Michael Buballa. Theoriezentrum, Institut für Kernphysik, TU Darmstadt

Transcription

1 Vacuum-fluctuation effects on inhomogeneous chiral condensates Michael Buballa Theoriezentrum, Institut für Kernphysik, TU Darmstadt International School of Nuclear Physics 38 th Course Nuclear matter under extreme conditions Relativistic heavy-ion collisions Erice, Sicily, September 16 24, 216 September 19, 216 Michael Buballa 1

2 Motivation QCD phase diagram (standard picture): T <qq> = <qq> = <qq> <qq> µ.. September 19, 216 Michael Buballa 2

3 Motivation QCD phase diagram (standard picture): T <qq> = <qq> = <qq> <qq> µ.. assumption: qq, qq constant in space September 19, 216 Michael Buballa 2

4 Motivation QCD phase diagram (standard picture): T <qq> = <qq> = <qq> <qq> µ.. assumption: qq, qq constant in space How about non-uniform phases? September 19, 216 Michael Buballa 2

5 NJL-model studies homogeneous phases only 1 T (MeV) qq = 2 qq = const µ (MeV) [D. Nickel, PRD (29)] September 19, 216 Michael Buballa 3

6 NJL-model studies including inhomogeneous phase 1 T (MeV) qq = 2 qq = const. inhom µ (MeV) [D. Nickel, PRD (29)] September 19, 216 Michael Buballa 3

7 NJL-model studies T (MeV) including inhomogeneous phase 1 8 qq = qq = const. inhom µ (MeV) 1st-order phase boundary completely covered by the inhomogeneous phase! Critical point Lifshitz point [D. Nickel, PRL (29)] [D. Nickel, PRD (29)] September 19, 216 Michael Buballa 3

8 NJL-model studies T (MeV) including inhomogeneous phase 1 8 qq = qq = const. inhom µ (MeV) [D. Nickel, PRD (29)] 1st-order phase boundary completely covered by the inhomogeneous phase! Critical point Lifshitz point [D. Nickel, PRL (29)] Inhomogeneous phase rather robust under model extensions and variations: vector interactions Polyakov-loop dynamics including strange quarks isospin imbalance magnetic fields [MB, S. Carignano, PPNP (215)] September 19, 216 Michael Buballa 3

9 Model limitations and open questions The NJL model is non-renormalizable. Are there cutoff artifacts? September 19, 216 Michael Buballa 4

10 Model limitations and open questions The NJL model is non-renormalizable. Are there cutoff artifacts? example: inhomogeneous continent September 19, 216 Michael Buballa 4

11 Model limitations and open questions The NJL model is non-renormalizable. Are there cutoff artifacts? example: inhomogeneous continent = second inhomogeneous phase at large µ [S. Carignano, MB, Acta Phys. Polon. Supp. (212)] T (MeV) µ (MeV) September 19, 216 Michael Buballa 4

12 Model limitations and open questions The NJL model is non-renormalizable. Are there cutoff artifacts? example: inhomogeneous continent = second inhomogeneous phase at large µ [S. Carignano, MB, Acta Phys. Polon. Supp. (212)] It this caused by the cutoff? T (MeV) µ (MeV) September 19, 216 Michael Buballa 4

13 Model limitations and open questions The NJL model is non-renormalizable. Are there cutoff artifacts? example: inhomogeneous continent = second inhomogeneous phase at large µ [S. Carignano, MB, Acta Phys. Polon. Supp. (212)] It this caused by the cutoff? use renormalizable model: QM model T (MeV) µ (MeV) September 19, 216 Michael Buballa 4

14 Model limitations and open questions The NJL model is non-renormalizable. Are there cutoff artifacts? example: inhomogeneous continent = second inhomogeneous phase at large µ [S. Carignano, MB, Acta Phys. Polon. Supp. (212)] It this caused by the cutoff? use renormalizable model: QM model T (MeV) µ (MeV) Long-term goal: Study role of fluctuations beyond mean field (FRG) September 19, 216 Michael Buballa 4

