QCD at finite density with Dyson-Schwinger equations

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1 QCD at finite density with Dyson-Schwinger equations Daniel Müller, Michael Buballa, Jochen Wambach Quark Gluon Plasma meets Cold Atoms Episode III August 3, 212 TU Darmstadt 1

2 Outline Motivation Dyson-Schwinger equations Color superconductivity Results Inhomogeneous phases Summary and outlook August 3, 212 TU Darmstadt 2

3 Motivation #, % -/. * *!!+ )(& "# $ % $ % * 1 # &'( 23!' -. What happens at high densities? color superconducting phases? weak coupling, effective models Dyson-Schwinger equations at T = (Nickel, Wambach, Alkofer (26)) inhomogeneous phases? S. Carignanos talk our aim: investigate phases with Dyson-Schwinger equations at finite T and µ K. Heckmann (211) August 3, 212 TU Darmstadt 3

4 Outline Motivation Dyson-Schwinger equations Color superconductivity Results Inhomogeneous phases Summary and outlook August 3, 212 TU Darmstadt 4

5 + Dyson-Schwinger equations (DSEs) Quark DSE 1 = 1 ( S 1 (p) = Z 2 S 1 (p) + Σ(p)) August 3, 212 TU Darmstadt 5

6 + Dyson-Schwinger equations (DSEs) Quark DSE 1 = 1 ( S 1 (p) = Z 2 S 1 (p) + Σ(p)) exact QCD equation need: gluon propagator and dressed quark gluon vertex gluon DSE, vertex DSE infinite tower of equations truncation August 3, 212 TU Darmstadt 5

7 Gluon Truncation Gluon propagator (Landau gauge) (data and fit from Fischer, Maas, Müller (21)): ( Dµν ab (k) = ZT (k 2 ) δab P T k 2 µν (k) + Z ) L(k 2 ) P L k 2 µν (k) Lattice data L Lattice data T fit Lattice data L Lattice data T fit L fit T Z[k] 1.5 Z[k] k [GeV] T = MeV k [GeV] T = 125 MeV August 3, 212 TU Darmstadt 6

8 Gluon Polarization 1 Gluon DSE (truncated) Effects of quark on the gluon propagator: 1 = 1 + Dµν 1,ab (k) = D 1,ab µν,ym (k) +Πab µν (k) D ab µν (k) = Z T (k 2 ) k 2 + G ab (k) PT µν (k) + Z L (k 2 ) k 2 + F ab (k) PL µν (k) August 3, 212 TU Darmstadt 7

9 Gluon Polarization 2 D ab µν (k) = Z T (k 2 ) k 2 + G ab (k) PT µν (k) + Z L (k 2 ) k 2 + F ab (k) PL µν (k) calculation with fully dressed quarks or HDL / HTL - like approximation: bare quark propagators large Temperatures or chemical potentials T,µ k (constant) vacuum parts absorbed in renormalization F (k 4, k ) = 2m 2 g (k)..., G(k 4, k ) = π 2 m2 g (k) k 4 k... ( m 2 g = α s(k 2 Nf µ 2 ) π + N ) f T 2 π 3 August 3, 212 TU Darmstadt 8

10 Vertex Truncation exact vertex: vertex DSE 4 point functions, non-abelian terms,... (complicated) Simplifications abelian vertex construction: Ansatz for the dressing function Γ(p, q) Γ a µ (p, q; k) Γ(p, q)γ λ a µ 2 perturbative QCD UV behaviour + phenomenological infrared strength August 3, 212 TU Darmstadt 9

11 Outline Motivation Dyson-Schwinger equations Color superconductivity Results Inhomogeneous phases Summary and outlook August 3, 212 TU Darmstadt 1

12 Color Superconductivity: pairing patterns Cooper instability fermionic system + attractive force Cooper pairs in QCD: diquarks q T COq most important phases: s s u d u d 2SC phase high m s / low µ CFL (-like) phase low m s / high µ August 3, 212 TU Darmstadt 11

13 Color superconductivity in the Dyson- Schwinger framework Nambu Gor kov formalism ( ) ψ define bispinors Ψ = C ψ T, Ψ = ( ψ ψ T C ) ( ) ( ) ( ) ( ) S + S = S + T S, S Σ + Φ = T + S, Σ Γ + = Φ + Σ, Γ = + Γ T, Φ: anomalous propagators / self energies, representing color superconducting phases August 3, 212 TU Darmstadt 12

14 Gluon polarization with color superconductivity Properties of the gluon polarization Debye and Meissner masses: md,ab 2 = lim Π ab TL (ω m =, p) p m 2 M,ab = lim Π ab TT (ω m =, p) p August 3, 212 TU Darmstadt 13

