lattice qcd and string theory julius kuti university of california, san diego international...

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