D-Branes and Topological Charge in Lattice QCD

Hank Thacker University of Virginia

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). Ilgenfritz, et al. Nuc. Phys Proc. Suppl. (2006). MILC Collaboration, Nuc. Phys. Proc. Suppl. (2005).

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, arXiv:0804.1534 [hep-lat] .

INT String/Gauge Workshop, Seattle, May, 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 2

Direct measurement of hairpin diagramDirect measurement of hairpin diagram

In large N chiral Lagrangian (Witten-Venziano) description, quenched hairpin propagator should look like:

H(p) = 5 5

1 12 2 0

2

2 2p mm

p m

'

57 335 6 684 33

59 225 6 701 19

118 180

0

0

1

. . ( )

. . ( )

. .

:

:

using and respectively

m m MeV

m m MeV

a GeV GeV

glue

glue

Results:

(Bardeen, et al, Phys. Rev. D 2004)

m m t e m t02 1( )

=

-- 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 expression for q derived from structure of the chiral anomaly 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.

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:

),( 2

1)()(

~

48 52

2

xxDTrxqxFFTrg

Results of first study of overlap q(x) distribution in 4D QCD (Horvath, et al, Phys. Rev. D (2003)): 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 spacings thick in transverse direction.

-- In each gauge configuration, two extended sheets of opposite charge are found, which are everywhere close to each other - - Possibly a single membrane with a dipole layer of topological charge

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

(Note: <q(x)q(0)> correlator must be ) 0 for all |x| 0

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) )

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

*

, ( ) ( , ) ( ) .

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.

To normalize expectations, first look for connected structures on random q(x) distributions, then compare with q(x) distributions in CPN-1 Monte Carlo configurations:

0.00.20.40.60.81.01.2

0

10

20

30

40

0

10

20

30

40

Z Data

X D

ata

Y Data

Largest coherent structure from a random TC distribution:

Another random TC distribution:

0.00.20.40.60.81.01.2

0

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40

0

10

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Z Data

X D

ata

Y Data

3D Graph 1

0.0 0.2 0.4 0.6 0.8 1.0

0.00.20.40.60.81.01.2

0

10

20

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40

0

10

20

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Z Data

X D

ata

Y Data

Coherent structure: (CP3 50 50 12 20 . )

0.00.20.40.60.81.01.2

0

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40

0

10

20

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Z Data

X D

ata

Y Data

Another CP(3) configuration:

0.00.20.40.60.81.01.2

0

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0

10

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30

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Z Data

X D

ata

Y Data

Still another CP(3) configuration:

-1.5-1.0-0.50.00.51.01.5

0

5

10

15

20

25

0

5

10

15

20

25

Z Data

X D

ata

Y Data

3D Graph 1

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

“Backbone” of coherent 1D regions is only 1 to 2 sites thick (~range of nonultralocality). Positive and negative regions everywhere close.

Plot sign(q(x)) for CP3 config:

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( )

large Nc

instantons

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

Confirmed by AdS/CFT duality (Witten, 1998)

Theta dependence, phase transitions, and domain walls:

= (free energy)

2

( cos )1

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

-vacuum =

Wilson loop

Q F d x A dxV C

z z( / ) ( / )2 22

/ 2

For CPN-1 can be obtained from area law for fractionally charged Wilson loops

(P. Keith-Hynes and HT, arXiv:0804.1534 [hep-lat] .)

E( )

log exp ( ) ( ) ( )i A dl V R T

2

0 z

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

CP5 for loop charge = 0.3 :

V(R)

R

( ) from fractionally charged Wilson loops

CP1 CP5

CP9

( )

/ 2

( ) ( )

/ 2 / 2

large N

Instanton gas

Digression on “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)

2

CP2, beta=1.8, Q = 1

CPN-1 instantons from overlap topological charge:

CP1, beta=1.6, Q = 1

CP1, beta=1.6, Q = -2

CP9, beta=0.9, Q = -1

cp1 cp2 cp3

cp5 cp9

Plot integrated q(x) in highest structure (within 2 sites of highest peak) for all configs with Q = +-1

(End of digression on melting instantons.)

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 “topological sandwich’’ of alternating sign membranes. This topological structure of the gauge fields presumably induces chiral symmetry breaking via surface near-zero modes with qL and 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

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 = fundamental 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.

eff 2 k

R S R R D S4 1 5 4 4

5D Chern-Simons term 4D term

Holographic view of domain wall between k-vacua:In Witten’s brane construction, 4D Yang-Mills is viewed from 6 dimensions = R4 x D, where D = S1 x radial coordinate of black-hole metric. Radius of S1 is an ultraviolet cutoff, analogous to lattice spacing.

Analog in (1+1)-D is (3+1)-dimensional solid cylinder.

(1+1)-D U(1) case is equivalent to Laughlin’s gedankenexperiment for topological understanding of integer quantum Hall effect.

zA dy 0 zA dy 2Dirac string

monopole

Longitudinal component of monopole field (B) is dual to topological charge (= longitudinal E field) in CP(N-1) model.

x-axis

domain wall

(Corbino disk)

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 integer multiples 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:

2

q q 0 vac 2 vac

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

Precise analogy between CPN-1 models in 2D and QCD 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

Wilson line integral over 3- surface ("Wilson bag")

charged particle charged membrane

(= domain wall) (= domain wall)

RSTUVW

3

2 [ ]

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

Quarks and D8-branes:

-- In Witten’s construction of 4D YM, gluons are world volume gauge fields of Nc D4-branes-- Topological charge membranes are D6-D6bar brane pairs (= 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

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 on D8-D8bar branes reduces to the NfxNf

chiral field of QCD. Tachyon effective potential related to chiral Lagrangian. D8 brane tachyon theory has solitonic vortex solutions which are D6 branes (A. Sen, 2004). The topological charge lamination of the QCD vacuum can be related to tachyonic instability of D8-D8bar pairs, thereby to SXSB.

