# Presentation: 134959

## QCD phase diagram at large Nc

- The standard lore:
- QCD Phase Diagram vs temperature, T, and quark chemical potential, μ
*One *transition, chiral = deconfined, “semicircle”
- Large Nc:
-      *Two* transitions, chiral ≠ deconfinement
-      Not just a critical end point, but a new “*quarkyonic*” phase:
- Confined, chirally symmetric baryons: *massive*, parity doubled.
- Work exclusively in rotating arm approximation...
- McLerran & RDP, 0706.2191, to appear in NPA.

## ρBaryon ↑

- The first semicircle

- ρBaryon ↑

- Cabibbo and Parisi ‘75: Exponential (Hagedorn) spectrum limiting temperature,
-      *or* transition to new, “unconfined” phase.  One transition.
- Punchline today: below for chiral transition, deconfinement splits off at finite μ.

- T →

## T ↑

- Phase diagram, ~ ‘06

- Lattice, T ≠ 0, μ = 0: two possible transitions; one crossover, same T.   Karsch ’06
- Remains crossover for μ ≠ 0?  Stephanov, Rajagopal, & Shuryak ‘98:
-      Critical end point where crossover turns into first order transition

- μ →

## μBaryon →

- Experiment: freezeout line

- Cleymans & Redlich ‘99: Line for chemical equilibriation at freezeout
-      ~ semicircle.
- N.B.: for T = 0, goes down to ~ nucleon mass.

- μBaryon →

- T ↑

## μquark →

- Experiment vs. Lattice

- μquark →

- Lattice “transition” appears *above* freezeout line?  Schmidt ‘07
- N.B.: small change in Tc with  μ?

- T ↑

## Slide 6

## T ↑

- Lattice Tc , vs μ

- Rather small change in Tc vs μ?  Depends where μc is at T = 0.  Fodor & Katz ‘06

- μquark →

## T →

- Lattice pressure

- (e-3p)/T4 ↑

- For all μ, pressure fits well with (Cheng et al., 0710.0354)

## E/A ↑

- EoS of nuclear matter

- Akmal, Panharipande, & Ravenhall ‘98: Equation of State for nuclear matter, T=0
-      E/A = energy/nucleon.  Fits to various nuclear potentials
- Anomalously small: binding energy of nuclear matter 15 MeV!
- Calc’s reliable to ~ twice nuclear matter density.

- ρBaryon →

## Expansion in large Nc

- Expansion in large Nc

- ‘t Hooft ’74: let Nc → ∞, with λ = g2 Nc fixed.
- ~ Nc2 gluons in adjoint representation, vs ~ Nc quarks in fundamental rep. ⇒
-      large Nc  dominated by *gluons *(iff Nf = # quark flavors *small*)
- “Double line” notation.  Useful even at small Nc (Yoshimasa Hidaka & RDP)

## Slide 11

## Large Nc : “planar” diagrams

- Large Nc : “planar” diagrams

- ~ g2 Nc = λ

- Planar diagram, ~ λ2

- Non-planar diagram, ~ λ2 /Nc
- Suppressed by 1/Nc

## Quark loops suppressed at large Nc 

- Quark loops suppressed at large Nc

- Quark loops are suppressed at large Nc *if* Nf , # quark flavors, is held fixed* *
-      Thus: limit of: large Nc , *small * Nf
- Quarks introduced as external sources.
- Analogous to “quenched” approximation, expansion about Nf = 0.
-      Veneziano ‘78: take both Nc and Nf  large.  Even more difficult.

## Form factors at large Nc 

- J ~ (gauge invariant) mesonic current

- Infinite # of planar diagrams for < *J J* >:

- Confinement => sum over mesons, form factors ~ Nc1/2

## Slide 15

## Mesons & glueballs free at Nc = ∞

- Mesons & glueballs *free *at Nc = ∞

- With form factors ~ Nc1/2 , 3-meson couplings ~ 1/Nc1/2 ; 4-meson, ~ 1/Nc
- For glueballs, 3-glueball couplings ~ 1/Nc , 4-glueball ~ 1/Nc2
- Mesons and glueballs don’t interact at Nc = ∞.
-      Large N limit *always* (some) classical mechanics Yaffe ‘82

## Baryons at large Nc 

- Baryons at large Nc

- Witten ‘79: Baryons have Nc quarks, so nucleon mass MN ~ Nc ΛQCD .
- Baryons like “solitons” of large Nc limit ( ~ Skyrmion)
- Leading correction to baryon mass:

- Appears ~ g4 Nc4 ~ λ2 Nc2 ?
- No, iteration of average potential,
- mass still ~ Nc .

