quarkyonic chiral spirals
DESCRIPTION
1/21. Quarkyonic Chiral Spirals. Toru Kojo (RBRC). In collaboration with. Y. Hidaka (Kyoto U.), L. McLerran (RBRC/BNL), R. Pisarski (BNL). 2/21. Contents. 1, Introduction . - Quarkyonic matter, chiral pairing phenomena. 2, How to formulate the problem. - PowerPoint PPT PresentationTRANSCRIPT
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Quarkyonic Chiral Spirals
Y. Hidaka (Kyoto U.), L. McLerran (RBRC/BNL), R. Pisarski (BNL)
Toru Kojo (RBRC)
In collaboration with
1/21
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Contents 1, Introduction
2, How to formulate the problem
3, Dictionaries
- Quarkyonic matter, chiral pairing phenomena
- Dimensional reduction from (3+1)D to (1+1)D
- Mapping (3+1)D onto (1+1)D: quantum numbers
4, Summary
2/21
- Dictionary between μ = 0 & μ ≠ 0 condensates
- Model with linear confinement
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1, Introduction Quarkyonic matter, chiral pairing phenomena
3/21
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Dense QCD at T=0 : Confining aspects
T ~ 0 confined hadrons
MD, μ << ΛQCD deconfined quarks
μ, MD >> ΛQCD μ
MD << ΛQCD << μquark Fermi sea with confined excitations
Colorless
E E
4/21
Allowed phase space differs → Different screening effects
MD 〜 Nc–1/2 x f(μ)e.g.)
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Quarkyonic Matter
As a total, color singlet
→ deconfined quarks
Quark Fermi sea + baryonic Fermi surface → Quarkyonic
(MD << ΛQCD << μ)McLerran & Pisarski (2007)
hadronic excitation
・ Large Nc: MD 〜 Nc–1/2 → 0
(so we can use vacuum gluon propagator ) Quarkyonic regime always holds.
5/21
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Quarkyonic Matter near T=0 ・ Bulk properties: deconfined quarks in Fermi sea
・ Phase structure: degrees of freedom near the Fermi surface
(All quarks contribute to Free energy, pressure, etc. )
cf) Superconducting phase is determined by dynamics near the Fermi surface.
6/21
Is chiral symmetry broken in a Quarkyonic phase ?If so, how ?
What is the excitation properties near the Fermi surface ?Confined.
Chiral ??Quarkyonic matter → excitations are
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E
Pz
Dirac Type
・ Candidates which spontaneously break Chiral Symmetry
PTot=0 (uniform)
L
R
Chiral Pairing Phenomena 7/21
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E
Pz
Dirac Type
・ Candidates which spontaneously break Chiral Symmetry
PTot=0 (uniform)
L
R
It costs large energy, so does not occur spontaneously.
Chiral Pairing Phenomena 7/21
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E
Pz
Chiral Pairing Phenomena
Dirac Type
・ Candidates which spontaneously break Chiral Symmetry
PTot=0 (uniform)
L
R
E
Pz
E
Pz
Exciton Type Density wave
PTot=0 (uniform) PTot=2μ (nonuniform)
L
R R
L
Long
〜 μ We will identify the most relevant pairing: Exciton & Density wave solutions will be treated and compared simultaneously.
Trans
7/21
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2, How to solveDimensional reduction from (3+1)D to (1+1)D
8/21
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Preceding works on the chiral density waves 9/21
・ Quark matter:
Deryagin, Grigoriev, & Rubakov ’92: Schwinger-Dyson eq. in large Nc
・ Nuclear matter or Skyrme matter:
Shuster & Son, hep-ph/9905448: Dimensional reduction of Bethe-Salpeter eq.
Rapp, Shuryak, and Zahed, hep-ph/0008207: Schwinger-Dyson eq.
Perturbative regime with Coulomb type gluon propagator:
Effective model:Nakano & Tatsumi, hep-ph/0411350, D. Nickel, 0906.5295
(Many works, so incomplete list)
Nonperturbative regime with linear rising gluon propagator:→ Present work
Migdal ’71, Sawyer & Scalapino ’72.. : effective lagrangian for nucleons and pions
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Set up of the problem・ Confining propagator for quark-antiquark:
(ref: Gribov, Zwanziger) cf) leading part of Coulomb gauge propagator
(linear rising type)
But how to treat conf. model & non-uniform system??
10/21
・ 2 Expansion parameters in Quarkyonic limit1/Nc → 0
ΛQCD/μ → 0 : Vacuum propagator is not modified : Factorization approximation
(ref: Glozman, Wagenbrunn, PRD77:054027, 2008; Guo, Szczepaniak,arXiv:0902.1316 [hep-ph]).
