July 25, 2005
Trinity College, Dublin
AntiQuark
Quark
String theorists interested in QCD string problem
Quenched, but relevant in confinement/string connection large N ADS/CFT
On-lattice QCD string excitation spectrum
Casimir energy of ground state
Goldstone modes
Æ collective string variables
Effective string theory?

Polyakov
..
1977 strings from field theory
Luscher,Symanzik,Weisz 1980 Wilson loops and QCD string
Luscher
1980 QCD picture of gluon excitation on all scales
1981 universal Casimir energy
Michael et al., Ford 1990 first on-lattice string excitations
Polchinski and Strominger 1991 effective non-critical string (fixes Nambu-Goto)
Gliozzi et al.
Munster et al.
1996 high precision Z(2) free energy
1997 two-loop Z(2) interface
Juge, JK, Morningstar 1997 first comprehensive QCD string spectrum
Teper 1998 large N
This talk: review of recent work on Casimir energy, the excitation spectrum of the Dirichlet string, and the closed string with unit winding (string-soliton)
Will not discuss: large N and AdS/CFT connection Neuberger, Teper , Brower string breaking
Hagedorn temperature
Bali
K-strings
`t Hooft flux quantization
OUTLINE
1. String formation in field theory real time string formation from simulations
Z(2) gauge field theory and LW effective string action benchmark checklist
2. Poincare and conformal effective string theory
Nambu-Goto string and its problems unless D = 26
Polyakov string and its problems unless D = 26
Poincare invariant and conformal effective string theory any D
3. Ground state Casimir energy
D=3 SU(3), SU(2), Z(2), Z(3)
D=4 SU(3)
4. Dirichlet string spectrum
D=3 Z(2), SU(2)
D=4 SU(3)
5. Closed string spectrum (torelon with unit winding)
D=4 SU(3) and D=3 Z(2)
6. Conclusions

R = 0.675 fm
D=3 Z(2) lattice gauge theory
R=6.75 fm
R=0.675 fm R=6.75 fm
Y quantum string simulation in real time
Z(2) lattice gauge theory D=3
(x, y, t) is plotted
X

Similar picture expected in QCD
β = −
1
2
Z(2) gauge Ising duality confining phase mass gap gapless surface
0 smooth
β
R roughening transition
Kosterlitz-Thouless role of skrew dislocations in Wilson surface universality class
β deconfined critical region continuum limit (QCD)
φ
Effective Schrodinger equation based on fluctuation matrix of string soliton M
= −∇ + "
U (
φ soliton
) effective potential x y long flux limit: spectrum expected to factorize translational zero mode Æ Goldstone spectrum zero energy bound state
φ
(y) exp(iqx) q
=
π
L n, n
= approximate here, but exact for torelon
1, 2,3...
P x
= +-1 and P z
= +-1 two symmetry quantum numbers mean field energy eigenvalues and wave functions are determined numerically exact ones from simulations shape and end effects distort the transfer to collective geometric variables

‘classical’ torelon ground state classical solutions suggest to introduce collective coordinates x( x,t) which describe the undulating motion in y-direction:
φ
(x, y, t) classical soliton
= φ s
(y
− ξ
(x, t))
+ η
(x, y
− ξ
(x, t), t) quantum field collective variables
(quantized) fluctuation field x y
Path integral can be written in terms of massless x field and massive h field interaction Lagrangian, FP determinants worked out
‘classical’ ground state, fixed end sources flipped at symmetry axis x y typical derivative interaction term a labels
( x,t) world sheet coordinate indices
To get effective string theory, we have to integrate out the massive field h
Most important step in deriving correction terms in effective action of Goldstone modes in Z(2) D=3 gauge model:
Goldstone
ξ massive scalar
η
Goldstone
ξ
~
µ
1 q 2 + M 2 µ when q =
π
R n << M n << ⋅ π
ξ is the D-2 dimensional displacement vector (collective string variables)
Boundary operators set to zero in open-closed string duality
S
LW
S
0
=
= σ
RT T S
0
+
S
1
+
S
2
+
S
3
+ …
1
2 πα '
T
0
R d τ d σ
0
⎪
⎪
1
2
S
1
=
1
4
T
0
τ { ( ∂ ∂
1
= σ R + µ −
)
σ = 0
π
24 R
+ ( ∂ ∂ )
σ = R
}
( D − 2)(1 + b
R
)
∆ E =
π
(1 + b
R R
)
S
2
=
1
4 c
2
T
0
R
0 d σ ⎨
⎪
1
2
( ∂ ∂ )( ∂ ∂ ) S
3
=
1
4 c
3
T
0
R
0 d σ ⎨
⎩
1
2
( ∂ ∂ )( ∂ ∂ )
⎫
⎭ higher dimensional ops O(1/R 3 ) c
3
term is not independent in D=3

