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

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

  3. 2/11 Chiral restoration T Chiral restoration line (Lattice) Ch. freeze out 〜MN /Nc μB/Nc (〜300 MeV)

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

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

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

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

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

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

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

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

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

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

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

  15. 5/11 Single Chiral Spiral pz ・ Choose one particular direction : ・ Two kinds of condensates appear : space-dep. P-odd linear comb. ・ Chiral rotation with fixed radius : V radius (for 1-pair) 〜ΛQCD3 period of rotation Δ Z 〜 1/2pF

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

  17. 7/11 (2+1) D Example Z2Np U(1) Rotational Sym. : (Np : Num. of patches ) • Θ SSB pF • Q • angleΘ~ 1/Np Variational parameter : • Q→Q(Θ, pF) We use canonical ensemble : ・ We will optimize the angle Θ

  18. 8/11 Energetic gainv.s. cost ・Cost : Deformation (dominant for largeΘ) pF Q equal vol. (particle num.) • Θ

  19. 8/11 Energetic gainv.s. cost ・Cost : Deformation (dominant for largeΘ) pF Q equal vol. (particle num.) • Θ ・Gain : Mass gap origin M E Condensation effects p Q

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

  21. 9/11 A schematic model 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. 9/11 A schematic model Strength of interactions is determined by Momentum transfer, NOT by quark momenta. → Even at high density, int. is strong for some processes. Therefore we use the int. with the following properties: strength IR enhancement Q UV suppression gluon exchange ? Q

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

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

  25. 10/11 Energy Landscape (for fixed pF) • Θ ( ΛQCD /pF )3/5 δEtot. ( ΛQCD /pF )1/2 ΛQCD /pF Θ − M × ΛQCD Q deformation energy too big gap too small ×

  26. 10/11 Energy Landscape (for fixed pF) • Θ ( ΛQCD /pF )3/5 δEtot. ( ΛQCD /pF )1/2 ΛQCD /pF Θ − M × ΛQCD Q deformation energy too big gap too small × Np~1/Θ~( pF / ΛfQCD)3/5 ・ Patch num. depends upon density.

  27. 11/11 (2+1) dim. ICS Nuclear physics CSC μq Nc1/2 ΛQCD ΛQCD

  28. 11/11 (2+1) dim. ICS Nuclear physics CSC μq Nc1/2 ΛQCD ΛQCD ・Deeply inside: (perturbative quarks) Very likely Chiral sym. restored

  29. 11/11 (2+1) dim. ICS Nuclear physics CSC μq Nc1/2 ΛQCD ΛQCD ・Deeply inside: (perturbative quarks) Very likely Chiral sym. restored ・Near the Fermi surface: QuarkyonicChiral Spirals Local violation of P & Chiral sym. Quarks acquire the mass gap , delaying the deconf. transition at finite density.

  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. V ・1, The low energy effective Lagrangian→ coming soon. ・2, Temperature effects & Transport properties ( → hopefully next CPOD ) My guess : T=0 (inhomogeneous) 0 << T < Tc(homogeneous) T 〜 Tc (linear realization) V V

  32. Appendix

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

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

  35. How useful is such regime ? ・ Two approximations compete : Vacuum: Large μ : small fraction large fraction So gluon sector will be eventually modified. (Large Nc picture is no longer valid.) ・ When modified? : larger phase space 〜ΛQCD gluon d.o.f: Nc2 Nc2 quark d.o.f: Nc Nc× (μ/ΛQCD)d-1 For (3+1)D, μ 〜 Nc1/2ΛQCD .

  36. Strategy Quark matter with pert. gluons Nuclear Vac This work Large Nc Bad Good Large μ Good Bad μ ΛQCD Nc1/2ΛQCD We will 1, Solve large Nc & μ, theoretically clean situation. 2, Construct the pert. theory of ΛQCD /μ expansion. 3, Infer what will happen in the low density region.

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

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

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

  40. 16/29 Comparison with other form factor models Typical model Ours quark mom. mom. transfer function of : weaken unchange (at large Nc) Strength at large μ : ・As far as we estimate overall size of free energy, two pictures would not differ so much, because: Typical int. : Hard Hard quarks (dominant in free energy) ・However, if we compare energy differenceb.t.w. phases, typical partlargely cancel out, so wemust 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) (Composite objects with internal momenta) Mom. space Decouple if p & k arevery different

  42. Relevant domain of Non-pert. effects Σm(p) Vac. restored made of low energy quark - antiquark |p| Λc( Λf) made of low energy quark - quark hole Finite Density Σm(p) restored restored broken (Fermi sea) |p| pF

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

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

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

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

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

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

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

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