We will perform the dimensional reduction of nonperturbative self-consistent equations, Schwinger-Dyson & Bethe-Salpeter eqs.
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e.g.) Dim. reduction of Schwinger-Dyson eq.
PL 〜 μ
PT 〜 0PL 〜 μ
PT 〜 0
quark self-energy
11/21
including ∑
・ Note1: Mom. restriction from confining interaction.
small momentaΔk 〜 ΛQCD
PL 〜 μ
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e.g.) Dim. reduction of Schwinger-Dyson eq.quark self-energy
12/21
・ Note2: Suppression of transverse and mass parts:
〜 μ 〜 ΛQCD
・ Note3: quark energy is insensitive to small change of kT :
ΔkL 〜 ΛQCD
ΔkT 〜 ΛQCD E= const. surface
along E = const. surface
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Schwinger-Dyson eq. in (1+1) D QCD in A1=0 gauge
Bethe-Salpeter eq. can be also converted to (1+1)D
factorization
⊗smearing
confining propagator in (1+1)D:
e.g.) Dim. reduction of Schwinger-Dyson eq.insensitive to kT
13/21
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Catoon for Pairing dynamics before reduction 14/21
ΔkT 〜 ΛQCD
〜 μ
quark
holeinsensitive to kT
sensitive to kT gluon
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1+1 D dynamics of patches after reduction 14/21
ΔkT 〜 ΛQCD
〜 μ
bunch of quarksbunch of holes
smeared gluons
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3, DictionariesMapping (1+1)D results onto (3+1)D phenomena
15/21
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Flavor Doubling
2D QCD in A1=0 gauge 4D “QCD” in Coulomb gauge
・ At leading order of 1/Nc & ΛQCD/μ
Dimensional reduction of Non-pert. self-consistent eqs :
(confining model)
・ One immediate nontrivial consequence: PT/PL → 0 Absence of γ1, γ2 → Absence of spin mixing
16/21
spin SU(2) x SU(Nf)(3+1)-D side
suppression of spin mixing
SU(2Nf)(1+1)-D side
no angular d.o.f in (1+1) D
cf) Shuster & Son, NPB573, 434 (2000)
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Flavor Multiplet
R-handed
L -handed
spin
17/21
particle near north & south pole
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Flavor Multiplet
R-handed
L -handed
spin
mass term
17/21
particle near north & south pole
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Flavor Multiplet
R-handed
L -handed
spin
mass term
17/21
particle near north & south pole
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Flavor Multiplet
R-handed
L -handed
right-mover (+)
left-mover (–)
left-mover (–)
right-mover (+)
spin
“flavor: φ” “flavor: χ”
Moving direction: (1+1)D “chirality”(3+1)D – CPT sym. directly convert to (1+1)D ones
ψ spin doublet
17/21
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Relations between composite operators・ 1-flavor (3+1)D operators without spin mixing:
Flavor singlet in (1+1)D
・ All others have spin mixing:
Flavor non-singlet in (1+1)D
ex) (They will show no flavored condensation)
18/21
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2nd Dictionary: μ = 0 & μ ≠ 0 in (1+1)D
( = 0 )
・ Dictionary between μ = 0 & μ ≠ 0 condensates:μ = 0 μ ≠ 0
19/21
( = 0 )induced by anomaly
“correct baryon number”
・ μ ≠ 0 2D QCD can be mapped onto μ = 0 2D QCD: chiral rotation (or boost)
(μ ≠ 0) (μ = 0)(due to special geometric property of 2D Fermi sea)
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Chiral Spirals in (1+1)D・ At μ ≠ 0: periodic structure (crystal) which oscillates in space.
20/21
・ `tHooft model, massive quark (1-flavor)
z
・ Chiral Gross Neveu model (with continuous chiral symmetry) Schon & Thies, hep-ph/0003195; 0008175; Thies, 06010243
Basar & Dunne, 0806.2659; Basar, Dunne & Thies, 0903.1868
B. Bringoltz, 0901.4035
cf)
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Quarkyonic Chiral Spirals in (3+1)D
z
( not conventional pion condensate in nuclear matter)
・ Chiral rotation evolves in the longitudinal direction:
20/21
・ No other condensates appear in quarkyonic limit. ・ Baryon number is spatially constant.
zL-quark
R-hole
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Summary
σ
π
V
21/21
vacuum value
Spatial distribution
σ T
V
Quarkyonic Chiral Spiral breaks chiral sym. locally but restores it globally.
0 z
σ
Num. of zero modes 〜 4Nf 2 + .... (not investigated enough)
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Topics not discussed in this talk
・ Coleman’s theorem on symmetry breaking.
・ Explicit example of quarkyonic matter:
・ Beyond single patch pairing.
