Lattice QCD and String Theory

Julius KutiUniversity of California, San Diego

International Conference on QCD and Hadronic PhysicsJune 19, 2005

Peking University

Collaborators:

Jimmy Juge DublinFrancesca Maresca UtahColin Morningstar Carnegie MellonMike Peardon Dublin

Early work: PolyakovLuscherPolchinski, StromingerBaker et al.Michael Teper Gliozzi et al.Hasenbusch, PinnJKM (old)Munster…

Juge, JK, Morningstar fixed end spectrum with fine structureHEP-LAT 0207004, PRL 90 (2003) 161601

Juge, JK, Maresca, Morningstar, Peardon closed winding string with fine structureHEP-LAT 0309180, Nucl.Phys.Proc.129:703-705,2004

Luscher, Weisz ground state Casimir energy JHEP 0207 (2002) 049, JHEP 0407:014,2004 open-closed string duality

Juge et al. and Caselle et al. Z(2) gauge model in 3 dimensions new work (first presented here)

This talk: review on the excitation spectrumof the Dirichlet string and the closed stringwith unit winding (string-soliton)

Recent work in QCD and Z(2):

OUTLINE

1. String formation in field theory - picture in space and time - main physical properties of the string

2. Dirichlet Strings in D=4 and D=3 dimensions - fixed end D=3 Z(2) string and SU(2) QCD string new results - fixed end D=4 SU(3) QCD string spectrum

3. Dirichlet Casimir Energy - origin of Casimir energy and effective string description - Luscher-Weisz results - Z(2) - paradox ? - 1+1 dimensional toy model insight from quantum mechanics

4. Closed String (torelon) with unit winding - D=4 SU(3) QCD spectrum new results - Closed string Casimir energy new results - D=3 Z(2) spectrum new results

5. Conclusions

Will not discuss: high spin Glueball spectrum

Casimir scaling of the flux

`t Hooft flux quantization de Forcrand baryon string configuration

finite temperature phase transition

Teper and collaborators

Lattice will be used as a theoretical tool

QuarkAntiQuark

Confining Force What is this confining fuzz?

String in QCD ?

String theorists interested in QCD string problem

Quenched, but relevant in confinement/string and large N

1. On-lattice QCD string spectrum

2. D=3 Z(2) gauge model microscopic loop equations (Polyakov) macroscopic string 3d Ising interface

Casimir energy of ground stateExcitation spectrumGoldstone modes and collective variablesEffective theory?Microscopic variables (loop equation)?Geometric interpretation?

AdS/CFT

scale of string formation?

Wilson Surface of 3d Z(2) Gauge Model

smooth ro 0 =0.407 =0.2216 ug h R c

mass gap

roughening transitionKosterlitz-Thouless universality class

gapless surface

confining phase

deconfined

critical regioncontinuum limit (QCD)effective field theory

Z(2) gauge Ising duality

Similar picture expected in QCD

4

1ln(tanh )

2

Semiclassical Loop ExpansionSoliton Quantization (string)

role of skrew dislocationsin Wilson surface

-in long flux limit spectrum is expected to factorize

- translational zero mode of soliton

- Goldstone spectrum

Effective Schrodinger equation based on fluctuation matrix of string soliton

2 "solitonM U ( )

effVL 30

xz

(z) exp(iqx)

q n, n 1,2,3...L

zero energybound state

quantized momenta of Goldstone modesin box of length L

shape and end effects distort!

good analytic/numerical handle on the Z(2) model

in addition to MC

Effective Schrodinger potential

Px = +-1 and Pz = +-1

two symmetry quantum numbers

X

Z

a = 0.04 fm

“Bag”

D=3 Z(2) Gauge Group

N=1 “massless excitation”

120x120 spatial lattice R=8 ~0.08 fm

bag-like dipole flip-flop

Analytic (soliton quantization and loop expansion)

D=3 Z(2) Gauge Group

N=3 massless excitation

120x120 spatial lattice R=60 ~ 6 fm

massless string-like oscillations

- Massless Goldstone modes?

