Domain Walls, Tachyonic Crystals, and large N QCD

Hank Thacker University of Virginia

INT Workshop on Large N Gauge Theories , Seattle, Feb. 3-6, 2009

References:QCD Results:I. Horvath, et al. Phys.Rev. D68:114505(2003);Phys.Lett. B612:

21(2005);B617:49(2005).P.Keith-Hynes and HT, Phys. Rev. D75:085001 (2007). D. Vaman, E. Barnes, and HT, (in progress).

CPN-1 Results: J. Lenaghan, S. Ahmad, and HT Phys.Rev. D72:114511 (2005), Y. Lian and HT, Phys. Rev. D75:065031 (2007),P. Keith-Hynes and HT, Phys. Rev. D78:025009 (2008).

Topological Charge Studies in Lattice QCD:

A variety of lattice calculations have provided an increasingly clear picture of the structure and role of topological charge fluctuations in QCD (quenched chiral logs, eta-prime mass insertion, topological susceptibility, direct studies of local q(x) distributions in MC configurations).

The advent of exactly chiral lattice fermions (ala Ginsparg-Wilson) has provided a new and improved “fermionic”definition of topological charge on the lattice. This has revealed long range coherent structure (Horvath et al, Phys Lett. B, 2003) without the use of invasive “cooling” procedures.

The observed structure consists of extended, thin, codimension 1 surfaces of coherent (same sign) TC in a laminated, alternating sign “topological sandwich.” Analogous codimension 1 structures are observed in 2D CPN models.

The lattice topological charge sheets have a natural holographic interpretation as wrapped D6 branes of IIA string theory = domain walls between k-vacua with . kloc πθθ 2+=

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) )

Horvath, et al (2003): vacuum of 4D SU(3) gauge theory dominated by layered codimension one sheets of topological charge.

CP(N-1) models on the lattice (Lenaghan, Ahmad, and HT, PRD 2005)

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

S N z x U x x z x h cx

= + + +∑β μ μμ

*

, $( ) ( , $ ) ( $ ) .

As in QCD, we study the topological charge distribution using q(x) constructed from overlap Dirac operator.

To exhibit coherent structure, look for nearest-neighbor-connected structures. Plot largest simply connected structure.

For best visualization plot 1 for sites on structure, 0 otherwise.

Full TC distribution for CP3:

Topological charge correlator for CP(3)

meson corr length changes by factor of 5 over this range

QCD

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

. E (θ )

θπ− π

large Nc

instantons

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

Clarified by AdS/CFT duality (Witten, 1998)

θ π=θ

Theta dependence, phase transitions, and domain walls:

= (free energy)

∝θ 2

∝ −( cos )1 θ

AdS/CFT Holography and QCD topological charge:

4D QCD IIA String Theory in a black hole metric.

Among other things, ADS/CFT confirms Witten’s (1979) large-Nc view of topological charge--Instantons “melt” and are replaced by (Witten, PRL 98):

Multiple vacuum states (“k-vacua”) with

Local k-vacua separated by domain wall = membrane

Domain wall = D6-brane of IIA string wrapped around S4

= Wilson line around D=disk with BH singularity at center ~ Aharonov-Bohm phase around “Dirac string”

k=integer is a Dirac quantization of 6-brane RR charge

TC is dual to Ramond-Ramond charge in string theory.

θ θ πeff = + 2 k

R S R R D S4 1 5 4 4× × → × ×

θ

5D Chern-Simons term 4D termθ

≈

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

qθ = 0 vacθ π= 2 vac

θ =

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

q

Wilson loops in 2D and Wilson membranes (“bags”) in 4D:

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

θ -vacuum =Wilson loop

θ Q = (θ / 2π ) εμν

V∫ Fμνd 2x = (θ / 2π ) Aμ dxμ

C—∫

θ π/ 2

For CPN-1 can be obtained from area law for fractionally charged Wilson loops (P. Keith-Hynes and HT, Phys. Rev, D78:025009 (2008) .)

E( )θ⇒

Precise analogy between CPN-1 models in 2D and QCD in 4D (Luscher, 1978):

Identify Chern-Simons currents for the two theories.

Aμ → Aμνσ ≡ −Tr AμAνAσ +32

A[μ∂νAσ ]⎧⎨⎩

⎫⎬⎭

jμCS = εμνAν → jμ

CS = εμνστ Aνστ

Q = ∂μ jμCS → Q = ∂μ jμ

CS

Wilson line → integral over 3-surface ("Wilson bag")charged particle → charged membrane(=domain wall) (=domain wall)

This analogy suggests that the coherent 1D structures in CPN-1 are charged particle world lines, and the 3D coherent structures in QCD are Wilson bags=excitation of Chern-Simons tensor on a 3-surface.

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

log exp i

θ2π

A ⋅ dl—∫ = − ε (θ ) − ε (0)[ ] V (R )T

To calculate theta-vacuum energy density in CPN-1, measure slope of linear potential:

CP5 for loop charge = 0.3 :

V(R)

R

ε θ( ) from fractionally charged Wilson loops

CP1 CP5

CP9

ε θ( )

θ π/ 2

ε θ( ) ε θ( )

θ π/ 2θ π/ 2

large N

Instanton gas

“Melting” instantons (Y. Lian and HT, hep-lat/0607026):

CP1 and CP2 are dominated by small instantons with radii of order a (so correlator remains negative for nonzero separation in continuum limit). Small instantons are easily seen with overlap q(x).

CP3 is on the edge of the instanton melting point – has some instantons but mostly coherent line excitations.

