1. Interweaving Chiral Spirals Toru Kojo (Bielefeld U.) with: K. Fukushima, Y. Hidaka, L. McLerran, R.D. Pisarski 1/11 (Confined) (arXiv: 1107.2124)

2. (Confined) Interweaving Chiral Spirals Toru Kojo (Bielefeld U.) (arXiv: 1107.2124) with: K. Fukushima, Y. Hidaka, L. McLerran, R.D. Pisarski 1/11 This talk (T=0)

3. Chiral restoration Ch. freeze out T 2/11 μ B /Nc Chiral restoration line (Lattice) 〜 M N / Nc ( 〜 300 MeV)

4. Chiral restoration ? NJL, PNJL, PQM, etc. Ch. freeze out T 2/11 μ B /Nc Chiral restoration line (Models) Conf. model ( Schwinger-Dyson eq. ) Chiral restoration line (Lattice) ( 〜 300 MeV) quark Fermi sea is formed (Glozman ) 〜 M N / Nc

5. Chiral restoration ? Ch. freeze out T 2/11 μ B /Nc Chiral restoration line (Lattice) ( 〜 300 MeV) quark Fermi sea is formed 1, In conventional models, chiral restoration happens quickly after the formation of the quark Fermi sea. 2, Assumption: Chiral condensate is const. everywhere. NJL, PNJL, PQM, etc. Chiral restoration line (Models) Conf. model ( Schwinger-Dyson eq. ) (Glozman ) 〜 M N / Nc

6. If we allow non-uniform condensates... Ch. freeze out T 3/11 μ B /Nc Chiral restoration line (Lattice) M N / Nc ( 〜 300 MeV) quark Fermi sea is formed Chiral restoration line (Models) ・ GSI-Frankfurt ・ Stony Brook

7. If we allow non-uniform condensates... Ch. freeze out T 3/11 μ B /Nc Chiral restoration line (Lattice) M N / Nc ( 〜 300 MeV) quark Fermi sea is formed Chiral restoration line (Models) Deconf. line Deconfinment line would be also shifted because: ・ GSI-Frankfurt ・ Stony Brook

8. If we allow non-uniform condensates... Ch. freeze out T 3/11 μ B /Nc Chiral restoration line (Lattice) M N / Nc ( 〜 300 MeV) quark Fermi sea is formed Chiral restoration line (Models) Deconf. line Non-uniform chiral condensate creates the mass gap of quarks near the Fermi surface. → The pure glue results are less affected by massive quarks. Deconfinment line would be also shifted because: ・ GSI-Frankfurt ・ Stony Brook

9. 4/11 Why restoration? (T=0) E Pz Dirac Type P Tot =0 (uniform) L R ・ Candidates of chiral pairing

10. 4/11 E Pz Dirac Type P Tot =0 (uniform) L R It costs large energy, so does not occur spontaneously. ・ Candidates of chiral pairing Why restoration? (T=0)

11. 4/11 E Pz Dirac Type P Tot =0 (uniform) L R E Pz E Exciton TypeDensity wave P Tot =0 (uniform) P Tot =2μ (non-uniform) L R R L Why non-uniform? (T=0) ・ Candidates of chiral pairing

12. 4/11 E Pz Dirac Type P Tot =0 (uniform) L R E Pz E Exciton TypeDensity wave P Tot =0 (uniform) P Tot =2μ (non-uniform) L R R L ・ Kinetic energy: comparable ・ Potential energy: Big difference Why non-uniform? (T=0) ・ Candidates of chiral pairing

13. 5/11 Single Chiral Spiral ・ Choose one particular direction : pzpz

14. 5/11 Single Chiral Spiral ・ Choose one particular direction : ・ Two kinds of condensates appear : linear comb. pzpz space-dep. P-odd

15. 5/11 Single Chiral Spiral ・ Choose one particular direction : ・ Two kinds of condensates appear : linear comb. ・ Chiral rotation with fixed radius : pzpz V period of rotation Δ Z 〜 1/2p F radius (for 1-pair) 〜 Λ QCD 3 space-dep. P-odd

16. 6/11 So far we have considered only the Chiral Spiral in one direction. YES! Is it possible to have CSs in multiple directions? Interweaving Chiral Spiral pzpz Then, the free energy becomes comparable to the S-wave color super conductor. Pairs around the entire Fermi surface can condense.