from fine structure in the spectrum
OUTLINE
1. String formation in field theory real time string formation from simulations
Z(2) gauge field theory and LW effective string action benchmark checklist
2. Poincare and conformal effective string theory
Nambu-Goto string and its problems unless D = 26
Polyakov string and its problems unless D = 26
Poincare invariant and conformal effective string theory any D
3. Ground state Casimir energy
D=3 SU(3), SU(2), Z(2), Z(3)
D=4 SU(3)
4. Dirichlet string spectrum
D=3 Z(2), SU(2)
D=4 SU(3)
5. Closed string spectrum (torelon with unit winding)
D=4 SU(3) and D=3 Z(2)
6. Conclusions

One-dimensional string sweeps out two-dimensional world-sheet described by
µ τ σ
( t,s) parameters mapped into the string coordinates: 0 τ σ 1 τ σ … d τ σ = +
1.
Consistent relativistic quantum theory requires parametrization invariance: action should only depend on embedding in spacetime characterized by the induced metric h ab
= ∂ a
X
µ ∂ b
X
µ a,b run over ( t,s) simplest choice: Nambu-Goto string action S
NG
= d d
M
L
NG
, L
NG
= −
1
2
πα
'
M designates world-sheet symmetries: D-dimensional Poincare group
Two-dimensional coordinate (Diff) invariance
− ab
1/ 2 string tension problems: Light cone quantization which has the correct D-2 oscillators spoils
Lorentz invariance outside the critical dimension D=26
Covariant (Virasoro) quantization leads to D-1 oscillators unless D=26
Z(2) vortex, Nielsen-Olesen vortex, and QCD string require Poincare invariance and D-2 oscillators (longitudinal oscillator is not protected as a Goldstone mode of symmetry breaking)
2.
S [X, ]
P
γ = −
1
4
πα
'
M d d ( )
1/ 2 γ ∂ a
X
µ ∂ b
X
µ
γ = det
γ ab classically S
NG and S
P are equivalent
S
P has three symmetries: D-dimensional Poincare invariance two-dimensional coordinate (Diff) invariance two-dimensional Weyl invariance new
Weyl equivalent metrics correspond to the same embedding in space-time
µ
X ' ( , )
µ
X ( , )
γ ab
τ σ = ω τ σ ⋅ γ τ σ ab
ω ar problem : Polyakov quantization contains additional Liouville mode (scalar field) and leads again to D-1 oscillators
3.
Polchinski and Strominger: we need a new effective string description
For strings emerging from field theory we would like to require D-2 oscillators, Poincare invariance, and dim invariance
Goldstone theorem does not protect longitudinal mode from acquiring a mass (breathing mode of Z(2) vortex is an example)