15 Quark-meson model Lagrangian: L QM = L mes + L q Lmes = 1 2 ( µ σ µ σ + µ π µ π ) U(σ, π), U(σ, π) = λ 4 ( σ 2 + π 2 v 2) 2 hσ, chiral limit: h = Lq = ψ ( i / g(σ + iγ 5 τ π) ) ψ September 19, 216 Michael Buballa 5

16 Quark-meson model Lagrangian: L QM = L mes + L q Lmes = 1 2 ( µ σ µ σ + µ π µ π ) U(σ, π), U(σ, π) = λ 4 ( σ 2 + π 2 v 2) 2 hσ, chiral limit: h = Lq = ψ ( i / g(σ + iγ 5 τ π) ) ψ Thermodynamic potential: Ω(T, µ) = T V log DσD πd ψdψ exp ( d 4 x E (L QM + µ ψγ ψ) ) [, 1 T ] V September 19, 216 Michael Buballa 5

17 Quark-meson model Lagrangian: L QM = L mes + L q Lmes = 1 2 ( µ σ µ σ + µ π µ π ) U(σ, π), U(σ, π) = λ 4 ( σ 2 + π 2 v 2) 2 hσ, chiral limit: h = Lq = ψ ( i / g(σ + iγ 5 τ π) ) ψ Thermodynamic potential: Ω(T, µ) = T V log DσD πd ψdψ exp ( d 4 x E (L QM + µ ψγ ψ) ) [, 1 T ] V Mean-field approximation: σ(x) σ(x) S( x), π a (x) π a (x) P( x) δ a 3 S( x), P( x) time independent classical fields Retain space dependence! September 19, 216 Michael Buballa 5

18 Mean-Field Approximation Mean-field Lagrangian: L MF = L MF mes + ψ S 1 ψ mesonic part: L MF mes = (( 1 S) 2 + ( ) P) 2 U(S, P) 2 (entirely classical) quark part bilinear in ψ and ψ quark fields can be integrated out! inverse dressed quark propagator: S 1 (x) = i / g ( S( x) + iγ 5τ 3P( x) ) γ ( i H MF [S, P] ) September 19, 216 Michael Buballa 6

19 Mean-Field Approximation Mean-field Lagrangian: L MF = L MF mes + ψ S 1 ψ mesonic part: L MF mes = (( 1 S) 2 + ( ) P) 2 U(S, P) 2 (entirely classical) quark part bilinear in ψ and ψ quark fields can be integrated out! inverse dressed quark propagator: S 1 (x) = i / g ( S( x) + iγ 5τ 3P( x) ) γ ( i H MF [S, P] ) Thermodynamic potential: Ω MF = Ω mes + Ω q d 3 x (( σ) 2 + ( π) ) ) 2 + U(σ, π) Ωmes = 1 V V ( 1 2 Ωq = T V Tr Log [ 1 (i T H MF + µ) ] [ )] = 1 E λ µ + T log (1 + e E λ µ T, V 2 λ E λ = eigenvalues of H MF [S, P] in general difficult September 19, 216 Michael Buballa 6 straightforward

20 Condensate modulations Difficulty: H QM is nondiagonal in momentum space has been diagonalized only for certain condensate functions so far Generalized constituent quark mass functions: M( x) := g ( S( x) + ip( x) ) Modulations with analytically known eigenvalues: homogeneous matter: M = const. chiral density wave (CDW): M(z) = e iqz real kink crystal (RKC): M(z) = ν sn( z ν) September 19, 216 Michael Buballa 7

21 Condensate modulations Difficulty: H QM is nondiagonal in momentum space has been diagonalized only for certain condensate functions so far Generalized constituent quark mass functions: M( x) := g ( S( x) + ip( x) ) Modulations with analytically known eigenvalues: homogeneous matter: M = const. chiral density wave (CDW): M(z) = e iqz real kink crystal (RKC): M(z) = ν sn( z ν) With this one gets: { [ Ω q = de ρ(e; {S, P}) E + T log 1 + e E µ T ] [ + T log ρ(e; {S, P}) : analytically known density of states 1 + e E+µ T ] } September 19, 216 Michael Buballa 7

22 Standard and extended MFA Ω q = { [ de ρ(e; σ, π) E + T log 1 + e E µ T ] [ + T log 1 + e E+µ T separates into vacuum ( E) and medium ( T log[... ]) parts vacuum contribution ( Dirac sea ) divergent ] } September 19, 216 Michael Buballa 8