・ Origin of self-energy divergences and how to avoid it.- Quark pole need not to disappear in linear confinement model.
- 1+1 D large Nc QCD has a quark Fermi sea but confined spectra.
- No problem in our case.
- Issues on rotational invariance, working in progress.
(Please ask in discussion time or personally during workshop)
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Phase Fluctuations & Coleman’s theorem ・ Coleman’s theorem:
σ
T
V
25/30
≠ 0
≠ 0
(SSB)
No Spontaneous sym. breaking in 2D
(physical pion spectra)
・ Phase fluctuations belong to: Excitations
σ
T
V
= 0 (No SSB)
ground state properties
IR divergence in (1+1)D phase dynamics
(No pion spectra)
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Non-Abelian Bosonization in quarkyonic limit 26/30
+ gauge int.
・ Fermionic action for (1+1)D QCD:
+ gauge int.
conformal inv. dimensionful
・ Bosonized version: U(1) free bosons & Wess-Zumino-Novikov-Witten action :
(Non-linear σ model + Wess-Zumino term)
“Charge – Flavor – Color Separation”
gapless phase modes gapped phase modes
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Quasi-long range order & large Nc 27/30
・ Local order parameters:
⊗⊗〜 finite
: symmetric phase
: long range order
: quasi-long range order (power law)
・ Non-Local order parameters:
〜 (including disconnected pieces)
gapless modes gapped modes
due to IR divergent phase dynamics 0 0
But this does not mean the system is in the usual symmetric phase!
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How neglected contributions affect the results? 29/30
・ Neglected contributions in the dimensional reduction:
〜 μ 〜 ΛQCD
spin mixing → breaks the flavor symmetry explicitly mass term → acts as mass term
(3+1)D (1+1)D
Expectations:1, Explicit breaking regulate the IR divergent phase fluctuations, so that quasi-long range order becomes long range order. 2, Perturbation effects get smaller as μ increases, but still introduce arbitrary small explicit breaking, which stabilizes quasi-long range order to long range order.
(As for mass term, this is confirmed by Bringoltz analyses for massive `tHooft model.)
Final results should be closer to our large Nc results!
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In this work, we will not discuss the interaction between different patches except those at the north or south poles.
15/30
(Since we did not find satisfactory treatments)
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2nd Dictionary: μ = 0 & μ ≠ 0 in (1+1)D
R(+)L(–)P
E
〜 μ or
&
fast slow
Then
( = 0 )
・ Dictionary between μ = 0 & μ ≠ 0 condensates:μ = 0 μ ≠ 0
20/30
( = 0 )
induced by anomaly
“correct baryon number”
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T ~ 0 confined hadrons
MD, μ << ΛQCD deconfined
quarks
μ, MD >> ΛQCD μ
MD << ΛQCD << μ: Quarkyonic
Confining aspects
Summary
T ~ 0 Dirac Type Broken Restored
Chiral aspects
μ
Locally Broken, But Globally Restored
30/30
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T ~ 0 confined hadrons
MD, μ << ΛQCD deconfined
quarks
μ, MD >> ΛQCD μ
MD << ΛQCD << μ: Quarkyonic
T ~ 0 Dirac Type Broken Restored
Confining aspects
Chiral aspectsLocally Broken, But Globally Restored
Chiral Spiral μ
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Quasi-long range order & large Nc 25/28
・ Local order parameters:
⊗⊗〜 0 0 finite
: symmetric phase
: long range order
: quasi-long range order (power law)
・ Non-Local order parameters:
〜 (including disconnected pieces)
gapless modes gapped modes
large Nc limit (Witten `78)
due to IR divergent phase dynamics
But this does not mean the system is in the usual symmetric phase!
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Dense 2D QCD & Dictionaries
・ μ=0 2D QCD is solvable in large Nc limit!
・ μ ≠ 0 2D QCD can be mapped onto μ = 0 2D QCD: chiral rotation (or boost)
(μ ≠ 0) (μ = 0)(due to geometric property of 2D Fermi sea)
μ=0 2D QCD μ ≠ 0 2D QCD μ ≠ 0 4D QCD
・ We have dictionaries:
dressed quark propagator, meson spectra, baryon spectra, etc..
(solvable!) exact at quarkyonic limit
18/25
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Quarkyonic Chiral Spiral
& No other condensate appears.
x3
(1+1)D spiral partner:
(3+1)D spiral partner:
19/25
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4, A closer look at QCS
24/30
Coleman’s theorem & Strong chiral phase fluctuations
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Summary
σ
π
V
21/21
σ π
V
Conventional chiral restoration occurs both locally and globally.
σ
0 z
vacuum value
Spatial distribution