- Local derivative expansion for their interactions? from fine structure in the spectrum

- Massive excitations?

- Breathing modes in effective Lagrangian? - String properties ? Bosonic, NG, rigid, …?

In rough phase (close to bulk critical point)

Most important step in deriving correction terms in effective actionof Goldstone modes in Z(2) D=3 gauge model:

Goldstone

Goldstone

massive scalar

2 2

1~

q M

2 2

0 0

1 1( )( )

4 2

T R

a a b bS c d dt s x x x xì üï ïï ï= ¶ ¶ ¶ ¶ +×××í ýï ïï ïî þ

ò ò

when q n MR

n R M/

resonancenon-local terms

0 0

1 12 ' 2

T R

eff a aS d dt s x xpa

ì üï ïï ï= ¶ ¶ +×××í ýï ïï ïî þ

ò òMassless Goldstone field collective string coordinate

{ }1 1 1 0 1 1

0

1( ) ( )

4

T

RS b d s st x x x x= == ¶ ¶ + ¶ ¶ò Boundary operators set to zeroin open-closed string duality

( ) ( 2)(1 )24

bV R R d

R Rp

s m= + - - +

(1 )b

ER Rp

D = +

higher dimensional ops O(1/R3)

Small wavelengths unstable! => glueball emission

22

0 0

{ ( )( )2

T R

a a b b

cS d dt s x x x x= ¶ ¶ ¶ ¶ +ò ò 3 ( )( ) ...}

2 a b a b

cx x x x¶ ¶ ¶ ¶ +

term is not independent in D=33c

is the D-2 dimensional displacement vector (collective string variables)x

symbol:circles

SU(2) and center Z(2) exhibit nearly universal behavior

includes next termin NG prediction

universal

first NG term

first termin field theory

2 2

dimensionless scale variable

2 2/ 112N

x R

D NE xx x

Summary of main results on the spectrum of the fixed end Z(2) string

NG

Expand energy gaps for large x

First correction to asymptotic spectrumappears to be universal

Higher corrections code new physicslike string rigidity, etc.

Similar expansion for string-solitonwith unit winding

Data for R < 4 fermi prefers field theorydescription which incorporates end effects naturally

02 4

( ) ( ) ( )NR E E a N b NNx x

D=3 Z(2) Gauge Group

N=1 “massive excitation”

120x120 spatial lattice R=8 ~ 0.08 fm

Bag-like breathing mode

D=3 Z(2) SU(2) Center Group

N=2 massive excitation

120x120 lattice R=60 ~ 6 fm

massive string-like oscillations

Fixed color source(quark)

Opposite fixed color source (antiquark)

angular momentum projected along quark-antiquark axis

CP

Three exact quantum numbers characterize gluon excitations:

Angular momentumwith chirality

Chirality, or reflectionsymmetry for = 0

g (gerade) CP evenu (ungerade) CP odd

S states ( =0) g

P states ( =1)

D states ( =2)

RSU(3) D=4

Gluon excitations are projected out withgeneralized Wilson loop operators on time sclices

the spatial straight line is replaced bylinear combinations of twisted paths

Three length scales in energy spectrum:R < 0.5 fm R < 0.5 fm

Short distance QCDShort distance QCDBag-like nearly spherical symmetryBag-like nearly spherical symmetry

OPE (Soto et al.)

R ~ 0.5 fm – 1.5 fm

Crossover (model sensitive)

R ~ 2 fm - 3 fm

Onset of string ordering

'u u u

'g g g g

u u

~ ~

~ ~ ~

~

Nambu-Goto levelsin black Fine structure

LW C~1

string?Casimir energy puzzle:

seen around R ~ 0.5 fm ?

