why non-analytic in g 2? Consider 3-loop contribution to the pressure (superficially g 4 ), and n = 0 modes. Now the middle loop is IR divergent: T n d 3 p n=0 dp p 2 [ p 2 + (2πnT ) 2 ] 2 0 [p 2 ] 2. Need to resum. Consider -loop polarization (of electric gluon modes) m 2 D g 2 T n d 3 p p 2 + (2πnT ) 2 g 2 T 2 + UV-divergent Electric modes are screened by Debye mass m D. Insert m D on all propagators: [g 2 T 2 ] 2 T n=0 d 3 p [ p 2 + m 2 D ]2 g 4 T 5 /m D g 3 T 4 K. Rummukainen (Oulu & CERN) EoS in pqcd Trento 06 / 22

12 Effectively, this means that we need to (re)sum over all diagrams of type All of these contribute to order g 3! K. Rummukainen (Oulu & CERN) EoS in pqcd Trento 06 2 / 22

13 Why g 6 T 4 is non-perturbative? Debye mass m D does not regulate magnetic sector! Let us consider vacuum diagram at finite T, where we add a (fictitious) mass term m to keep track of the scale: [Linde 80] N loops { (N ) 4-vertices (2N 2) propagators [ T [ ] d 3 p] N (g 2 ) N 2N 2 [ g q 2 + m 2 g 6 T 4 2 T m ] N 4 If m = g 2 T ( magnetic scale), all loops contribute to pressure at g 6! Perturbatively m = 0 for magnetic gauge modes Loop expansion fails when k< g 2 T (3d confinement). Note: Loop expansion OK when k πt (hard scales, n 0) k gt (Debye, electric scales) K. Rummukainen (Oulu & CERN) EoS in pqcd Trento 06 3 / 22

14 3d effective theories An efficient way to implement resummations and organize the perturbative (and lattice) calculations 2 g T g T π T Energy scales: non perturbative perturbative QCD 3D "electric" theory _ 2 3D "magnetic" theory L = F 4 ij Hierarchy of effective theories [Braaten,Nieto 95] Integrate over πt (fermions!) 3-dim. effective theory L E for gt, g 2 T -modes Integrate over gt 3-dim. effective theory L M for g 2 T -modes ( integrate 2-loop optimized matching of theories) K. Rummukainen (Oulu & CERN) EoS in pqcd Trento 06 4 / 22

15 Electric effective theory L E is 3d adjoint Higgs model: L E = 2 Tr F 2 ij + Tr[D i, A 0 ] 2 + m 2 D Tr A2 0 + λ A (Tr A 2 0) 2 For N f = 0 (pure gauge), µ = 0 the couplings are g 2 3 = g 2 T = m 2 D g 2 T 2 λ A g 4 T 8π 2 ln(6.742t /Λ MS ) T L E can be analyzed using perturbation theory or lattice simulations. Pressure is accurate to order g 6! Magnetic L M is just 3d QCD: L M = 2 Tr F 2 ij with coupling constant g 2 3 g 2 T. K. Rummukainen (Oulu & CERN) EoS in pqcd Trento 06 5 / 22

16 Pressure: pert. theory p = + g 2 + g 3 + g 4 ln p SB g + g 4 + g 5 + g 6 ln g + g 6 + g L E L M The relation between physical pressure and 3d free energy is [(Braaten,Nieto)] p = 5 λ A p SB 2 g ( ) g π 2 (F E + (known)) T where F E = Vg 6 3 ln F E can now be calculated perturbatively g 6 ln g non-perturbatively higher order [da] exp[ d 3 xl E ] K. Rummukainen (Oulu & CERN) EoS in pqcd Trento 06 6 / 22

17 (skeletons) = loop graphs for g 6 ln g : , # of diagrams 3:6:47:490 at :2:3:4 loops One 4-loop diag. with 6 3-point vertices and 9 propagators contains terms, to be integrated over! (2 9 (3 2) 6 ) Computer algebra program is a must to sort out the integrals (FORM) (rings) = , = ,, = = + 2, 2 = , + 2 =, 2 = , , 2 = +, 2 = = + K. Rummukainen (Oulu & CERN) EoS in pqcd Trento 06 7 / 22

18 Non-perturbative g 6 -contribution g 6 coefficient is of interest: first genuinely non-perturbative coefficient; can have significant quantitative effect. The cleanest way to obtain the genuinely 3-dimensional g 6 T 4 contribution is to use L M, i.e. 3d pure gauge QCD. It is directly related to the condensate F 2 [Braaten, Nieto 95; Karsch, Lütgemeier, Patkos, Rank 96] ij. This can be determined on the lattice. [Hietanen et al. 04] However, this requires 4-loop lattice continuum matching: Fij 2 lat = a 3 + g 2 T a 2 + (g 2 T ) 2 a + (g 2 T ) 3 (ln g 2 + [matching coeff.] + [non-pert. physics]) Ta This has not been done yet (stochastic perturbation theory?) K. Rummukainen (Oulu & CERN) EoS in pqcd Trento 06 8 / 22

19 Lattice simulations of L E Lattice simulations of the electric theory L E give information about higher order contributions: g 7 + g p/p d lattice T/Λ MS g 6 has to be tuned matching to 4d lattice Not good enough accuracy yet [Kajantie, Laine, K.R., Schroeder 0] K. Rummukainen (Oulu & CERN) EoS in pqcd Trento 06 9 / 22

20 Hard Thermal Loop Perturbation Theory Lagrangian: δ = is QCD L = [ L QCD + ( δ)l HTL ]g 2 δg 2 L HTL is constructed so that already the free propagator (δ = g 2 = 0) gives thermal quasiparticles, screening (m D ), Landau damping... more physical starting point, better behaved expansion Aim: bring validity of expansion close to T c Calculations technically very difficult already at 2-loop level! K. Rummukainen (Oulu & CERN) EoS in pqcd Trento / 22

21 N f = 0 QCD with HTLPT at (light gray) and 2 (dark gray) loops: Compared with 4d lattice (dots) and 3d lattice (L E, dotted lines) Good convergence, overshoots the 4d lattice result Formally accurate to order g 4 ln /g [Andersen, Braaten, Petitgirard, Strickland 02] K. Rummukainen (Oulu & CERN) EoS in pqcd Trento 06 2 / 22

22 Conclusions P.T. has been pushed as far as it goes (for p(t, µ)) Slow convergence; large order-by-order fluctuations System strongly coupled Not close to Stefan-Bolzmann (0T c : 2%, 00T c : 7%) Large perturbative deviations from ideal gas (can also be non-perturbative) To be done: g 6 non-perturbative contribution mq 0 K. Rummukainen (Oulu & CERN) EoS in pqcd Trento / 22