Yang-Mills Propagators in Landau Gauge at Non-Vanishing Temperature

Transcription

1 Yang-Mills Propagators in Landau Gauge at Non-Vanishing Temperature Leonard Fister, Jan M. Pawlowski, Universität Heidelberg... work in progress ERG Corfu - September 2

2 Motivation ultimate goal: computation of physical observables from microscopic dynamics experiment: thermodynamic potential, pressure, entropy, screening masses, etc. study QCD phase diagram fully non-perturbatively: functional renormalisation group credits: GSI Darmstadt applicable for all temperatures

3 Outline Motivation Yang-Mills Flow Equation Thermal Flow Equation Propagators at Non-Vanishing Temperature Summary 6 the Gs. utvia the off, ion hat tes on Z(p 2 ) 2 Bowman (24) Sternbeck (26) scaling (DSE) decoupling (DSE) scaling (FRG) decoupling (FRG) gluon dressing function G(p 2 ) Sternbeck (26) scaling (DSE) decoupling (DSE) scaling (FRG) decoupling (FRG) ghost dressing function S ent s in ive cus in ] 2 p [GeV] Bowman (24) Sternbeck (26) scaling (DSE) decoupling (DSE) p [GeV] taken from: Fischer, Maas, Pawlowski, Annals Phys. 324 (29).

4 Yang-Mills Theory - Basics Yang-Mills action: S YM = d 4 x 4 F a µνf a µν + 2ξ ( µa a µ) 2 + c a µ D ab µ c b Landau gauge: ξ covariant derivative: D a µν = δ ab µ + gf abc A c µ field-strength tensor: F a µν = µ A a ν ν A a µ gf abc A b µa c ν effective action Γ[A, c,c] Z[J, η, η] e W [J,η, η] = Γ[A, c, c] = sup J,η, η DAD cdc e S[A, c,c]+ (J A+ η c c η) (J A + η c c η) W [J, η, η]

5 Flow Equation (for Yang-Mills Theory) Wetterich, Phys. Lett. B3 (993) t Γ k [A, c, c] = 2 Tr t R k t C k [A, c, c]+r k

6 Flow Equation (for Yang-Mills Theory) Wetterich, Phys. Lett. B3 (993) t Γ k [A, c, c] = 2 Tr t R k t C k [A, c, c]+r k t = k ln k full propagator regulator t Γ[A, c, c] = 2

7 Yang-Mills Propagators obtained from generating flow equation via functional derivation wrt the in-/out-going fields t = + /2 + t = /2 +

8 Yang-Mills Propagators ( truncated ) obtained from generating flow equation via functional derivation wrt the in-/out-going fields t = + /2 + t = /2 +

9 Yang-Mills Propagators - Parametrisation zero temperature: ghost propagator gluon propagator Dgh(p) ab = G(p) p 2 δab Dgl,µν(p ab 2 Z(p 2 ) ) = Π µν p 2 δ ab, Π µν...transversal 4d-projector

10 Yang-Mills Propagators - Parametrisation zero temperature: ghost propagator gluon propagator Dgh(p) ab = G(p) p 2 δab Dgl,µν(p ab 2 Z(p 2 ) ) = Π µν p 2 δ ab, Π µν...transversal 4d-projector finite temperature (Matsubara formalism): p =2πTn p, n p... Matsubara modes ghost propagator gluon propagator D ab gl,µν(p 2, p 2 ) = δ ab P T µν Z T (p 2, p 2 ) p 2 + p 2 + δ ab P L µν Z L (p 2, p 2 ) p 2 + p 2 transversal- longitudinal- magnetic 3d projector electric

11 Flow Equation for Thermal Fluctuations at non-vanishing temperature: quantum and thermal fluctuations idea: () calculate quantum fluctuations at zero temperature (2) project onto thermal fluctuations and add to () thermal flow: Γ k,t [A, c, c] =Γ k,t Γ k,t = Litim, Pawlowski, arxiv: hep-th/9963. Litim, Pawlowski, JHEP (26) 26. advantages: limit trivially satisfied truncations for () and (2) may differ