1. Very few stable modes

2. Non string-like distortions

Short distance region

23 3

aCb gauge a1 2

1 1 2 2 1 2

singlet

octet

g Q QH H g (Q Q) d rA (r, t)J(r, t)

4 | r r |

1 1 1 14 J(r) ( ) ( )

| r r | | r r | | r r | | r r |

Q Q

FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFF

FFHF

431

+6

Multipole operator product expansion

of A(r,t) and J(r,t):FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF

Crossover transition regiondifficult to interpretmodel dependent

R<0.5fm approximate spherical symmetryR<0.5fm approximate spherical symmetryBag-like “non-string” pictureBag-like “non-string” picture

'u u u

'g g g g

u u

~ ~

~ ~ ~

~

Luscher-Weisz Casimir Energy

31( ) '( )

2LWC r r F r

SU(3)

Short distance

QCD runningasymptotic Casimir energy-> string formation

V(r) = r + const - (d-2)/24r

F(r) = V’(r)

CLW(r)

(d-2)/24

asymptotic r infinity

Evidence for stringformation in QCD?

Loop equationsADS string theoryQuark loops ?

b=0

b=0.08 fm

is this significant?

NG

NG

is this significant?

Casimir paradox similar to QCD

Z(2) Casimir energy

LW

R=0.4fm

Two basic questions:

• Why the precocious onset of Ceff ~ 1?

• Where does the central charge C=1 reside? On a geometric string?

Or distributed between massless Goldstone modes and the bulk?

Answer to second question will determine whether early

onset of Ceff ~1 is a true signal of string formation,

or just an accident

12 241

n

n

smart enoughfor string theory?

We turn to the D=1+1 lattice for learning how to do the sum:

22

2 1( , ) ( )

2 24reg

LE L a O a

a a L

This is NOT a paradox

2 2reg 2

2L 1E (L,a) O(a /L )

a 2a 24L

2 2

2a

128 L

2

2d E(L,a)312

eff dLC (L) L

String tension

end effects

Casimir

1D example:J(x) is represented by infinite Dirichlet walls at two ends

Vanishing correctionin a -> 0 limit

31

2 241

n

n

This IS a paradox

How to eliminate problematic end effects?

string-solitons with unit winding

(x L)

square well

(x)

L-M2

0

1

10

barrier

spectrum

L=50

M=0.5

Energy spectrum (world box size varied)Ceff=0.99 Ceff=0.85

=10barrie

r

=0no barriersquare well

sharp resonanceslattice string spectrum

avoidedlevel crossings

no resonances, butCeff is missing only 15% !!

World size with PBC

4. Closed String (torelon) with unit winding

- D=4 SU(3) QCD spectrum new results

- Closed string Casimir energy new results

- D=3 Z(2) spectrum new results

2 22 2 2

2 2 2 2

4 4{1 ( ) }

3 ( )N N

nE R N N

R R R

Relativistic excitation energies of D=3 string soliton

2

n N N

p nR

Exact in NG

O(R-4) corrections in Polchinski-Strominger

Francesca Maresca, PhD thesis, Dublin, 2004

15 basic torelon operators translated and fuzzed in large correlation matrices

Point group notation for string states

2 2 2 20

2( )

3E L O L

Casimir energy

Exact in NG string no O(L-2) correction

Polchinski-Strominger expect correctionseven without rigidity term

Crossover from short distance behavior to string level ordering

a

~ 0.2 fmsa

Expected string behavior

2 2 21 0 3

3

4

2

E E p

pL

2 21 0 8E E

Large L?

Large L?

2 2 21 0 3

3

8

4

E E p

pL

Large L?Fine structure not Nambu-Goto

2 2 21 0 312E E p

Perfect string degeneracies

Perfect string degeneracies

- Massless Goldstone modes

- Local derivative expansion for their interactions from fine structure in the spectrum

- Massive excitations

- Breathing modes in effective Lagrangian - String properties ? Bosonic, NG, rigid, …

QCD String check list

?

WITHIN REACH of

LATTICE GAUGE THEORY

Conclusions:

1. Fine structure in QCD string spectrum Progress on string-soliton spectrum 2. Casimir energy paradox: low energy Goldstone

modes geometric string theory?

3. What is the large N limit ? (Herbert Neuberger)

4. Effective low-energy string theory? Universality class of QCD string ?

neither was seen before