CP4 and higher have no instantons – only line excitations.

Crude estimate (lower bound) for instanton melting point = “tipping point” of integral over instanton size in semiclassical calculation (Luscher, 82):

for CPN-1, Ncrit=2 for QCD, Ncrit=12/11

So SU(3) QCD should be well above instanton melting point, and topological charge should come in the form of membranes or domain walls. (Figuratively speaking, instantons become arbitrarily large and hollow like soap bubbles. No force between walls of bubble because vacuum inside and vacuum outside differ by .) (c.f. quantization of D6 brane RR charge ala Polchinski (Witten, 98)– discussed later)

⇒

Δθ π= ±2

CP1, beta=1.6, Q = 1

CP2, beta=1.8, Q = 1

CPN-1 instantons from overlap topological charge:

CP1, beta=1.6, Q = -2

CP9, beta=0.9, Q = -1

Summary of Monte Carlo results: In both 4D QCD and 2D CPN-1 (for N>3), topological charge comes in the form of extended membranes of codimension 1, with opposite sign sheets (or lines in CPN-1) juxtaposed in dipole layers.

(Exception -- CP1 and CP2 models are dominated by small instantons)

The QCD vacuum is a laminated “topological sandwich’’of alternating sign membranes. This topological structure of the gauge fields presumably induces chiral symmetry breaking via surface near-zero modes with qLand qR attached to + and – topological charge membranes.

(MILC collaboration: evidence from IPR of quark near-zero modes gives effective dimensionality of , indicating that modes are delocalized along 2+1 D membrane-like surfaces.)

0.3≈

Quarks and D8-branes (Sakai-Sugimoto):

-- In Witten’s construction of 4D YM, gluons are world volume gauge fields of Nc D4-branes-- Topological charge membranes are D6 and D6bar branes (= Luscher’s Wilson bags).-- Introduce quarks in the “probe” (~ quenched) approximation, via Nf D8-D8bar brane pairs. (Sakai & Sugimoto; Antonyin, Harvey, Jensen, Kutasov).

0 1 2 3 4 5 6 7 8 9

D4 branes x x x x x - - - - -

D6 branes x x x - - - x x x x

D7 brane x x x x - - x x x x

D8 branes x x x x - x x x x x

space-time disk S4

the 5 dimensions transverse to D4 branes.

Angular coordinates around black hole

-- In D4/D8 model, tachyonic mode of D8-D8bar string (and/or gravitational effects induced by D4 branes) leads to chiral symmetry breaking = joining together of D8 and D8bar into a single brane

)()()( fff NUNUNU →⊗⇒

→4-5 disk

Interplay between topological charge and chiral symmetry breaking (work in progress with D. Vaman and E. Barnes): .

Tachyon effective field theory on D8-D8bar branes (similar to DBI action) has kink solutions which are (non-BPS) D7 branes. (Sen, 2002)

D7 brane has tachyonic mode which makes it unstable. Decay of D7 brane into D6 branes suggests a stringy view of TC fluctuations in QCD which predicts the laminated, alternating-sign array of topological charge membranes seen in Monte Carlo configurations.

D7 brane decay: Tachyonic crystals and the laminar instability of the perturbative QCD vacuum

In Witten’s D4 brane construction of HQCD, RR sources (D6 branes) are required for nonzero theta.

Consider 1-flavor HQCD: D6 branes can arise from sequential brane annihilation/decay ( annihilation)

D7 brane is localized in x5 (holographic direction) but fills 4D spacetime homogenously.

D7 decay is described (ala Sen) by a solvable BCFT with exactly marginal boundary perturbation.

qq

D8D8 → D7 → D6

Open string propagating in a background of Dbranes is described by a CFT with a boundary interaction on the world sheet.

Homogeneous decay of (non-BPS) D7 brane described in real time by time-dependent solution . In Euclidean time, this corresponds to an exactly marginal (dimension 1) boundary perturbation in CFT, first solved by Callan, et al and Polchinski, et al:

e− x0

LI =1

8π(∂X)2

0

l

∫ dσ + λ cos X(0) − cos X(l)( )

This BCFT interpolates between Neumann ( ) and Dirichlet ( ) boundary conditions ! !

λ = 0λ =

12

⇒ At the critical coupling the ends of the string are stuck at the extrema of the boundary perturbation:

X(0) = (2n +1)π , X(l) = 2nπ

At the critical coupling value , ends of the string are firmly stuck to the minima of the boundary perturbation.

Boundary perturbation

Background array for λ =

12

Going from to on the string worldsheet corresponds in target space to going from a homogeneous D7 brane to a “tachyonic crystal” array of codimension one D6 branes = string theory version of a topological sandwich.

D6 D6 D6 D6 D6 D6 D6 D6

string

λ = 0 λ =12

λ =12

STRING: homogeneous D7 brane tachyon crystal of D6 branes

GAUGE: perturbative vacuum physical gauge vacuum

→→

Conclusions:

Topological charge distribution in the QCD vacuum is dominated by codimension one membranes arranged in an alternating sign array (topological sandwich).

In Witten’s D4 brane construction of holographic QCD, topological charge membranes correspond to D6 branes of IIA string theory (wrapped around S4). On the gauge theory side they are described by Wilson line (D=2) or Wilson bag (D=4) operators = D-1 dimensional surface integral of Chern-Simons tensor.

The introduction of quarks via D8 branes (Sakai-Sugimoto) combined with Sen’s BCFT description of tachyonic brane annihilation/decay suggests a dynamical explanation of the laminar instability of the perturbative QCD vacuum which leads to the topological sandwich structure.