12 Flow Equation for Thermal Fluctuations technique make a guess for the finite temperature result and iterate around it: does not change the final result numerical stability in iteration procedure as one starts closer to physics k,t = Γ(2) k, + k,t Γ(2) k, thermal contribution + k,t,trial Γ(2) k,t,trial

13 Flow Equation for Thermal Fluctuations technique make a guess for the finite temperature result and iterate around it: does not change the final result numerical stability in iteration procedure as one starts closer to physics k,t = Γ(2) k, + k,t Γ(2) k, thermal contribution + k,t,trial Γ(2) k,t,trial re-ordering k,t = Γ(2) k, + k,t,trial Γ(2) k, + k,t Γ(2) k,t,trial k,t,trial = k T n p k,t =

14 Flow Equation for Thermal Fluctuations technique make a guess for the finite temperature result and iterate around it: does not change the final result numerical stability in iteration procedure as one starts closer to physics k,t = Γ(2) k, + k,t Γ(2) k, thermal contribution + k,t,trial Γ(2) k,t,trial re-ordering k,t = Γ(2) k, + k,t,trial Γ(2) k,... -th iteration + k,t Γ(2) k,t,trial

15 p r e l i m i n a r y Results

16 Electric Gluon-Propagator (after th iteration) T = T= MeV T=2 MeV T=3 MeV 2 p r e l i m i n a r y spatial momentum p

17 Electric Gluon-Propagator (after th iteration) 3 T = T=3 MeV p r e l i m i n a r y m 2 spatial momentum p thermal 2πT

18 Magnetic Gluon-Propagator (after th iteration) 3 T = T=3 MeV p r e l i m i n a r y spatial momentum p

19 Electric Gluon-Propagator (higher iterations) 3 T = th iteration (T=3 MeV) st iteration (3 MeV) p r e l i m i n a r y spatial momentum p

20 Electric Gluon-Propagator (higher iterations) T = th iteration (T=3 MeV) st iteration (3 MeV) 2nd iteration (3 MeV) 2 p r e l i m i n a r y spatial momentum p

21 Lattice Results Temperature dependence: Transverse propagator Temperature dependence: Longitudinal propagator ] -2 D T (p) [GeV p [GeV] T/T c Longitudinal propagator for SU(3).5 ] D L (p) [GeV 2 T/T c.5 Electric screening mass A. Maas, arxiv: [hep-lat]. [GeV] SU(2) p [GeV].5 ] -2 (p) [GeV 3 2 -/2 D L () SU(3) (.2+.9 T/T c -)MeV

22 Summary / Outlook motivation: idea: Yang-Mills propagators at non-vanishing temperature introduce thermal flow equation and add to zero temperature part flow at finite temperature zero temperature result ( input ) (purely) thermal flow have seen: first results for non-perturbative propagators (preliminary) outlook: couple to full QCD calculation talk J. M. Pawlowski talk L. M. Haas Braun, Gies, Pawlowski, Phys. Lett. B684 (2) Braun, Haas, Marhauser, Pawlowski, arxiv: 98.8 [hep-ph].

23 Dyson-Schwinger Approx. for the Ghost-Eq. t = t (DSE) = t + t = + t = with = t G c = = t G A = t t

24 Vertices

25 Full Method k,t = Γ(2) k,t = + Λ,T Γ(2) Λ, + dk k ˆ Γ (2) k,t k,t = + dk k Γ (2) k,t k,t, Γ(2) k,t = T Λ ˆ Γ (2) k,t = T n d 3 p (2) Γ (2π) 3 k, d 4 p (2) Γ (2π) 4 k, Γ (2) k,t = T n d 3 p (2π) 3 Γ(2) k,t (2) Γ k,t =