## Baryons are not free at Nc = ∞ 

- Baryons are *not *free at Nc = ∞

- Baryons interact strongly.  Two baryon scattering ~ Nc :

- Scattering of three, four... baryons also ~ Nc
- Mesons also interact strongly with baryons, ~ Nc0 ~ 1

## Skyrmions and Nc = ∞ baryons

- Witten ‘83; Adkins, Nappi, Witten ‘83: Skyrme model for baryons

- Baryon soliton of pion Lagrangian: fπ ~ Nc1/2 ,  κ ~ Nc , mass  ~ fπ2 ~ κ ~ Nc .
- Single baryon: at r = ∞, πa = 0, U = 1.  At r = 0,  πa =  π ra/r .
- Baryon number topological: Wess & Zumino ’71; Witten ’83.
- Huge degeneracy of baryons: multiplets of isospin and spin, I = J: 1/2 ... Nc/2.
-      Obvious as collective coordinates of soliton, coupling spin & isospin
- Dashen & Manohar ’93, Dashen, Jenkins, & Manohar ‘94:
-      Baryon-meson coupling ~ Nc1/2,
-      Cancellations from extended SU(2 Nf) symmetry.

## Slide 20

## Towards the phase diagram at Nc = ∞

- As example, consider gluon polarization tensor at zero momentum.
-      (at leading order, ~ Debye mass2 , gauge invariant)

- For μ ~ Nc0 ~ 1, at Nc = ∞ the gluons are blind to quarks.
- When μ ~ 1, deconfining transition temperature Td(μ) = Td(0)
- Chemical potential only matters when larger than mass:
-      μBaryon > MBaryon.  Define mquark = MBaryon/Nc ; so μ > mquark .
- “Box” for T < Tc ; μ < mquark: confined phase baryon free, since their mass ~ Nc
- Thermal excitation ~ exp(-mB/T) ~ exp(-Nc) = 0 at large Nc.
-      So hadronic phase in “box” = mesons & glueballs only, *no* baryons.

## Phase diagram at Nc = ∞, I

- At *least* three phases.  At large Nc, can use pressure, P, as order parameter.
- Hadronic (confined): P ~ 1.  Deconfined, P ~ Nc2.  Thorn ’81; RDP ’84...
- P ~ Nc: quarks or baryons = “quark-yonic”.  Chiral symmetry restoration?
-      N.B.: mass threshold at mq neglects (possible) nuclear binding, Son.

## Nuclear matter at large Nc 

- μBaryon = √kF2 + M2 , kF = Fermi momentum of baryons.
- Pressure of ideal baryons density times energy of non-relativistic baryons:

- This is small, ~ 1/Nc .  The pressure of the I = J tower of resonances is as small:

- Two body interactions are huge, ~ Nc in pressure.

- At large Nc , nuclear matter is dominated by potential, not kinetic terms!
- Two body, three body... interactions *all* contribute ~ Nc .

## Window of nuclear matter

- Balancing Pideal baryons ~ Ptwo body int.’s, interactions important very quickly,

- For such momenta, only two body interactions contribute.
- By the time kF ~ 1, *all *interactions terms contribute ~ Nc to the pressure.
- But this is *very* close to the mass threshold,

- Hence “ordinary” nuclear matter is only in a *very* narrow window.
- One quickly goes to a phase with pressure P ~ Nc.
-      So are they baryons, or quarks?

## Perturbative pressure

- At high density, μ >> ΛQCD,  compute P(μ) in QCD perturbation theory.
- To ~ g4, Freedman & McLerran (’77)4; Ipp, Kajantie, Rebhan, & Vuorinen ‘06

- At μ ≠ 0, only diagrams with at least one quark loop contribute.  Still...

- For μ >> ΛQCD, but μ ~ Nc0 ~ 1, calculation reliable.
- Compute P(μ) to ~ g6 , g8... ?  No “magnetic mass” at μ ≠ 0, well defined ∀ (g2)n.

## ΛQCD 

- As gluons blind to quarks at large Nc, for μ ~ Nc0 ~ 1, *confined* phase for T <  Td
- This includes μ >> ΛQCD!  Central puzzle.  We suggest:

- To left: Fermi sea.
- Deep in the Fermi sea, k << μ ,
-       looks like quarks.
- But: within ~ ΛQCD of the Fermi surface,
-      confinement => *baryons *
- We term combination “quark-yonic”

- ΛQCD

- OK for μ >> ΛQCD.  When μ ~ ΛQCD, baryonic “skin” entire Fermi sea.
- But what about chiral symmetry breaking?

## Skyrmion crystals

- At low density, chiral symmetry broken
- by Skyrme crystal, as in vacuum.
- Chiral symmetry *restored* at
- nonzero density: < U > = 0 in each cell.
- Goldhaber & Manton ’87: due to “half” Skyrmion symmetry in each cell.
- Forkel, Jackson et al, ’89: excitations *are* chirally symmetric.
- Easiest to understand with “spherical” crystal, KPR ’84, Manton ’87.
- Take same boundary conditions as a single baryon, but for sphere of radius R:
-      At r = R: πa = 0.  At r = 0,  πa =  π ra/r .  Density one baryon/(4 π R3/3).
- At high density, term ~ κ dominates, so energy density ~ baryon density4/3.
-      Like perturbative QCD!  Accident of simplest Skyrme Lagrangian.

