Topological Charge Membranes in QCD

Hank Thacker

Collaborators:QCD Study:I. Horvath, K.F. Liu, T. Draper, S.J. Dong, N. Mathur (UKY), F.X.

Lee (GWU/JLAB), J.B. Zhang (Adelaide)

CP(N-1) Study: J. Lenaghan (BNL-PRD), S. Ahmad, Y. Lian (UVA)

JLAB Users Meeting – June 21, 2005

Long distance structure of the QCD vacuum state:Gauge field topology (TC)

Confinement (QC) χSB

Strong phenomenological evidence for all three:

QC : No free quarks or any other color states

: Pions and kaons behave like Goldstone bosons. Success of chiral Lagrangian analysis of low energy QCD.

TC : Heavy mass of eta-prime meson (i.e. resolution of “U1 problem”) requires topological charge fluctuations in QCD vacuum:

χSB

instantons ?

bags ?

strings ? center vortices?

m t′ ∝ ≡η χ2 top. susc.

Large N ? branes ?

Topological Charge Fluctuations in QCD:

We have several different but incomplete views of the QCD vacuum

1. “Strong coupling” view of vacuum: color phases are disordered at large distances (viz. SC expansion on the lattice)

-- Naturally incorporates quark confinement

and other stringy phenomena

-- chiral SB and mass mechanism

is obscure (Aoki phases…)

2. Instanton liquid model populates vacuum with localized, self-dual and anti-self-dual quantized lumps of topological charge.

-- Naturally incorporates chiral SB and mass.

-- Does not incorporate

confinement. (Too much

long-range order.)

′η

′ηFF~

qq

Consider pure glue QCD free energy as a function of parameter ( ).

Witten (1979): Large-Nc arguments require a phase transition (cusp) at Contradicts instantonexpansion, which gives smooth - dependence.

. E ( )θ

θπ− π

large Nc

instantonsLarge Nc behavior conjectured from chiral Lagrangianarguments (Witten, 1979)

Confirmed by AdS/CFT duality (Witten, 1998)

θ π=θ

θS S i Q x d x→ + zθ ( ) 4

4D QCD = IIA String Theory in a black hole metric(See E. Witten, “Black Holes and Quark Confinement”, Current Science ,V.81(2002))

e.g. Confinement “explained” by

QCD flux tube = holographic projection of fundamental string

ADS/CFT also confirms Witten’s (1979) large-Nc view of topological charge (E. Witten, PRL (1998)):

Mutiple vacuum states (“k-vacua”) with

Local k-vacua separated by domain wall = membrane

Domain wall = fundamental D6-brane of IIA string theory wrapped around S4, (TC in QCD is dual to Ramond-Ramond charge).

k=integer is a Dirac-type quantization of 6-brane charge in 9+1 D (Note: 6+2 = 9-1, hence 2D surface surrounds 6-brane in 9 space dimensions.)

Fundamental new development in QCD: ADS/CFT duality (see talk by Jeff Harvey)

θ θ πeff = + 2 k

-- Definition of Q(x) in terms of overlap Dirac operator provides an effective tool for studying local topological charge structure without modifying (e.g cooling) the gauge field. (Horvath et al, PRD (2003) )

-- Overlap construction of Q derived from structure of the chiralanomaly on the lattice. Chiral symmetry and non-ultralocality of D lead to a smoothing of short-range fluctuations allowing the possibility of observing long-range coherent structure.

g48

Tr FF(x) Q(x) 12

2

2πγ~ ( , )≡ → tr D x x5

Exactly chiral Dirac operators: A new method for studying topological charge on the lattice:

Discovery of exactly chiral Ginsparg-Wilson fermions provided a new definition of topological charge on the lattice:

Result of first study of overlap Q(x) distribution in 4D QCD (Horvath, et al, Phys. Rev. D (2004)):

Extended coherent 3-dimensional sheets in 4-D space !

Results:-- Only small 4D coherent structures found with sizes of O(a) and integrated Q(x)<<1. (No instantons.)

-- Large coherent structures are observed which are locally 3-D sheets in 4-D space (surfaces of codimension 1), typically only ~1 or 2 lattice spacingsthick in transverse direction.

-- In each configuration (~1.5 fm4 with PBC’s), two sheets of opposite charge are found, which are everywhere close to each other and “crumpled,” occupying the bulk of 4D space - - Possibly a single membrane with a dipole layer of topological charge

⇒ Short range, negative TCh correlator (required by spectral decomposition).

(Note: TCh correlator must be )≤ ≠0 for all |x| 0

Fractional size of largest coherent 4D structures:

Fractional size of largest coherent 3D structures:

2D slice of Q(x) distribution for 4D QCD Note: Topological charge distributed more-or-less uniformly throughout membrane, not concentrated in localized lumps. (Horvath, et al, Phys.Lett. B(2005) )

The ADS/CFT holographic view of topological charge in the QCD vacuum has an analog in 2D U(1) theories:

--Multiple discrete k-vacua characterized by an effective value of which differs from the in the action by integermultiples of .