23 Standard and extended MFA Ω q = { [ de ρ(e; σ, π) E + T log 1 + e E µ T ] [ + T log 1 + e E+µ T separates into vacuum ( E) and medium ( T log[... ]) parts vacuum contribution ( Dirac sea ) divergent standard mean-field approximation (smfa): neglect the vacuum part completely ( no-sea approximation ) standard procedure for a long time hope: Dirac-sea effects can be absorbed in the meson-potential but: artifacts for the phase diagram [Skokov et al., PRD (21)] ] } September 19, 216 Michael Buballa 8

24 Standard and extended MFA Ω q = { [ de ρ(e; σ, π) E + T log 1 + e E µ T ] [ + T log 1 + e E+µ T separates into vacuum ( E) and medium ( T log[... ]) parts vacuum contribution ( Dirac sea ) divergent standard mean-field approximation (smfa): neglect the vacuum part completely ( no-sea approximation ) standard procedure for a long time hope: Dirac-sea effects can be absorbed in the meson-potential but: artifacts for the phase diagram [Skokov et al., PRD (21)] ] } extended mean-field approximation (emfa): include the Dirac sea September 19, 216 Michael Buballa 8

25 Standard and extended MFA Ω q = { [ de ρ(e; σ, π) E + T log 1 + e E µ T ] [ + T log 1 + e E+µ T separates into vacuum ( E) and medium ( T log[... ]) parts vacuum contribution ( Dirac sea ) divergent standard mean-field approximation (smfa): neglect the vacuum part completely ( no-sea approximation ) standard procedure for a long time hope: Dirac-sea effects can be absorbed in the meson-potential but: artifacts for the phase diagram [Skokov et al., PRD (21)] ] } extended mean-field approximation (emfa): include the Dirac sea This talk: emfa study of inhomogeneous (and homogeneous) phases [S. Carignano, MB, B.-J. Schaefer, PRD (214); S. Carignano, MB, W. Elkamhawy, PRD (216)] September 19, 216 Michael Buballa 8

26 Fixing the parameters Fit g, λ, and v to three vacuum observables : M vac = g σ, m σ, f π September 19, 216 Michael Buballa 9

27 Fixing the parameters Fit g, λ, and v to three vacuum observables : M vac = g σ, m σ, f π smfa: m 2 σ = 2 U σ 2 σ= σ, π= m 2 σ,tree, f π = σ (Goldberger-Treiman: M vac = g π f π ) September 19, 216 Michael Buballa 9

28 Fixing the parameters Fit g, λ, and v to three vacuum observables : M vac = g σ, m σ, f π smfa: m 2 σ = 2 U σ 2 σ= σ, π= m 2 σ,tree, Including the Dirac sea (usual identification): f π = σ (Goldberger-Treiman: M vac = g π f π ) m 2 σ = 2 Ω σ 2 σ= σ, π= m 2 σ,curv, f π = σ September 19, 216 Michael Buballa 9

29 Fixing the parameters Fit g, λ, and v to three vacuum observables : M vac = g σ, m σ, f π smfa: m 2 σ = 2 U σ 2 σ= σ, π= m 2 σ,tree, Including the Dirac sea (usual identification): f π = σ (Goldberger-Treiman: M vac = g π f π ) m 2 σ = 2 Ω σ 2 σ= σ, π= m 2 σ,curv, f π = σ Correct procedure meson propagator with loop corrections: D j (q 2 ) = 1 q 2 m 2 j,tree +g2 Π j (q 2 )+iɛ m σ m σ,pole, f π = Mvac g π,ren = 1 Zπ σ = Z j + reg. terms, j = σ, π q 2 m 2 j,pole +iɛ September 19, 216 Michael Buballa 9

30 Fixing the parameters M 2 vac 2 σ 4Mvac 2 g 2 =, λ = fπ M2 vac L 2() 2g2 m [1 12 g2 ( 1 4M2 vac m 2 σ ) ] L 2(mσ) 2, v 2 = M2 vac g2 L 1 g 2 λ ; L 1 = 4iN f N c d 4 p (2π) 4 1 p 2 M 2 vac +iɛ, L2(q2 ) = 4iN f N c d 4 p (2π) 4 1 [(p+q) 2 M 2 vac +iɛ][p2 M 2 +iɛ] September 19, 216 Michael Buballa 1