- Skyrmion crystal: soliton periodic in space.
- Kutschera, Pethick & Ravenhall (KPR) ’84;  Klebanov ’85 + ...
- Lee, Park, Min, Rho & Vento, hep-ph/0302019 =>

## Schwinger-Dyson equations at large Nc: 1+1 dim.’s

- Schwinger-Dyson equations at large Nc: 1+1 dim.’s

- ‘t Hooft ‘74: as gluons blind to quarks at large Nc, S-D eqs. simple for quark:
-      Gluon propagator, and gluon quark anti-quark vertex unchanged.
- To leading order in 1/Nc, only quark propagator changes:

- ‘t Hooft ‘74: in 1+1 dimensions, single gluon exchange generates linear potential,

- In vacuum, Regge trajectories of confined mesons.  Baryons?
- Solution at μ ≠ 0?  Should be possible, not yet solved.
- Thies et al ’00...06: Gross-Neveu model has crystalline structure at μ ≠ 0

## Schwinger-Dyson eqs. at large Nc: 3+1 dim.’s

- Glozman & Wagenbrunn 0709.3080: in 3+1 dimensions,
-      confining gluon propagator, 1/(k2)2 as k2 -> 0:

## Asymptotically large μ

- For μ ~ (Nc)p, p > 0, gluons no longer blind to quarks.  Perturbatively,

- First two terms from quarks & gluons, last only from gluons.  Two regimes:
- μ ~ Nc1/4 ΛQCD : Nc μ4 F0 ~ Nc2 F2 ~ Nc2 >> Nc μ2 F1 ~ Nc3/2.
-         Gluons & quarks contribute equally to pressure; quark cont. T-independent.
- μ ~ Nc1/2  ΛQCD : New regime: m2Debye ~ g2 μ2 ~ 1, so gluons feel quarks.
-      Nc μ4 F0 ~ Nc3 >> Nc μ2 F1 , Nc2 F2 ~ Nc2 .
-      Quarks dominate pressure, T-independent.
- Eventually, first order deconfining transition can either:
- end in a critical point, or bend over to T = 0: ?

## T↑

- Td

- μ→

- mq

- ↓1st order

- Phase diagram at Nc = ∞, II

- χ sym.
- broken

- Chiral transition

- Quarkyonic

- Deconfined

- Hadronic
- “Box”

- Chirally symmetric

- We suggest: quarkyonic phase includes chiral trans.  Order by usual arguments.
- Mocsy, Sannino & Tuominen ‘03: splitting of transitions in effective models
- But: quarkyonic phase confined.  Chirally symmetric baryons?

## Chirally symmetric baryons

- B. Lee, ‘72; DeTar & Kunihiro ’89; Jido, Oka & Hosaka, hep-ph/0110005; Zschiesche et al nucl-th/0608044.  Consider *two* baryon multiplets.  One usual nucleon, other parity partner, transforming *opposite* under chiral transformations:

- With two multiplets, can form chirally symmetric (parity even) mass term:

- Generalized model at μ ≠ 0: D. Fernandez-Fraile & RDP ’07...

## Anomalies?

- ‘t Hooft, ‘80: anomalies rule *out* massive, parity doubled baryons in vacuum:
-     No massless modes to saturate anomaly condition
- Itoyama & Mueller’83; RDP, Trueman & Tytgat ‘97:
- At T ≠ 0 , μ ≠ 0 , anomaly constraints *far* less restrictive (many more amplitudes)
-    E.g.: anomaly unchanged at T ≠ 0 , μ ≠ 0, but Sutherland-Veltman theorem *fails*
*Must* do: show parity doubled baryons consistent with anomalies at μ ≠ 0.
-     At T ≠ 0 , μ = 0 , no massless modes.  Anomalies probably rule out model(s).
-     But at μ ≠ 0 , *always* have massless modes near the Fermi surface.
- Casher ‘79: heuristically, confinement => chiral sym. breaking in vacuum
-     Especially at large Nc, carries over to T ≠ 0 , μ = 0 .
-     Does *not* apply at μ ≠ 0: baryons strongly interacting at large Nc.
- Banks & Casher ’80: chiral sym. breaking from eigenvalue density at origin.
- Splittorff & Verbaarschot ‘07: at μ ≠ 0, eigenvalues spread in complex plane.
-      (Another) heuristic argument for chiral sym. restoration in quarkyonic phase.

## T↑

- Confined

- Quarkyonic

- Deconfined

- Td

- μ→

- M/Nc

## Guess for phase diagram in QCD

*Pure* guesswork: deconfining & chiral transitions split apart at critical end-point?
- Line for deconfining transition first order to the right of the critical end-point?
- Critical end-point for deconfinement, or continues down to T=0?

## ?

- Hadronic

- T↑

- μB→

- MN

- Deconfined

- Quarkyonic

- χ sym.
- broken

- Chirally symmetric

- Chiral transition

## T↑

- Hadronic

- μB→

- MN

- Deconfined

- Quarkyonic

- ?