-- Interpretation of effective similar to Coleman’s discussion of 2D massive Schwinger model (Luscher(1978), Witten (1979,1998)), where background E field.

In 2D U(1) models (CP(N-1) or Schwinger model): Domain walls between k-vacua are world lines of charged particles:

θθ2π

θ

q qθ = 0 vacθ π= 2 vac

θ =

What are these coherent sheets of TC? Are they D-branes?

Precise analogy between U(1) in 2D and SU(N) in 4D (Luscher, 1978):

Identify Chern-Simons currents for the two theories.

A A Tr A A A A A

j A j A

Q j Q j

CS CS

CS CS

μ μνσ μ ν σ μ ν σ

μ μν ν μ μνστ νστ

μ μ μ μ

ε ε

Wilson line integral over 3- surfacecharged particle charged membrane(= domain wall) (= domain wall)

→ ≡ − + ∂RSTUVW

= → =

= ∂ → = ∂

→→

32 [ ]

This analogy suggests that the coherent 3D structures found in 4D QCD should have an analog of 1D coherent structures in CP(N-1) sigma models.

In both cases, CS current correlator has massless pole ~1/q2

CP(N-1) models on the lattice (with J. Lenaghan and S.Ahmad)

Here z = N-component scalar, and U = U(1) gauge field.

Monte Carlo is done with a Cabibbo-Marinari heat bath for the z’s and 10-hit Metropolis for the U links.

Overlap Dirac op is defined on the U(1) field and solved exactly using the LAPACK singular value decomposition routine. SVD on

overlap is

D U U

D UUW − =

= +

1

1

Λ ~

~

x μ,

S z x U x x z x h c= + + +∑β μ μ*

$

( ) ( , $ ) ( $ ) .

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X Da

ta

Y Data

3D Graph 1

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“Backbone” of coherent 1D regions is only 1 to 2 sites thick (~range of nonultralocality). Positive and negative regions everywhere close (to give negative correlator for |x|>0). .

Topological Charge Correlator from Overlap Q

• In continuum limit, spectral property (reflection positivity) requires :

• Correlator must have positive contact term at x=0, so that topological susceptibility is >0.

• At small x, correlator consists of 2 pieces: (1) a (zero-range) positive contact term, and (2) a negative short range term

• OPE predicts that short range term

with for 4D QCD and for 2D CPN-1

⇒ ≡ < >G x Q x Q x( ) ( ) ( ) | |0 0 0 for all

χ t G x d x= >z ( ) 2 0

~ . | |x

const x→

−− ×0

ν

ν = 8 ν = 4

Horvath, et al (Phys. Lett. 2005):

Positive contact term and negative short-distance term in QCD:

Short distance behavior of overlap TCh correlator for CP(3):

S. Ahmad, J. Lehaghan, and H. Thacker (in preparation)

Wilson lines in 2D and Wilson membranes in 4D:

On an open 2D surface with boundary, a theta term is equivalent to a Wilson line of charge around the boundary

θ -vacuum =Wilson line

θ θ π ε θ πμνμν μ

μ Q F d x A dxV C

= =z z( / ) ( / )2 22

As increases from 0 to , background electric field increases. ( linear Coulomb potential between external charges.) For a + - pair pops out of the vacuum and screens one unit of electric flux. More precisely, a collective motion of the sea of + and –charges screens the external charge from

( / ) ( / )θ π θ π2 2 1→ −

θ π/ 2

θ ππ θ π< < 2

⇒

As increases from to , the background electric fielddecreases back to 0.

At the Wilson line around V is completely screened. (No force between external charges.) Similar to screened quasiparticles in a superconductor (Kivelson and Rokhsar, 1988).

θ 2π

θ π= 2

π

⇒ Quasiparticle excitations in CPN-1 are electrically neutral (screened) particles = composite of charged z-particle and screening charge.

+ + + + + + + + + + ++ +

- - - - - - - - - - -- -

Topological charge (electric field) distribution around a Wilson line excitation:

This is just the kind of one-dimensionally coherent TC distribution seen in the CPN-1 Monte Carlo configurations.

Following Luscher’s analogy, we expect the elementary excitations in 4D QCD to consist of completely screened Wilson bag surfaces representing 2+1 dimensional excitations of the 3-index Chern-Simons tensor . Topological charge distribution (curl of CS tensor) around bag surface is a 3-dimensional “dipole sheet,” just like what is seen in the QCD Monte Carlo configurations !