31 Fixing the parameters M 2 vac 2 σ 4Mvac 2 g 2 =, λ = fπ M2 vac L 2() 2g2 m [1 12 g2 ( 1 4M2 vac m 2 σ ) ] L 2(mσ) 2, v 2 = M2 vac g2 L 1 g 2 λ ; L 1 = 4iN f N c d 4 p (2π) 4 1 p 2 M 2 vac +iɛ, L2(q2 ) = 4iN f N c d 4 p (2π) 4 1 [(p+q) 2 M 2 vac +iɛ][p2 M 2 +iɛ] Hands-on renormalization : Regularize L 1 and L 2 (here: Pauli-Villars). Increase cutoff Λ, keeping M vac, m σ, and f π fixed. September 19, 216 Michael Buballa 1

32 λ Fixing the parameters M 2 vac 2 σ 4Mvac 2 g 2 =, λ = fπ M2 vac L 2() 2g2 m [1 12 g2 ( 1 4M2 vac m 2 σ ) ] L 2(mσ) 2, v 2 = M2 vac g2 L 1 g 2 λ ; L 1 = 4iN f N c d 4 p (2π) 4 1 p 2 M 2 vac +iɛ, L2(q2 ) = 4iN f N c d 4 p (2π) 4 1 [(p+q) 2 M 2 vac +iɛ][p2 M 2 +iɛ] Hands-on renormalization : Regularize L 1 and L 2 (here: Pauli-Villars). Increase cutoff Λ, keeping M vac, m σ, and f π fixed. Results for M vac = 3 MeV, f π = 88 MeV, m σ = 6 MeV: g Λ (MeV) September 19, 216 Michael Buballa Λ (MeV) v MeV

33 Phase diagram for homogeneous matter September 19, 216 Michael Buballa 11

34 Phase diagram for homogeneous matter T (MeV) no sea µ (MeV) September 19, 216 Michael Buballa 11

35 Phase diagram for homogeneous matter T (MeV) no sea O µ (MeV) September 19, 216 Michael Buballa 11

36 Phase diagram for homogeneous matter T (MeV) no sea O3 3 PV µ (MeV) September 19, 216 Michael Buballa 11

37 Phase diagram for homogeneous matter T (MeV) no sea O3 3 PV 3 O µ (MeV) September 19, 216 Michael Buballa 11

38 Phase diagram for homogeneous matter T (MeV) no sea O3 3 PV 3 O3 6 PV µ (MeV) September 19, 216 Michael Buballa 11

39 Phase diagram for homogeneous matter T (MeV) no sea O3 3 PV 3 O3 6 PV 6 O µ (MeV) September 19, 216 Michael Buballa 11

40 Phase diagram for homogeneous matter T (MeV) no sea O3 3 PV 3 O3 6 PV 6 O3 1 PV µ (MeV) September 19, 216 Michael Buballa 11

41 Phase diagram for homogeneous matter T (MeV) no sea O3 3 PV 3 O3 6 PV 6 O3 1 PV 1 O µ (MeV) September 19, 216 Michael Buballa 11

42 Phase diagram for homogeneous matter T (MeV) no sea O3 3 PV 3 O3 6 PV 6 O3 1 PV 1 O3 2 PV µ (MeV) September 19, 216 Michael Buballa 11

43 Phase diagram for homogeneous matter 18 T (MeV) no sea O3 3 PV 3 O3 6 PV 6 O3 1 PV 1 O3 2 PV 2 O µ (MeV) September 19, 216 Michael Buballa 11

44 Phase diagram for homogeneous matter 18 T (MeV) no sea O3 3 PV 3 O3 6 PV 6 O3 1 PV 1 O3 2 PV 2 O3 5 PV µ (MeV) September 19, 216 Michael Buballa 11

45 Phase diagram for homogeneous matter T (MeV) no sea 12 O3 3 PV 3 O3 6 9 PV 6 O3 1 6 PV 1 O3 2 3 PV 2 O3 5 PV Convergence reached at Λ 2 GeV. µ (MeV) September 19, 216 Michael Buballa 11

