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 KFU Graz, January 3, 213 January 3, 213 TU Darmstadt 1

2 Outline Introduction: QCD phase diagram Dyson-Schwinger equations Color superconductivity Results Inhomogeneous phases Summary and outlook January 3, 213 TU Darmstadt 2

3 Phase diagram of water wikimedia.org January 3, 213 TU Darmstadt 3

4 QCD phase diagram p-t #, % -/. * *!!+ )(& "# $ % $ % * 1 # &'( 23!' -. January 3, 213 TU Darmstadt 4 K. Heckmann (211)

5 QCD phase diagram T-µ What happens at high densities? color superconducting phases? weak coupling, effective models Dyson-Schwinger equations at T = (Nickel, Wambach, Alkofer, PRD(26)) inhomogeneous phases? (Nickel, Buballa, PRD(29), Nickel, PRD(29)) our aim: investigate phases with Dyson-Schwinger equations at finite T and µ January 3, 213 TU Darmstadt 5

6 Outline Introduction: QCD phase diagram Dyson-Schwinger equations Color superconductivity Results Inhomogeneous phases Summary and outlook January 3, 213 TU Darmstadt 6

7 Dyson-Schwinger equations (DSEs) + Quark DSE 1 = 1 ( S 1 (p) = Z 2 S 1 (p) + Σ(p)) January 3, 213 TU Darmstadt 7

8 Dyson-Schwinger equations (DSEs) + Quark DSE 1 = 1 ( S 1 (p) = Z 2 S 1 (p) + Σ(p)) Propagator (in vacuum): S 1 (p) = i/pa(p) + B(p) exact QCD equation need: gluon propagator and dressed quark gluon vertex additional DSEs infinite tower of equations truncation January 3, 213 TU Darmstadt 7

9 Gluon Truncation Yang-Mills Gluon propagator (data and fit from Fischer, Maas, Müller, Eur.Phys.J.C(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 January 3, 213 TU Darmstadt 8

10 Gluon Polarization 1 Gluon DSE (truncated) Quark effects on the gluon propagator: 1 = 1 + Dµν 1,ab (k) = D 1,ab µν,ym (k) +Πab µν (k) Π ab µν (k) = Gab (k)p T µν (k) + F ab (k)p L µν (k) January 3, 213 TU Darmstadt 9

11 Gluon Polarization 2: HTL-HDL approximation 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) HTL-HDL - like approximation: bare quark propagators large Temperatures or chemical potentials T, µ k Debye screening and Landau damping F(k 4, k ) = 2m 2 g (k)..., G(k π 4, k ) = 2 m2 g (k) k 4 k... ( m 2 g = α s(k 2 Nf µ 2 ) π + N ) f T 2 π 3 January 3, 213 TU Darmstadt 1

12 Vertex Truncation 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 (Fischer, Müller, PRD(29)) Γ(k 2 ) = Z 2 Z 3 ( d 1 d 2 + k + k 2 2 ( ) β α(ν) ln(k 2 /Λ 2 2δ ) + 1). k 2 +Λ 2 4π January 3, 213 TU Darmstadt 11

13 Outline Introduction: QCD phase diagram Dyson-Schwinger equations Color superconductivity Results Inhomogeneous phases Summary and outlook January 3, 213 TU Darmstadt 12

14 Color Superconductivity: pairing patterns Cooper instability fermionic system + attractive force Cooper pairs in QCD: diquarks q T Cγ 5 Oq most important phases: s s u d u d 2SC phase high m s / low µ CFL (-like) phase low m s / high µ January 3, 213 TU Darmstadt 13

15 Color superconductivity in the Dyson- Schwinger framework Nambu Gor kov formalism ( define bispinors Ψ = ψ 1 2 C ψ T ( ) S + S = S, S = Coupled equations of S and T ) (, Ψ = 1 2 ψ ψ T C ) ( ) S + T T + S, Σ = non-color-superconducting solution T =, Φ = ( ) Σ + Φ Φ + Σ January 3, 213 TU Darmstadt 14

16 Color superconductivity in the Dyson- Schwinger framework Nambu Gor kov formalism ( define bispinors Ψ = ψ 1 2 C ψ T ( ) S + S = S, S = Coupled equations of S and T ) (, Ψ = 1 2 ψ ψ T C ) ( ) S + T T + S, Σ = non-color-superconducting solution T =, Φ = Color superconducting condensates ( ) Σ + Φ Φ + Σ q T Cγ 5 Oq = Z 2 d 4 p (2π) 4 Tr [γ 5OT (p)] January 3, 213 TU Darmstadt 14

17 Outline Introduction: QCD phase diagram Dyson-Schwinger equations Color superconductivity Results Inhomogeneous phases Summary and outlook January 3, 213 TU Darmstadt 15

18 HTL-HDL Condensates cond [a.u.] C CFL ud C CFL uds C 2SC ud µ [MeV] dependence of csc condensates on chemical potential (T = 1 MeV) cond [a.u.] T [MeV] C CFL ud C CFL uds C 2SC ud dependence of csc condensates on temperature (µ = 58 MeV) January 3, 213 TU Darmstadt 16