PS: to resolve the paradox we could start from path integral of field theory with collective coordinate quantization including measure terms
Convert path integral to covarian form with D unconstrained X m fields
S
0
=
1
2
πα
'
∫ τ τ ∂
+
X
µ ∂
−
X
µ
+ determinants
The measure derives from the physical motion of the underlying gauge fields and therefore should be built from physical objects, like the induced metric h ab
= ∂ a
X
µ ∂ b
X
µ same determinant as Polyakov, but built from induced metric:
S
L
=
26 D
48
π
∫ + d d
+ − e iS
L
Polyakov determinant in conformal gauge
S
'
L
=
48
π
∫ + d d
−
∂ 2
+
X
(
⋅∂ ∂
∂
+
X
⋅∂
+
−
X
X)
⋅∂
2
2
−
X h e
φ substituting induced conformal gauge metric for
S
PS
=
1
4
π
will fix b=0, c2, and c3 in Luscher-Weisz effective action, there will be higher order corrections with new unknown parameters
−
⎡
⎣
1
α
'/ 2
∂
+
X
µ ∂
−
X
µ
+ β
∂ 2
+
X
−
X
+
X
⋅ ∂ 2
−
X
(
∂
+
X
⋅ ∂
−
X)
2
+ −
3
O(R )
⎤
⎦
D
−
26
12 conform invariant
Poincare invariant
D-2 oscillators anomaly free
PS String-soliton with winding number w along the compact dimension
R is the length of the compact direction
E
N
2 = σ 2 2
R
⋅ w
2
π
3
(D 2) 4 (N
+
N )
+
4
N
− = ⋅ =
1 f or torelon
π
R
2 2
2 n p=
2
π ⋅ n
R n=0,1,2,…
+ 2 +
R
1 p O( )
T 3
Energy spectrum
J. Drummond
F. Maresca
JK p is the momentum along compact dimension by left and right movers
N
+
N sum of right and left movers
Poincare invariant conformal field theory
D-2 oscillator states and anomaly free
Dirichlet string open string attached to D
0 branes in modern language Caselle
E
N
R 1
−
π
2
(D 2)
σ
R
2 same as fixed end NG energy spectrum
N=0 ground state (Casimir energy)
Arvis highly degenerate

OUTLINE
1. String formation in field theory real time string formation from simulations
Z(2) gauge field theory and LW effective string action benchmark checklist
2. Poincare and conformal effective string theory
Nambu-Goto string and its problems unless D = 26
Polyakov string and its problems unless D = 26
Poincare invariant and conformal effective string theory any D
3. Ground state Casimir energy
D=3 SU(3), SU(2), Z(2), Z(3)
D=4 SU(3)
4. Dirichlet string spectrum
D=3 Z(2), SU(2)
D=4 SU(3)
5. Closed string spectrum (torelon with unit winding)
D=4 SU(3) and D=3 Z(2)
6. Conclusions
..
C eff
(r)
..
SU(3)
V(r) = σ r + const
π (D-2)/24r
F(r) = V’(r) b=0
D
Short distance
QCD running
C ( )
LW
=
1
2
3
'( ) asymptotic Casimir energy
-> string formation
D b=0.08 fm
−π (D-2)/24 asymptotic r
Æ infinity
Evidence for string formation in QCD?
Quark loops ?

Changing dimension D shows significant difference
It is D = 2+1 which is more tantalizing
NG universal subleading full PS should not need b
NG
?
LW invoked a boundary operator for D = 3+1
Later LW showed that b=0 is set from matching te
Polyakov loop correlator spectrum to open-closed string duality full PS universal subleading
SU(3)
Luscher
Weisz
SU(2)
Caselle also Majumdar
Pepe
Rago universal subleading full PS
Z(2) D=3 classical + quantum
Juge et al.
Caselle et al. global fit zero-point oscillators only
Conclusions: 1. Casimir energy does not fully match expected universal behavior
2. End distortions matter, sensitive to D, boundary operators?
3. limited global fit to torelon Casimir energy by Meyer and Teper consistency with universal subleading correction is reported
4. It is difficult to discover the string from Casimir energy

Casimir energy of baryon string
Takahashi, Suganuma
D=4 SU(3)
Phys. Rev. D70 (2004) 074506
Y shape is favored, first excitation is reported
β = de Forcrand and Jahn
Nuclear Physics A 755 (2005) 475
D=3 Z(3) gauge model in dual potts spin representation analytic Casimir term by de Forcrand et al.
β =
Y shape configuration is favored by Casimir energy
Holland, JK
β = 3
0.59 100 lattice
β = 3
0.56 100 lattice
D=3 Z(3) gauge model in dual potts spin representation work in progress: space-time picture, baryon string excitations
β = wetting at transition point
β = 3
0.5512 100 lattice

OUTLINE
1. String formation in field theory real time string formation from simulations
Z(2) gauge field theory and LW effective string action benchmark checklist
2. Poincare and conformal effective string theory
Nambu-Goto string and its problems unless D = 26
Polyakov string and its problems unless D = 26
Poincare invariant and conformal effective string theory any D
3. Ground state Casimir energy
D=3 SU(3), SU(2), Z(2), Z(3)
D=4 SU(3)
4. Dirichlet string spectrum
D=3 Z(2), SU(2)
D=4 SU(3)
5. Closed string spectrum (torelon with unit winding)
D=4 SU(3) and D=3 Z(2)
6. Conclusions
PS string

