Hank Thacker University of Virginia

Topological Charge Membranes, the Chiral Condensate, and Goldstone Boson Propagation

Chiral Dynamics Workshop, JLAB, Aug. 6-10, 2012

Numerical and theoretical evidence for topological charge membranes in the QCD vacuum:

Lattice QCD results (Horvath, et al; Ilgenfritz,et al,2003):

Studies of local TC distribution have revealed a “lasagna vacuum” of extended, thin, codimension 1 membranes of coherent TC in a laminated, alternating sign array. Analogous codimension 1 structures are observed in 2D CPN models.

Holographic QCD (Luscher, 1978; Witten, Sakai, Sugimoto,1998):

Topological charge membranes have a natural holographic interpretation as D6 branes of IIA string theory = “domain walls” between k-vacua with

Large-Nc Chiral Dynamics (Witten, Veneziano,1979):

Chiral Lagrangian form of axial U1 anomaly is an eta-prime mass term . Periodic under , but only by a change of branch for log multiple discrete k-vacua separated by domain walls. (Here is the chiral U(1) phase.)

kloc 2

∝ logDetU( )2

: (η ')2 η ' → η '+ 2π

⇒′η / fπ( ) → ′η

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

.

E()

large Nc

instantons

This large Nc behavior conjectured from chiral Lagrangian arguments (Witten, 1979)

Clarified by AdS/CFT duality (Witten, 1998)

Theta dependence, phase transitions:

= (free energy)

2

( cos )1

( ) from fractionally charged Wilson loops

CP1 CP5

CP9

( )

/ 2

( ) ( )

/ 2 / 2

large N

Instanton gas

(Keith-Hynes and HT, PRD 2008)

--Multiple discrete k-vacua characterized by an effective value of which differs from the in the action by integer multiples of = number of units of background Ramond-Ramond flux in the holographic framework.

-- Interpretation of effective similar to Coleman’s discussion of 2D massive Schwinger model (Coleman, 1976), where

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

2

q 0 vac 2 vac

q

background E field

The emerging picture -- A “laminated” vacuum: Vacuum is filled with alternating sign dipole layers of topological charge. Confinement scale set by correlation length of surface orientation vector .

The nature of the chiral condensate in this picture follows from the observation that, in the presence of quarks, becomes the U1 chiral field, so the axial vector current is given by . This gives a delta-function on the brane surface.

Conclusion: condensate consists of surface quark modes on the membranes, with left and right chiral densities living on topological charge sheets on opposite sides of each membrane.

ψ 1 ± γ 5( )ψ

Jμ5 =∂μ

Codimension-one TC dipole layers

+ - + - + - + - + - - + - + - + - + - +

+ - + - + - + - + - - + - + - + - + - +

+ - + - + - + - + - - + - + - + - + - +

±2π steps in θ

∂μ

CL

CR

Membranes are sheets of magnetic RR charge: fields on opposite sides of brane must be matched by a topologically nontrivial gauge transformation g defined on the brane surface.

The gauge transf on the brane surface can be separated into a 3D “in brane” transformation and the transf of the component transverse to the brane.

Wu-Yang matching defines gauge transf. g on brane

Aμ

The 3D in-brane g.t. represents currents on the brane surface.

g action on transverse A component represents transverse fluctuations of brane surface.

Currents on surface are related to transverse motion of surface by 4D gauge invariance.

⇒

⇒

Aμ

Brane dynamics, the descent eqns. and anomaly inflow:

Topological charge density is a total divergence where is the Chern-Simons current

Q =∂μKμ

Kμ

Kμ =μναβTr Aν∂αAβ +23

AνAαAβ⎛⎝⎜

⎞⎠⎟≡μναβK3

ναβ

Integral over 4D k-vacuum bubble reduces to 3D integral of over membrane surface. Effective action for membrane given by integral of gauge variation of Chern-Simons 3-form over (2+1)D brane surface.

Note that K3 is not gauge invariant. But (Faddeev, 1984)

⇒

δQ = δ dK3( ) = d δK3( ) = 0

So locally we can write as a total deriv. of a 2-form

δK3 = dK2

δK3

K3

δK3

But K2 is a WZW 2-form where is the gauge transformation that defines .

: Tr ω g−1∂μg g−1∂νg( ) +K

g =eiω δK3

Integral of K2 over a closed 2-surface is ambiguous mod . Depends on winding number of g in enclosed 3-volume = units of boundary current on 2-surface.

2

Anomaly inflow idea: Chiral anomaly is interpreted as current disappearing from the bulk theory but reappearing as a subdimensional boundary current, so that

∂μJμ5 ≠ 0

In QCD the bulk AV current is the chiral current . The boundary current on the brane is also chiral current, but it mixes with the Chern-Simons current via annihilation.

∂μ

Kμ qq

The anomaly inflow constraint at the brane surface is implemented by requiring to transform under a color gauge transformation in a way that cancels the variation of :

∂μ

Kμ

δ ∂μ( ) =−δKμ = ∂μ δθ( )

This defines the chiral field , with source = CS fluctuations on brane surface.

δ −δKμ

massless pole in . This is cancelled by massless Goldstone pole in . (4D Kogut-Susskind)

KμKν

∂μ ∂νθ

δKμ

δKν

CS brane

chiral q-qbar current

χt ≠ 0 ⇔

Pion propagation and membrane surface states:

In Nf = 1 QCD, quarks occupy 2D surface states and

becomes the chiral U(1) field . Because of the repeated annihilations induced by , the acquires a mass.

Without annihilation, is a massless excitation :

η '

δKμ ′ηqq

δ =∂μ ∂μδ⎡⎣ ⎤⎦bulk + ∂μδθ⎡⎣ ⎤⎦boundary( ) = 0

The boundary term represents the flow of the condensate. The anomaly inflow constraint still applies, but now on brane does not change winding number, so is globally true:

δ

∂μδ =−δKμ

δgδKμ = ε μναβ∂νK2

αβ ⇒ ∂μδKμ = 0

δ =0

qL

qR

Chiral eigenmodes are pairs of Landau orbitals on opposite sides of brane surface, described by 2+1 D Chern-Simons theory:

Surface modes can be interpreted as approximate topological zero modes associated with TC dipole layer.

Summary:

The QCD vacuum is a condensate of codimension one membranes where the value of jumps by across the brane. A brane is similar to a sheet of magnetic monopoles, in that the A fields on the two sides of the brane must be matched together by a topologically nontrivial gauge transformation on the brane surface.The matching gauge transformation describes surface color currents . Variations of the matching gauge transformation correspond to transverse fluctuations of the brane. The effective action of the brane is given by the gauge variation of the Chern-Simons 3-form on the surface (the “world sheet” action).

In pure YM without quarks, the bulk field is sourced by the

Chern-Simons current, . For Nf = 1, becomes the

U(1) chiral field . The axial current includes delta-function surface currents on the branes. The chiral condensate is formed from surface eigenmodes similar to 2D Landau levels in a transverse magnetic field.

±2π

g−1∂μg

∂μδ =−δKμ

∂μ

′η

In the absence of annihilation, gauge invariance and the anomaly inflow definition of constrain the combination of surface and bulk contributions to to be conserved, giving a massless Goldstone boson as a propagating surface wave along the brane.

qq

∂μ

Future: The description of the chiral condensate in terms of surface modes of a Chern-Simons/WZW theory defines a direct connection between QCD and the EFT that describes the chiral condensate. This should provide relations between fundamental low energy constants which will test the picture. e.g. relation between and .f χt