46 Phase diagram for inhomogeneous matter CDW modeluation: M( x) g ( S( x) + ip( x) ) = e iqz September 19, 216 Michael Buballa 12

47 Phase diagram for inhomogeneous matter CDW modeluation: M( x) g ( S( x) + ip( x) ) = e iqz smfa: Λ = T (MeV) µ (MeV) September 19, 216 Michael Buballa 12

48 Phase diagram for inhomogeneous matter CDW modeluation: M( x) g ( S( x) + ip( x) ) = e iqz Λ = 6 MeV T (MeV) µ (MeV) September 19, 216 Michael Buballa 12

49 Phase diagram for inhomogeneous matter CDW modeluation: M( x) g ( S( x) + ip( x) ) = e iqz T (MeV) renormalized (Λ = 5 GeV) µ (MeV) September 19, 216 Michael Buballa 12

50 Phase diagram for inhomogeneous matter CDW modeluation: M( x) g ( S( x) + ip( x) ) = e iqz T (MeV) renormalized (Λ = 5 GeV) µ (MeV) inhomogeneous island gets smaller but survives. Lifshitz point ˆ= critical point September 19, 216 Michael Buballa 12

51 Ginzburg-Landau analysis Expansion of the thermodynamic potential: Ω(M) = Ω() + 1 V d 3 x { 1 γ 2 2 M( x) 2 + 1γ 4 4,a M( x) 4 + 1γ 4 4,b M( x) }, September 19, 216 Michael Buballa 13

52 Ginzburg-Landau analysis Expansion of the thermodynamic potential: Ω(M) = Ω() + 1 V d 3 x { 1 γ 2 2 M( x) 2 + 1γ 4 4,a M( x) 4 + 1γ 4 4,b M( x) }, γi > restored phase (M ) γ2 <, γ 4b > homogeneous broken phase (M = const. ) γ4b < inhomogeneous phase September 19, 216 Michael Buballa 13

53 Ginzburg-Landau analysis Expansion of the thermodynamic potential: Ω(M) = Ω() + 1 V d 3 x { 1 γ 2 2 M( x) 2 + 1γ 4 4,a M( x) 4 + 1γ 4 4,b M( x) }, γi > restored phase (M ) γ2 <, γ 4b > homogeneous broken phase (M = const. ) γ4b < inhomogeneous phase Critical point: γ 2 = γ 4,a = Lifshitz point: γ 2 = γ 4,b = September 19, 216 Michael Buballa 13

54 Ginzburg-Landau analysis Expansion of the thermodynamic potential: Ω(M) = Ω() + 1 V d 3 x { 1 γ 2 2 M( x) 2 + 1γ 4 4,a M( x) 4 + 1γ 4 4,b M( x) }, γi > restored phase (M ) γ2 <, γ 4b > homogeneous broken phase (M = const. ) γ4b < inhomogeneous phase Critical point: γ 2 = γ 4,a = Lifshitz point: γ 2 = γ 4,b = NJL: γ 4,a = γ 4,b CP = LP [Nickel, PRL (29)] September 19, 216 Michael Buballa 13

55 Ginzburg-Landau analysis Expansion of the thermodynamic potential: Ω(M) = Ω() + 1 V d 3 x { 1 γ 2 2 M( x) 2 + 1γ 4 4,a M( x) 4 + 1γ 4 4,b M( x) }, γi > restored phase (M ) γ2 <, γ 4b > homogeneous broken phase (M = const. ) γ4b < inhomogeneous phase Critical point: γ 2 = γ 4,a = Lifshitz point: γ 2 = γ 4,b = NJL: γ 4,a = γ 4,b CP = LP [Nickel, PRL (29)] QM-model: γ 4,a = γ 4,b if m σ = 2M vac (as always in NJL!), but in general CP and LP do not coincide. September 19, 216 Michael Buballa 13

56 Sigma-mass dependence GL results and phase diagrams for m σ = 55, 59, 61, 65 MeV: T (MeV) µ (MeV) CP LP 65 T (MeV) µ (MeV) Size of the inhomogeneous phase very sensitive to m σ! September 19, 216 Michael Buballa 14