17. 7/11 pFpF U(1) Z 2Np (N p : Num. of patches ) Rotational Sym. : (2+1) D Example Θ Q Variational parameter : angle Θ ~ 1/N p We use canonical ensemble : Q → Q ( Θ, p F ) ・ We will optimize the angle Θ SSB

18. 8/11 Energetic gain v.s. cost ・ Cost : Deformation Θ Q pFpF equal vol. (particle num.) (dominant for large Θ )

19. 8/11 Energetic gain v.s. cost ・ Cost : Deformation Condensation effects ・ Gain : Mass gap origin Θ Q pFpF equal vol. (particle num.) M E Q p (dominant for large Θ )

20. 8/11 Energetic gain v.s. cost ・ Cost : Deformation Condensation effects ・ Gain : Mass gap origin ・ Cost : Interferences among CSs Θ Q pFpF equal vol. (particle num.) M E Q p Condensate – Condensate int. destroy one another, reducing gap (dominant for large Θ ) (dominant for small Θ ) ( Model dep. !! )

21. A schematic model 9/11 Strength of interactions is determined by Momentum transfer, NOT by quark momenta. → Even at high density, int. is strong for some processes. Q gluon exchange

22. A schematic model Q IR enhancement UV suppression strength 9/11 Strength of interactions is determined by Momentum transfer, NOT by quark momenta. Therefore we use the int. with the following properties: Q gluon exchange → Even at high density, int. is strong for some processes. ?

23. A schematic model Q IR enhancement UV suppression strength 9/11 Strength of interactions is determined by Momentum transfer, NOT by quark momenta. Therefore we use the int. with the following properties: ・ The detailed form in the IR region does not matter. Q gluon exchange → Even at high density, int. is strong for some processes. ?

24. A schematic model Q IR enhancement UV suppression strength 9/11 Strength of interactions is determined by Momentum transfer, NOT by quark momenta. Therefore we use the int. with the following properties: Q gluon exchange Λf Λf → Even at high density, int. is strong for some processes. ・ The detailed form in the IR region does not matter.

25. Energy Landscape (for fixed p F ) Λ QCD /p F ( Λ QCD /p F ) 1/2 ( Λ QCD /p F ) 3/5 × δE tot. Θ deformation energy too big gap too small Θ 10/11 − M × Λ QCD Q

26. Energy Landscape (for fixed p F ) Λ QCD /p F ( Λ QCD /p F ) 1/2 ( Λ QCD /p F ) 3/5 × δE tot. Θ deformation energy too big gap too small − M × Λ QCD Q N p ~ 1/Θ ~ ( p F / Λ fQCD ) 3/5 Θ ・ Patch num. depends upon density. 10/11

27. Λ QCD N c 1/2 Λ QCD μqμq 11/11 (2+1) dim. ICS

28. Λ QCD N c 1/2 Λ QCD μqμq Very likely Chiral sym. restored ・ Deeply inside: (perturbative quarks) 11/11 (2+1) dim. ICS

29. Λ QCD N c 1/2 Λ QCD μqμq Very likely Chiral sym. restored Local violation of P & Chiral sym. Quarkyonic Chiral Spirals ・ Near the Fermi surface: ・ Deeply inside: (perturbative quarks) Quarks acquire the mass gap, delaying the deconf. transition at finite density. (2+1) dim. ICS 11/11

30. Summary & Outlook The ICS has large impact for chiral restoration & deconfinement. ・ 1, The low energy effective Lagrangian → coming soon. ・ 2, Temperature effects & Transport properties ( → hopefully next CPOD )

