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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
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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):
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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
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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
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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?
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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
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-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!
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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
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a = 0.04 fm
“Bag”
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D=3 Z(2) Gauge Group
N=1 “massless excitation”
120x120 spatial lattice R=8 ~0.08 fm
bag-like dipole flip-flop
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Analytic (soliton quantization and loop expansion)
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D=3 Z(2) Gauge Group
N=3 massless excitation
120x120 spatial lattice R=60 ~ 6 fm
massless string-like oscillations
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- 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, …?
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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
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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
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symbol:circles
SU(2) and center Z(2) exhibit nearly universal behavior
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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
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D=3 Z(2) Gauge Group
N=1 “massive excitation”
120x120 spatial lattice R=8 ~ 0.08 fm
Bag-like breathing mode
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D=3 Z(2) SU(2) Center Group
N=2 massive excitation
120x120 lattice R=60 ~ 6 fm
massive string-like oscillations
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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
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Gluon excitations are projected out withgeneralized Wilson loop operators on time sclices
the spatial straight line is replaced bylinear combinations of twisted paths
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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
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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
~ ~
~ ~ ~
~
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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
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is this significant?
NG
NG
is this significant?
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Casimir paradox similar to QCD
Z(2) Casimir energy
LW
R=0.4fm
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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
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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
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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
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31
2 241
n
n
This IS a paradox
How to eliminate problematic end effects?
string-solitons with unit winding
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(x L)
square well
(x)
L-M2
0
1
10
barrier
spectrum
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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
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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
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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
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Francesca Maresca, PhD thesis, Dublin, 2004
15 basic torelon operators translated and fuzzed in large correlation matrices
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Point group notation for string states
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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
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Crossover from short distance behavior to string level ordering
a
~ 0.2 fmsa
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Expected string behavior
2 2 21 0 3
3
4
2
E E p
pL
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2 21 0 8E E
Large L?
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Large L?
2 2 21 0 3
3
8
4
E E p
pL
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Large L?Fine structure not Nambu-Goto
2 2 21 0 312E E p
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Perfect string degeneracies
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Perfect string degeneracies
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- 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
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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