57 Different parameter fixing RP : f π = σ Zπ, m σ = m σ,pole BC : f π = σ, m σ = m σ,curv September 19, 216 Michael Buballa 15

58 Different parameter fixing RP : f π = σ Zπ, m σ = m σ,pole BC : f π = σ, m σ = m σ,curv Parameters: g 2 λ v September 19, 216 Michael Buballa 15

59 Different parameter fixing RP : f π = σ Zπ, m σ = m σ,pole BC : f π = σ, m σ = m σ,curv Parameters: completely different September 19, 216 Michael Buballa 15

60 Different parameter fixing RP : f π = σ Zπ, m σ = m σ,pole BC : f π = σ, m σ = m σ,curv Parameters: completely different Homogeneous phase diagram: identical for mσ = 2M vac (moderate differences for m σ 2M vac ) reason: Ωhom depends only on the ratio λ g 4 September 19, 216 Michael Buballa 15

61 Different parameter fixing RP : f π = σ Zπ, m σ = m σ,pole BC : f π = σ, m σ = m σ,curv Parameters: completely different Homogeneous phase diagram: identical for mσ = 2M vac (moderate differences for m σ 2M vac ) reason: Ωhom depends only on the ratio λ g 4 Ginzburg-Landau: γ2 and γ 4,a UV finite in both schemes γ RP 2 γ2 BC = Mvac 2 η(mσ 2 ), γrp 4,a γbc 4,a = η(m2 σ ). η(m 2 σ ) UV finite, vanishes for m σ = 2M vac September 19, 216 Michael Buballa 15

62 Different parameter fixing Inhomogeneous phase diagram: qualitatively different! September 19, 216 Michael Buballa 16

63 Different parameter fixing Inhomogeneous phase diagram: RP 1 Λ = 2 MeV BC 1 September 19, 216 Michael Buballa 16

64 Different parameter fixing Inhomogeneous phase diagram: RP 1 Λ = 3 MeV BC 1 September 19, 216 Michael Buballa 16

65 Different parameter fixing Inhomogeneous phase diagram: RP 1 Λ = 4 MeV BC 1 September 19, 216 Michael Buballa 16

66 Different parameter fixing Inhomogeneous phase diagram: RP 1 qualitatively different! BC 1 September 19, 216 Michael Buballa 16

67 Different parameter fixing Inhomogeneous phase diagram: RP 1 Ginzburg-Landau: qualitatively different! BC 1 γ4,b = 2 g 2 L 2 () M= + UV finite terms RP: 2/g 2 = L 2 ()+ finite UV divergences cancel BC: g = const. UV divergence persists September 19, 216 Michael Buballa 16

68 Different parameter fixing Inhomogeneous phase diagram: RP 1 Ginzburg-Landau: qualitatively different! BC 1 γ4,b = 2 g 2 L 2 () M= + UV finite terms RP: 2/g 2 = L 2 ()+ finite UV divergences cancel BC: g = const. UV divergence persists no renormalized limit! September 19, 216 Michael Buballa 16

69 Different parameter fixing So far we discussed: RP : fπ = σ Zπ, m σ = m σ,pole BC : fπ = σ, m σ = m σ,curv September 19, 216 Michael Buballa 17

70 Different parameter fixing So far we discussed: RP : fπ = σ Zπ, m σ = m σ,pole BC : fπ = σ, m σ = m σ,curv How about the two other possibilities? BP : fπ = σ, m σ = m σ,pole RC : fπ = σ Zπ, m σ = m σ,curv September 19, 216 Michael Buballa 17

71 Different parameter fixing So far we discussed: RP : fπ = σ Zπ, m σ = m σ,pole BC : fπ = σ, m σ = m σ,curv How about the two other possibilities? BP : fπ = σ, m σ = m σ,pole RC : fπ = σ Zπ, m σ = m σ,curv Does not even work for homogeneous phases: γ BP 2 = m 2 σ 4 L 2 ()+ fiinite chiral symmetry gets never restored γ RC 2 = m2 σ 4 L 2 ()+ fiinite chiral symmetry is never broken September 19, 216 Michael Buballa 17