15 Gluon polarization with color superconductivity Properties of the gluon polarization Debye and Meissner masses: md,ab 2 = lim Π ab TL (ω m =, p) p m 2 M,ab = lim Π ab TT (ω m =, p) p transversality: k µ k ν D ab µν (k) = ensured by regularization and truncation requires off-diagonal (in Nambu Gor kov space) contributions to the quark-gluon vertex August 3, 212 TU Darmstadt 13

16 Outline Motivation Dyson-Schwinger equations Color superconductivity Results Inhomogeneous phases Summary and outlook August 3, 212 TU Darmstadt 14

17 Condensates cond [a.u.] cfl ud cfl uds 2sc µ [MeV] dependence of csc condensates on chemical potential (T = 1 MeV) q T COq = Z d 4 p 2 Tr [OT + ] (2π) 4 cond T/ cond cfl ud -.5 cfl uds 2sc T [MeV] dependence of csc condensates on temperature (µ = 58 MeV) August 3, 212 TU Darmstadt 15

18 Phase diagrams T [MeV] CP 1st order region 2SC high ms µ [MeV] 1st order region 2SC small ms CFL August 3, 212 TU Darmstadt 16

19 Gluon masses in the weak coupling limit m 2 cfl [GeV2 ] Self energy ansatz: Φ + (p) = φ i γ 5 M 2SC/CFL,i φ [MeV] m M md Debye and Meissner masses in the CFL phase (T = 1 MeV, µ = 1 GeV) m 2 2sc [GeV 2 ] (weak coupling results from Rischke (2)) 1.6 m M, m D, m M,8 1 m D, φ [MeV] Debye and Meissner masses in the 2SC phase (T = 1 MeV, µ = 1 GeV) August 3, 212 TU Darmstadt 17

20 Gluon masses - full calculation m 2 D [GeV 2 ] D 1-3 D 4-7 D µ [MeV] Debye masses (T = 1 MeV) m 2 M [GeV 2 ] M 1-3 M 4-7 M µ [MeV] Meissner masses (T = 1 MeV) August 3, 212 TU Darmstadt 18

21 Outline Motivation Dyson-Schwinger equations Color superconductivity Results Inhomogeneous phases Summary and outlook August 3, 212 TU Darmstadt 19

22 Inhomogeneous phases General remarks till now: spatially homogeneous matter chiral 1st order transition possibly covered by inhomogeneous condensates see S. Carignanos talk (NJL model) qq = qq (x) Dyson-Schwinger equations - approximations need some simplifications: HDL / HTL truncation 1-dimensional modulations chiral density wave (chiral spiral): qq (x) = qq e iqx August 3, 212 TU Darmstadt 2

23 Chiral spiral non-diagonal structure in momentum space Dirac decomposition requires 1 components Structure in p p space S 1 = August 3, 212 TU Darmstadt 21

24 Gap equations Effective action (HDL / HTL truncations) Γ = Tr ln S 1 Tr (1 S 1 S) Tr S(p)Γa µ, Dab µν (p q)s(q)γb ν Gap equations Solve both equations simultaneously! Γ S(p) = S 1 (p) = S 1 (p) +Σ(p) dγ dq = August 3, 212 TU Darmstadt 22

25 Mass and gap equation M() [MeV] µ = 3 MeV µ = 32 MeV µ = 41 MeV Q [MeV] solution for the mass function for given Q at T = 1 MeV gapq [a.u.] µ = 3 MeV µ = 32 MeV µ = 41 MeV Q [MeV] Gap equation for Q for given Q at T = 1 MeV August 3, 212 TU Darmstadt 23

26 M and Q M,Q [MeV] µ [MeV] M() Q dependence of the mass and wave vector on chemical potential (T = 1 MeV) M,Q [MeV] T [MeV] M() Q dependence of the mass and wave vector on temperature (µ = 32 MeV) August 3, 212 TU Darmstadt 24

27 Phase diagram T [MeV] Chiral spiral homogeneous µ [MeV] August 3, 212 TU Darmstadt 25

28 Summary and Outlook Summary Color superconductivity with Dyson-Schwinger equations CFL-phase for µ > 5 MeV 2SC phase at low densities and at finite T strange quark phase transition visible in 2SC condensates inhomogeneous phases: chiral spiral covers 1st order area Outlook improvement of the vertex inhomogeneous color superconducting phases... August 3, 212 TU Darmstadt 26

29 Summary and Outlook THANK YOU August 3, 212 TU Darmstadt 27