31. Summary & Outlook The ICS has large impact for chiral restoration & deconfinement. ・ 1, The low energy effective Lagrangian → coming soon. ・ 2, Temperature effects & Transport properties ( → hopefully next CPOD ) My guess : V T=0 (inhomogeneous) V 0 << T < Tc (homogeneous) T 〜 Tc (linear realization) V

32. Appendix

33. ( → McLerran’s talk) This talk (T=0) (Confined) Large Nc phase diagram (2-flavor)

34. Consequences of convolutional effects Nonpert. gluon dynamics Fermi surface effects emphasized by Large Nc emphasized by Large μ × We will discuss small fraction flatter for larger μ

35. How useful is such regime ? Large μ : small fraction Vacuum: large fraction So gluon sector will be eventually modified. ・ Two approximations compete : ・ When modified? : gluon d.o.f: Nc 2 quark d.o.f: Nc Nc 2 Nc × (μ/Λ QCD ) d-1 larger phase space For (3+1)D, μ 〜 Nc 1/2 Λ QCD. (Large Nc picture is no longer valid.) 〜 Λ QCD

36. Strategy Large Nc Large μ μ Good Bad Λ QCD Nc 1/2 Λ QCD Nuclear Quark matter with pert. gluons This work We will 1, Solve large Nc & μ, theoretically clean situation. 2, Construct the pert. theory of Λ QCD /μ expansion. 3, Infer what will happen in the low density region. Vac

37. Θ ~ QΘ ~ Λ QCD condensation region Interference effects Gap distribution will be small gap 12/29

38. 15/29 A crude model with asymptotic freedom Color Singlet p - k IR enhancement UV suppression ・ ex) Scalar - Scalar channel strength

39. 15/29 A crude model with asymptotic freedom Color Singlet ・ ex) Scalar - Scalar channel must be close must be close p - k G ΛfΛf IR enhancement UV suppression strength

40. 16/29 Comparison with other form factor models Typical model quark mom. Ours mom. transfer Strength at large μ : function of : weaken unchange ( at large Nc ) ・ As far as we estimate overall size of free energy, two pictures would not differ so much, because: Hard quarks Typical int. : Hard (dominant in free energy) ・ However, if we compare energy difference b.t.w. phases, typical part largely cancel out, so we must distinguish these two pictures.

41. A key consequence of our form factor. 1 Quark Mass Self-energy (vacuum case) At Large Nc, largely comes from Quark - Condensate int. (large amplitude 〜 Nc) Mom. space Decouple if p & k are very different (Composite objects with internal momenta)

42. Relevant domain of Non-pert. effects Σ m (p) |p| Λ c ( Λ f ) restored Σ m (p) |p| pFpF restored broken (Fermi sea) made of low energy quark - antiquark made of low energy quark - quark hole Vac. Finite Density

43. 3 Messages in this section ・ 1, Condensates exist only near the Fermi surface. ・ 2, Quark-Condensate int. & Condensate-Condensate int. are local in mom. space. ・ 3, Interferences among differently oriented CSs happens only at the patch-patch boundaries. Decouple Couple ( Range 〜 Λ f ) ~ Q Θ Q Θ >> Λ f If Boundary int. is rare process, and can be treated as Pert.

44. 20/29 One Patch : Bases for Pert. Theory Θ Particle-hole combinations for one patch chiral spirals

45. 21/29 Picking out one patch Lagrangian ・ Kin. terms: trivial to decompose ・ Int. terms: Different patches can couple : momentum belonging to i -th patch i i j j k k Patch - Patch int.All fermions belong to the i -th patch

46. 22/29 i ＋i ＋ i −i − eigenvalue: Moving direction “(1+1) D” “chirality” in i -th patch Dominant terms in One Patch, 1

47. 22/29 i ＋i ＋ i −i − eigenvalue: Moving direction “(1+1) D” “chirality” in i -th patch ・ Fact : “Chiral” Non - sym. terms suppressed by 1/Q Dominant terms in One Patch, 1