Conclusions:

The large Nc holographic view of topological charge structure in QCD is strongly supported by lattice results. Witten’s D6 branes (= Luscher’s Wilson bags) provide a plausible model for observed coherent sheets of topological charge.

Further detailed studies of relation between topological charge sheets and low Dirac eigenmodes should clarify their role in chiral symmetry breaking. Lattice results on spacetime structure of low eigenmodes suggest they are surface modes on the sheets. (A membrane version of the Dyakanov-Petrov instanton-driven chiral condensate, but fundamentally different because surface modes are delocalized along membranes, as compared to localized ‘tHooft zero modes in instanton model.)

The emerging picture -- A “laminated” vacuum: Alternating sign sheets (or lines) of topological charge:

+ - + - + - + -

Possible dynamics of vacuum lamination in CPN-1:

Spectrum consists of nonsinglet and singlet z-zbar pairs.

Msinglet > Mnonsinglet due to annihilation diagrams:

singlet pairs pop out of vacuum, but they can propagate farther by forming nonsinglet pairs with members of neighboring singlet pairs:

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

orTwo degenerate vacua with topological order (ala Wen and Zee in quantum hall eff.)

D6-D6bar strings

A “dimerization” (lamination) of the vacuum. Similar to Peierls transition in one-dimensional chain of atoms .

Like antiferromagnetic order, but not tied to even-odd sublattice (hence topological)

Note that singlet pairs must all polarize in the same direction to form nonsinglet mesons with nearest neighbors.

This mechanism is also compatible with long range behavior of large N solution :

+ 1

In large N, singlet pairs are treated individually, but with massive z-propagators.

nonzero top. susc. spontaneous generation of constituent z-mass

Vacuum polarization tensor

2 6 3

52

2

2

2

2

N

q q

q

M

qO

q

M

FHG

IKJ

FHG

IKJ

LNM

OQP

q M2 2 ( )q

Conformal field theory between the branes:

CPN-1 in Lorentz gauge A 0

2

A

J

q x

CS

2 ( )

Thickness of coherent lines of q(x) is of order a in continuum.

Consider an idealized brane vacuum configuration where q(x) is confined to one-dimensional subspaces (or zero-dimensional for small instantons).

In voids between branes, so can be written as the real part of a holomorphic fcn of z=x1+ix2

( , ) ( ) ( )x x z z1 2 -- has branch cuts at branes (and/or poles at small instantons)

2 0 ( , )x x1 2

( )z

0

string coordinate for flux tube between charged particles (= Dbranes = string endpoints) ( ) ~z ?

CPN Topological Charge Correlators from CFT

Static dilute brane approximation M2 >> q2 (in QCD, large ps glueball mass)

Effective theory with z’s integrated out:

OPE for Chern-Simons current correlator:

S F 1

42

J x J C x C x FCS CS ( ) ( ) ~ ( ) ( ) ( ) ...0 01 2

2

Form of OPE coefficients completely determined up to 2 overall constants by CFT arguments:

C x cx x x

x1 1

2

2 2

2

( )

( )

C x c x

x x

x2 22

2

2

( ) ln FHG

IKJ

Result for TC correlator is sum of contact terms:

G t dx q x t q c t c t c t( ) ( , ) ( , ) ' ' ( ) ( ) z , 0 0 1 2 2

Gives a good fit to all lattice correlators using

With lattice spacings, essentially independent of beta.

( ) exptd

t

d

FHG

IKJ

1 2

2

d 15.

Fit correlator to

SUSY Relics in QCD (Aharony, Shifman, and Veneziano (2002)):

(P. Keith-Hynes and HT, Phys. Rev. D, 2006)

Holographic projection of orientifold compactification of string theory predicts “planar (large N) equivalence” between SUSY YM and ordinary 1-flavor QCD.

A “SUSY relic” prediction that can be tested by Monte Carlo: Degeneracy between the scalar and pseudoscalar mesons in 1-flavor QCD (they belong to the same WZ multipltet in SUSY YM chiral Lagrangian (Veneziano, Yankielowicz)).

Prediction is tested using MC results for scalar and pseudoscalar valence and hairpin diagrams. (see poster).

D-branes of topological charge in ordinary QCD may be SUSY relics – related to domain walls between discrete chiral condensates in SUSY.

Role of vacuum Dbranes in breaking conformal symmetry points to a connection between chiral anomaly and conformal anomaly (which also belong to same WZ multiplet in SUSY YM).

N1

N1

Test of scalar-pseudoscalar degeneracy in 1-flavor QCD: (with Patrick Keith-Hynes)

(valence mass)2 (hairpin mass shift)2 (total mass)2

+ [315(6)]2 +[407(11)]2 +[515(13)]2

+[1416(14)]2 - [1350(90)]2 +[427+249-756]2

Mass (GeV)

Pseudoscalar massScalar mass

1.0

0.5