Aμνσ

D-branes (6-branes) in 4D QCD:

Complete screening = absence of force between bag walls is the gauge theory analog of Polchinski’s demonstration that closed string exchange force vanishes between D-branes.

D-branes and Closed String Exchange:

Polchinski (1995): Study interaction between 2 parallel d-branes due to closed string exchange.

Result:

(1) Net force is zero (d-brane=BPS state) due to exact cancellation between NS-NS and RR string exchange.

(2) Separating NS-NS and RR exchange reveals that the RR charge of the d-branes is exactly the minimal n=1 value required by Diracquantization condition. (Analogous to monopole quantization.)

There is considerable evidence that the coherent sheets of topological charge observed by Horvath, et al (PRD 2004) are none other than the “Wilson bags”= wrapped 6-branes of IIA string theory, as anticipated by Luscher (1978) and predicted by string/gauge holography (Witten (1998)).

“Coulomb” interaction NS-NS exchangeScreening (backflow) R-R exchange

Complete screening of force between bag surfaces for corresponds to the BPS property of d-branes and quantized Ramond-Ramond charge. (Recall Witten’s demonstration that jumps by integer multiples of because RR charge on 6-brane is quantized.)

≈≈θ π= 2

2π

Conclusions:In both 4D QCD and 2D CPN, topological charge

excitations exhibit long range order on surfaces of dimension D-1 (codimension 1).

Lattice results strongly support a “membraney” view of topological charge excitations in the QCD vacuum, similar to that implied by ADS/CFT duality.

Picture of QCD vacuum as a liquid of membranes may give a unified view of how Topological Charge, Quark Confinement, and XSB are related:

XSB: Goldstone bosons “skate” along membranes

QC: Unlike instantons, membranes separate space into two disjoint regions, so color phases can be incoherent on opposite sides of membrane.

disordered vacuum confinement ⇒ ⇒

End of Slide Show

Estimate Hausdorff dimension of largest structures on each configuration:

Starting from each point on structure, draw a circle of radius r and count n(r) = number of points on structure inside circle.

Fit to n(r) = const.x rd

Result: d=1.24

For comparison, generate spin configurations for 2D Ising model slightly above Tc and adjust T so that total volume of largest domains are equal to size of topological charge structures in CPN configs.

Result: Ising d=1.85

i.e. Ising domains are approximately round, while CPN structures are approximately one-dimensional.

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Ultralocal Q(x)

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Overlap Q(x)

Analyze configurations including only points where |q(x)|> some threshold s.t. a fraction f of all points are included:

Fraction of top susc from fraction f of points

Fractional size of largest structure

Average size of structure (weighted by number of points)

If instantons were important, would the overlap Q(x) “see” them in uncooled MC configs?

Show movie migration1_1_medium.avimigration2_0_final.avi

An excellent test case is the 2D U1 Higgs model:

Witten: Classical instantons “melt” from quantum fluctuations in both QCD and CPN, but not in 2D U1 Higgs, where instantons come in a fixed size.

⇒ Test overlap method on U1 Higgs. (preliminary results with Y. Lian)

Confinement in the strong-coupling expansion:• SC expansion analogous to high temperature expansion. We integrate vacuum link variables over all directions in group space, reflecting a totally disordered ground state.

• SC expansion gives area law for Wilson loops

•Linearly rising potential⇒

QCD vacuum contains disordered gauge fields which exclude color-electric flux. (“Anderson localization” of color flux.)

Unfortunately, SC expansion not good for quantitative predictions, because:

q

qq

q

strong coupling limit: continuum limit:

gg

2

2 0→∞

→

For an estimate of the thickness of the structures for CP(N-1):

Start at any point on a coherent structure

Walk in all 4 directions and find the shortest path to an opposite sign site

Call this ½ the thickness and average over all points on the structure

Result: The average thickness of a coherent structure is ~2.2 sites independent of beta. (z+z- meson correlation length varied by factor of 3 from 5 to 15.)

Conclusion: In continuum limit, coherent regions have zero thickness in physical units.

This seems strange, but is consistent with the fact that the TChcorrelator exhibits a positive delta-function contact term over a radius of ~1.5 sites, independent of beta.

Coherent structures are the origin of the contact term

Alternating-sign layering of vacuum gives negative short-range contribution (Form predicted by operator product expansion.)

Models of topological charge based on instantons suggest that XSB may be driven by topological charge fluctuations (Dyakonov-Petrov)

• ‘tHooft zero modes produce lumps of forming anonzero

• Goldstone boson propagation occurs via mixing of zero-modes producing delocalized quark eigenstates.

• Instanton picture suggests a more general connection between XSB and TC: i.e. Goldstone bosons propagate most easily through regions of large, coherent topological charge.

Possible connection between TC and XSB:

ψ γ ψ( )1 5±ψψ

Goldstone pions “hop” from instanton to instanton