48. 22/29 i ＋i ＋ i −i − eigenvalue: Moving direction “(1+1) D” “chirality” in i -th patch ・ Longitudinal Kin. (Sym.) ・ Transverse Kin. (Non-Sym.) ・ Fact : “Chiral” Non - sym. terms suppressed by 1/Q ex) free theory Dominant terms in One Patch, 1

49. 22/29 i ＋i ＋ i −i − eigenvalue: Moving direction “(1+1) D” “chirality” in i -th patch ・ Longitudinal Kin. (Sym.) ・ Transverse Kin. (Non-Sym.) excitation energy momentum measured from Fermi surface ・ Fact : “Chiral” Non - sym. terms suppressed by 1/Q ex) free theory Dominant terms in One Patch, 1

50. 23/29 Dominant terms in One Patch, 2 “Chiral” sym. part Non - sym. part 1/Q suppressed ( can be treated in Pert. ) IR dominant ( must be resummed → MF )

51. 23/29 Dominant terms in One Patch, 2 “Chiral” sym. part Non - sym. part 1/Q suppressed ( can be treated in Pert. ) IR dominant ( must be resummed → MF ) ・ IR dominant : Unperturbed Lagrangian Longitudinal Kin. + “Chiral” sym. 4-Fermi int. Transverse Kin. + Non - sym. 4-Fermi int. ・ IR suppressed : Perturbation Gap eq. can be reduced to (1+1) D ( P T - factorization )

52. 24/29 Quick Summary of 1-Patch results ・ Integral eqs. such as Schwinger-Dyson, Bethe-Salpeter, can be reduced from (2+1) D to (1+1) D. cf) kT factorization ・ Chiral Spirals emerge, generating large quark mass gap. (even larger than vac. mass gap) ・ Quark num. is spatially uniform. (in contrast to chiral density) Pert. corrections At leading order of Λ QCD /μ ・ Quark num. oscillation. ・ CSs : Plane wave → Solitonic approach to Baryonic Crystals

53. 25/29 Multi-patches & Optimizing Θ Θ Q(Θ)Q(Θ)

54. 26/29 Multi-Patches: Boundary Effects ・ Interferences among differently oriented CSs destroy one another, reducing the mass gap. ・ Such effects arise only around patch boundaries. (Remark: Such deconstruction effects are bigger if CS’swave vectors take closer value.) Phase space : 〜 N p × Λ f 2 (N p 〜 1/Θ ) reduction of gap : 〜 Λ f Energetic Cost : 〜 N p × Λ f 3 (Checked by Pert. Numerical study by Rapp.et al 2000)

55. 27/29 Energy Landscape Λ f /p F (Λ f /p F ) 1/2 (Λ f /p F ) 3/5 × δE tot. Θ deformation energy too big gap too small − M × Λ c Q N p ~ 1/Θ ~ ( p F / Λ f ) 3/5

56. Chiral Spirals (CSs) ・ One can find (1+1) D solution for the gap equation. (except boundaries of patches ) ・ The size of mass gap is ~ Λ f, if we choose G ~ 1 / Λ f. ・ The subdominant terms can be computed systematically as 1 / Q or Θ expansion. ・ The form of chiral condensates: Spirals & &

57. Energetic cost of deformed Fermi sea Θ Q pFpF ・ Constraint: Canonical ensemble → Fermi vol. fixed V.S. ・ Energetic difference : (deformation energy) δE deform. ~ N p × p F 3 × Θ 5 (1 + O(Θ 2 ) ) (This expression holds even if condensations occur.)