72 Fate of the inhomogeneous continent NJL model 1 8 T (MeV) µ (MeV) September 19, 216 Michael Buballa 18

73 Fate of the inhomogeneous continent NJL model QM model, Λ = T (MeV) 6 4 T (MeV) µ (MeV) µ (MeV) no continent September 19, 216 Michael Buballa 18

74 Fate of the inhomogeneous continent NJL model QM model, Λ = 6 MeV T (MeV) 6 4 T (MeV) µ (MeV) µ (MeV) The continent appears... September 19, 216 Michael Buballa 18

75 Fate of the inhomogeneous continent NJL model QM model, Λ = 5 GeV T (MeV) 6 4 T (MeV) µ (MeV) µ (MeV)... and melts away September 19, 216 Michael Buballa 18

76 Fate of the inhomogeneous continent NJL model QM model, Λ = 5 GeV T (MeV) 6 4 T (MeV) µ (MeV) µ (MeV)... and melts away Problem: Ω MF not bounded from below w.r.t. large amplitudes when λ < w.r.t. large wave numbers q when g 2 < September 19, 216 Michael Buballa 18

77 Vacuum instabilities Thermodynamic potential for T = µ = Λ = 6 MeV Ω vac ( ) - Ω vac () (MeV/fm 3 ) (MeV) Ω vac ( ) - Ω vac () (MeV/fm 3 ) Λ = 5 GeV (MeV) September 19, 216 Michael Buballa 19

78 Vacuum instabilities Thermodynamic potential for T = µ = Λ = 6 MeV Ω vac ( ) - Ω vac () (MeV/fm 3 ) (MeV) Ω vac ( ) - Ω vac () (MeV/fm 3 ) Λ = 5 GeV (MeV) known instability [Skokov et al., PRD 21] symptomatic of the renormalized one-loop approximation [Coleman,Weinberg, PRD (1973)]. The inclusion of higher order loop contributions is known to cure this problem. September 19, 216 Michael Buballa 19

79 Vacuum instabilities Thermodynamic potential for T = µ = Λ = 6 MeV Ω vac ( ) - Ω vac () (MeV/fm 3 ) (MeV) Ω vac ( ) - Ω vac () (MeV/fm 3 ) Λ = 5 GeV (MeV) known instability [Skokov et al., PRD 21] symptomatic of the renormalized one-loop approximation [Coleman,Weinberg, PRD (1973)]. The inclusion of higher order loop contributions is known to cure this problem. similar behavor in q direction [Broniowski, Kutschera, PLB (199)] September 19, 216 Michael Buballa 19

80 Vacuum instabilities Thermodynamic potential for T = µ = Λ = 6 MeV Ω vac ( ) - Ω vac () (MeV/fm 3 ) (MeV) Ω vac ( ) - Ω vac () (MeV/fm 3 ) Λ = 5 GeV (MeV) known instability [Skokov et al., PRD 21] symptomatic of the renormalized one-loop approximation [Coleman,Weinberg, PRD (1973)]. The inclusion of higher order loop contributions is known to cure this problem. similar behavor in q direction [Broniowski, Kutschera, PLB (199)] Can the problem be cured by including bosonic fluctuations ( FRG)? September 19, 216 Michael Buballa 19

81 Summary Inhomogeneous phases in the QCD phase diagram should be considered! September 19, 216 Michael Buballa 2

82 Summary Inhomogeneous phases in the QCD phase diagram should be considered! Investigation of inhomogeneous phases in the QM model with fermionic vacuum fluctuations (Dirac sea): Increasing cutoff: Inhomogeneous phase shrinks, but generally survives High sensitivity to the sigma-meson mass; m σ = 2M vac LP = CP Consistent loop corrections to mσ and f π crucial Vacuum instabilities w.r.t. large amplitudes and wave numbers September 19, 216 Michael Buballa 2

83 Outlook Towards including bosonic fluctuations: Repeat present analysis of fermionic fluctuations within the FRG framework [Carignano, Schaefer, MB; work in progress] Identify phase boundary of the inhomogeneous phase as onset of p-wave pion condensation [ talk by R.A. Tripolt]: Dπ 1 (ω =, p = q) = Analyze bosonic excitation spectrum in the inhomogeneous phase [ M. Schramm, next talk] September 19, 216 Michael Buballa 21