19 HTL-HDL Phase diagrams T [MeV] st order region CP 2SC µ [MeV] CFL January 3, 213 TU Darmstadt 17

20 Gluon Polarization 3 Gluon DSE (recap) 1 = 1 + Π ab µν (k) = g2 Tr ( Γ a, µ 2 S(p)Γb ν (p, q)s(q)) q now: calculation of the quark loop with dressed NG propagators requires regularization and renormalization quark-gluon vertex: Slavnov-Taylor identity as a guide off-diagonal (in Nambu Gor kov space) contributions to the quark-gluon vertex January 3, 213 TU Darmstadt 18

21 Condensates cond [a.u.] C CFL ud C CFL uds C 2SC ud µ [MeV] dependence of csc condensates on chemical potential (T = 1 MeV) cond [a.u.] C CFL ud Cuds CFL Cud 2SC T [MeV] dependence of csc condensates on temperature (µ = 58 MeV) January 3, 213 TU Darmstadt 19

22 Phase diagrams T [MeV] CP 1st order region 2SC high ms µ [MeV] 1st order region 2SC small ms CFL January 3, 213 TU Darmstadt 2

23 Gluon polarization with color superconductivity Properties of the gluon polarization Debye and Meissner masses: m 2 D,ab m 2 M,ab = lim Π ab TL (ω m =, p) p = lim Π ab TT (ω m =, p) p January 3, 213 TU Darmstadt 21

24 Gluon polarization with color superconductivity Properties of the gluon polarization Debye and Meissner masses: m 2 D,ab m 2 M,ab = lim Π ab TL (ω m =, p) p = lim Π ab TT (ω m =, p) p m M = for non-csc phases m M for csc phases (broken gauge symmetry Higgs mechanism) January 3, 213 TU Darmstadt 21

25 Gluon masses - full calculation m 2 D [GeV 2 ] D 1-3 D 4-7 D 8 m 2 M [GeV 2 ] M 1-3 M 4-7 M µ [MeV] Debye masses (T = 1 MeV) µ [MeV] Meissner masses (T = 1 MeV) January 3, 213 TU Darmstadt 22

26 Outline Introduction: QCD phase diagram Dyson-Schwinger equations Color superconductivity Results Inhomogeneous phases Summary and outlook January 3, 213 TU Darmstadt 23

27 Inhomogeneous phases General remarks till now: spatially homogeneous matter chiral 1st order transition possibly covered by inhomogeneous condensates (Nickel, PRD(29)) allow spatial dependence of chiral condensate: qq = qq (x) Dyson-Schwinger equations - approximations HTL-HDL truncation 1-dimensional modulations Q = Q e z ( chiral density wave (chiral spiral): B(x, y) = B(x y) 1 2 e iqx + e iqy) ( B(p, p ) = 1 2 B(p) + B(p ) ) δ(p p + Q) January 3, 213 TU Darmstadt 24

28 Chiral spiral Dirac decomposition requires 1 components non-diagonal structure in momentum space S(p) S(p, p ) Structure in p z p z space... S 1 =... January 3, 213 TU Darmstadt 25

29 Gap equations Effective action (HTL-HDL truncations) Gap equations Γ eff = Tr ln S 1 Tr (1 Z 2 S 1 g2 S) + 2 Tr ( ) SΓ a Dab µν SΓb ν Γ eff S(p, p ) = S 1 (p, p ) = Z 2 (S 1 (p, p ) +Σ(p, p )) Solve both equations simultaneously! dγ eff dq = January 3, 213 TU Darmstadt 26

30 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 January 3, 213 TU Darmstadt 27

31 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) January 3, 213 TU Darmstadt 28

32 Phase diagram T [MeV] Chiral spiral homogeneous µ [MeV] January 3, 213 TU Darmstadt 29

33 Summary and Outlook Summary QCD at finite density with Dyson-Schwinger equations: CFL-phase for µ 5 MeV 2SC phase at lower densities and at finite T strange quark phase transition visible in 2SC condensates inhomogeneous phases: chiral spiral covers 1st order area Outlook improvement of vertex and gluon truncation inhomogeneous color superconducting phases... January 3, 213 TU Darmstadt 3 THANK YOU

34 End January 3, 213 TU Darmstadt 31

35 End M()[MeV], qq [a.u.] cond. full cond. HDL / HTL M() full M() HDL / HTL µ [MeV] January 3, 213 TU Darmstadt 32

36 Gluon masses in the weak coupling limit m 2 2SC [GeV2 ] 1.6 m M, m D, m M,4 7 1 m D,4 7 m M,8.8 m D, φ [MeV] Debye and Meissner masses in the 2SC phase (T = 1 MeV, µ = 1 GeV) Self energy ansatz: Φ + (p) = φ i γ 5 M 2SC/CFL,i m 2 CFL [GeV2 ] (weak coupling results from Rischke, PRD (2)) φ [MeV] m M md Debye and Meissner masses in the CFL phase (T = 1 MeV, µ = 1 GeV) January 3, 213 TU Darmstadt 33