58. Condensation effects 1. ・ Gain : Less single particle contributions (due to mass gap generated by condensates) M E E Q Q Q p p Fermions occupy energy levels only up to Q − M. δE 1particle ~ − M × Λ c × Q (phase space) ΛcΛc (after adding condensation energy)

59. Condensation effects 2. ・ Cost : Induced interactions b.t.w. CSs i i Θ i-1 QΘ Patch-Patch boundary 1, Int. between CSs happen only within phase space, 〜 Λ f 2 2, The strength becomes smaller with smaller size of MBMB (mass gap near the boundary) 3, The sign is positive. δE int. ~ + N p × f int. (M B, Θ) (Num. of boundary points) ( f → 0 as M B → 0 )

60. 6/21 Consequences of form factor. 1 For quarks – condensates int. to happen, their momentum domains must be close each other. p k k (+ q) p (+ q) loop (condensate) ・ Schwinger-Dyson eq. for mass gap: (q=0 for vacuum)

61. 6/21 Consequences of form factor. 1 their momentum domains must be close each other. p k k (+ q) p (+ q) loop must be close (condensate) ・ Schwinger-Dyson eq. for mass gap: (q=0 for vacuum) For quarks – condensates int. to happen,

62. 6/21 Consequences of form factor. 1 their momentum domains must be close each other. p k k (+ q) p (+ q) loop must be close (condensate) ・ Schwinger-Dyson eq. for mass gap: (q=0 for vacuum) ・ UV cutoff for k is measured from p, NOT from 0. ・ Condensate created by fermions around momenta k can couple only to fermions with momenta p ~ k. For quarks – condensates int. to happen,

63. 7/21 Consequences of form factor. 2 Dominant contributions to condensates : Low energy modes When p → ∞ : ・ k must also go to ∞, so ε(k) → ∞. (for vacuum) ・ Phase space is finite : Nothing compensates denominator.

64. 7/21 Consequences of form factor. 2 Dominant contributions to condensates : Low energy modes When p → ∞ : ・ Phase space is finite : Nothing compensates denominator. ・ k must also go to ∞, so ε(k) → ∞. Σ m (p) p p Λ c ( Λ f ) ー ＜ψ (p)ψ (p)＞ー ＜ψ (p)ψ (p)＞ ー ・ finite density: Low energy modes appear near the Fermi surface. Remark) (for vacuum) tr S(p)

65. 3/21 Our goal (MF treatments) ・ Large Nc (high density expansion, T=0) ・ Large density (simple shape of the Fermi surface) ・ (2+1) D Simple analytic insights Θ ・ To express the energy density as a function of theta, ・ 4-Fermi int. with a strong form factor and to determine the best shape. Approximations to be used 2D

66. 8/21 At very high density: P F >> Λ f ~ Λ c pFpF ・ Low energy modes: particle - holes (near the Fermi surface) ・ Domain of condensations: limited to Fermi surface region ・ Decoupling: For Δ P >> Λ f Quarks do not couple to condensates in very different momentum domain. ΔPΔP Quark-condensate int. is local in momentum space.

67. 10/21 Do we need to treat many CSs simultaneously ? ~ Q Θ ~ Λ f Θ Domain of condensation ~ Λ c Λ f ~ Λ c Q Θ phase space ( for 1-boundary ) ( for 1- patch ) Q

68. 10/21 ~ Q Θ ~ Λ f Θ Domain of condensation ~ Λ c Λ f ~ Λ c Q Θ phase space ( for 1-boundary ) ( for 1- patch ) We consider Λ f /p F << Θ << 1 where Phase space: 1- patch >> 1- boundary Boundary effects Small Perturbations to the 1-patch problem ( Patch-Patch interactions ) Q Do we need to treat many CSs simultaneously ?

69. 14/21 Chiral Spirals (CSs) ・ One can find (1+1) D solution for the gap equation. (except boundaries of patches ) ・ The size of mass gap is ~ Λ f, if we choose G ~ 1 / Λ f. ・ The form of chiral condensates: Spirals & &

70. 1/1 What is the best shape ? ・ Gain : Less single particle contributions (due to mass gap generated by condensates) M E E pFpF pFpF pFpF p p Fermions occupy levels only up to p F − M. δE 1-paticle 〜 − M × Λ × Q