PS string
D=3 SU(2) and Z(2) exhibit similar behavior symbol:circles end distortion
PS universal subleading
PS

end distortions in field theory first NG term first subleading
Summary of main results on the spectrum of the fixed end Z(2) string full PS x
= σ
R dimensionless scale variable
E
N
/
σ = x 1
− D
12
− x 2
2 π + 2
π x 2
N NG
Expand energy gaps for large x
(
N
π
−
E
0
) ( ) x 2
+ ( ) x 4
+ …
First correction to asymptotic spectrum appears to be universal
Higher corrections code new physics like string rigidity, etc.
Similar expansion for string-soliton with unit winding
Data for R < 4 fermi prefers field theory description which incorporates end effects naturally full PS
Juge et al.
universal subleading
Majumdar
?
Juge et al.
Juge et al.
Juge et al.

Surprise: data point approach the PS prediction from above universal subleading term suggested the approach from below data follows the full PS curve more closely than expected
Juge et al.
Juge et al.

Opposite fixed color source (antiquark)
R
Fixed color source
(quark)
Æ angular momentum projected along quark-antiquark axis
S states (
Λ
=0)
P states (
Λ
=1)
D states (
Λ
=2)
..
.
g
+ −
+ −
+ −
Three exact quantum numbers characterize gluon excitations:
+ −
Angular momentum with chirality
CP
Chirality, or reflection symmetry for
Λ
= 0 g (gerade) CP even u (ungerade) CP odd
Juge, JK, Morningstar SU(3) Dirichlet spectrum with fine structure
PRL 90 (2003) 161601
Gluon excitations are projected out with generalized Wilson loop operators on time sclices the spatial straight line is replaced by linear combinations of twisted paths effective mass plot analysis from large correlation matrix

+
,
, u
Σ − u
LW
C~1 string?
Π
U
Multipole states adiabatically evolve into string states with expected level ordering in large R limit
Short distance multipole expansion:
H
Cb
=
H gauge
+
⋅
π
1
− r |
−
{ −
+
6
4
singlet
3
1
octet
× a
∫ d rA (r,t)J(r,t)
Bali an Pineda

OUTLINE
1. String formation in field theory real time string formation from simulations
Z(2) gauge field theory and LW effective string action benchmark checklist
2. Poincare and conformal effective string theory
Nambu-Goto string and its problems unless D = 26
Polyakov string and its problems unless D = 26
Poincare invariant and conformal effective string theory any D
3. Ground state Casimir energy
D=3 SU(3), SU(2), Z(2), Z(3)
D=4 SU(3)
4. Dirichlet string spectrum
D=3 Z(2), SU(2)
D=4 SU(3)
5. Closed string spectrum (torelon with unit winding)
D=4 SU(3) and D=3 Z(2)
6. Conclusions full PS universal subleading
Juge et al.
no end effects

full PS universal subleading full PS universal subleading
Juge et al.
no end effects
F. Maresca and M. Peardon
Trinity, 2004
15 basic torelon operators translated and fuzzed in large correlation matrices
Typical effective mass plots show good plateaus

Perfect string degeneracies
E
1
2 =
E
2
0
+ p
3
=
2
π
L
4
πσ + p
3
2
Expected string behavior
E
1
2
=
E 2
0
+
8
πσ
E
1
2
=
E 2
0
+ p
3
=
4
π
L
8
πσ + p
3
2 more lattice systematics is needed required technology is in place
E
2
1
=
E
2
0
+
12
πσ p
3
=
2
π
L
+ p
3
2

1. Poincare invariant conformal effective string provides the right framework to analyze the lattice QCD simulations
2. Fine structure in spectrum tests the string properties
3. End effects in Casimir energy and Dirichlet spectrum remain problematic for string theory interpretation
4.
Precocious approach to Poincare string spectrum more accurate data is needed to test higher terms
5. It remains a challenge for String Theory to explain the
Poincare invariant and conformal effective QCD string