Download Lattice QCD and String Theory Lattice 2005 Julius Kuti Confining Force

Lattice QCD and String Theory
Julius Kuti
University of California, San Diego
Lattice 2005
July 25, 2005
Trinity College, Dublin
Confining Force
Quark
What is this confining fuzz?
AntiQuark
String in QCD ?
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?
Some early developments:
Polyakov
..
Luscher,Symanzik,Weisz
P. Hasenfratz et al.
..
Luscher
Michael et al., Ford
Polchinski and Strominger
Gliozzi et al.
..
Munster
et al.
Juge, JK, Morningstar
Teper
1977
1980
1980
1981
1990
1991
1996
1997
1997
1998
strings from field theory
Wilson loops and QCD string
QCD picture of gluon excitation on all scales
universal Casimir energy
first on-lattice string excitations
effective non-critical string (fixes Nambu-Goto)
high precision Z(2) free energy
two-loop Z(2) interface
first comprehensive QCD string spectrum
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
string breaking
Hagedorn temperature
K-strings
`t Hooft flux quantization
Neuberger, Teper, Brower
Bali
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
R=6.75 fm
D=3 Z(2) lattice gauge theory
R=0.675 fm
R=6.75 fm
quantum string simulation in real time
Z(2) lattice gauge theory D=3
Y
X
φ(x, y, t) is plotted
Wilson Surface of 3d Z(2) Gauge Model
Similar picture expected in QCD
1
2
β = − ln(tanh β )
Z(2) gauge
Ising duality
confining phase
mass gap
0
gapless surface
smooth β R =0.407
ro ug h
deconfined
β c =0.2216
roughening transition
Kosterlitz-Thouless
universality class
role of skrew dislocations
in Wilson surface
Loop Expansion
Soliton Quantization (string)
Effective Schrodinger equation
based on fluctuation matrix of string soliton
critical region
continuum limit (QCD)
effective φ4 field theory
M = −∇2 + U" (φsoliton )
effective potential
long flux limit: spectrum expected to factorize
translational zero mode Æ Goldstone spectrum
zero energy
bound state
x
y
φ(y) ⋅ exp(iqx)
approximate here, but exact for torelon
π
q = n, n = 1, 2,3...
L
Px = +-1 and
Pz = +-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:
classical soliton
φ(x, y, t) = φs (y − ξ(x, t)) + η(x, y − ξ(x, t), t)
y
x
collective variables fluctuation field
(quantized)
quantum field
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
y
x
∂ a ξ∂ a ξ∂ y η∂ y φs
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:
massive scalar
η
~ ∂ µ ξ∂ µ ξ
Goldstone
ξ
Goldstone
ξ
π
n << M
R
n << R ⋅ M / π
when q =
∂ a ξ∂ a ξ∂ y η∂ y φs
ξ is the D-2 dimensional displacement
Boundary operators set to zero in
open-closed string duality
vector (collective string variables)
S LW = σRT + µT + S 0 + S1 + S 2 + S 3 + …
T
R
⎧
⎫
1
⎪1
⎪
S0 =
d τ ∫ d σ ⎨ ∂ a ξ∂ a ξ ⎬
∫
⎪2
⎪
2πα ' 0
⎪
⎪
⎩
⎭
0
T
1
∂ µ ξ∂ µ ξ
q + M2
2
V ( R ) = σR + µ −
π
b
(D − 2)(1+ )
R
24R
π
b
∆E = (1+ )
1
S1 = b ∫ d τ {(∂1ξ∂1ξ )σ=0 + (∂1ξ∂1ξ )σ=R }
R
R
4 0
T
R
T
R
⎧
⎫
⎧1
⎫
1
1
⎪1
⎪
⎪
S2 = c2 ∫ d τ ∫ d σ ⎨ (∂ a ξ∂ a ξ )(∂ b ξ∂ b ξ )⎬ S3 = c3 ∫ d τ ∫ d σ ⎪
⎨ (∂ a ξ∂ b ξ )(∂ a ξ∂ b ξ )⎬
⎪2
⎪
⎪
⎪
4 0
4 0
⎪
⎪
⎩
⎭
⎪
⎪
⎩2
⎭
0
0
higher dimensional ops O(1/R3)
c term is not independent in D=3
3
Benchmark checklist:
- 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, …?
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
Poincare and conformal effective string action
One-dimensional string sweeps out two-dimensional world-sheet described by X µ (τ, σ)
(t,s) parameters mapped into the string coordinates: (X 0 ( τ, σ), X1 ( τ, σ),…, X d (τ, σ)) D = d + 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)
1.
simplest choice: Nambu-Goto string action SNG = ∫ dτdσLNG ,
LNG = −
M
1
(− det h ab )1/ 2
2πα '
M designates world-sheet
symmetries: D-dimensional Poincare group
Two-dimensional coordinate (Diff) invariance
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.
Polyakov introduced world-sheet metric γ ab (τ, σ) into the action:
SP [X, γ ] = −
1
dτdσ(−γ )1/ 2 γ ab∂ a X µ∂ b X µ
4πα ' M∫
classically SNG and SP are equivalent
γ = det γ ab
SP 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
X 'µ (τ, σ) = X µ (τ, σ)
γ 'ab ( τ, σ) = exp(2ω( τ, σ)) ⋅ γ ab (τ, σ)
ω arbitrary
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 Xm fields
S0 =
1
dτ+ dτ−∂ + X µ ∂ − X µ
2πα ' ∫
+ 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:
26 − D
SL =
dτ+ dτ−∂ + φ∂ −φ
eiSL Polyakov determinant in conformal gauge
48π ∫
S'L =
S PS =
2
2
26 − D
+
− ∂ + X ⋅ ∂ − X∂ + X ⋅ ∂ − X
d
τ
d
τ
48π ∫
(∂ + X ⋅ ∂ − X) 2
substituting induced conformal gauge metric h +− for
⎤
∂ +2 X ⋅ ∂ − X∂ + X ⋅ ∂ −2 X
1
+
− ⎡ 1
µ
d
d
X
X
τ
τ
∂
∂
+
β
+ O(R −3 ) ⎥
⎢
+
− µ
2
∫
4π
(∂ + X ⋅ ∂ − X)
⎣ α '/ 2
⎦
will fix b=0, c2, and c3 in Luscher-Weisz effective action,
there will be higher order corrections with new unknown
parameters
β = βc ≡
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
Energy spectrum
π
4π 2 n 2
1
2
E N 2 = σ 2 R 2 ⋅ w 2 − σ ⋅ (D − 2) + 4πσ ⋅ (N + N) +
+
p
+
O(
)
J. Drummond
T
3
R2
R3
F. Maresca
N − N = n ⋅ w, winding number w = 1 for torelon
JK
2π ⋅ n
n=0,1,2,…
p=
R
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 D0 branes in modern language
π
2πN
E N = σR ⋅ 1 −
(D − 2) +
2
12σR
σR 2
highly degenerate
same as fixed end NG energy spectrum
N=0 ground state (Casimir energy)
Arvis
eφ
Caselle
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
..
Luscher-Weisz Casimir Energy
Ceff(r)
SU(3)
V(r) = σ r + const
- π(D-2)/24r
..
D
b=0
Short distance
QCD running
F(r) = V’(r)
CLW (r ) =
asymptotic Casimir energy
-> string formation
D
b=0.08 fm
1 3
r F '(r )
2
−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
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
should not need b
NG
?
full PS
full PS
universal
subleading
universal
subleading
SU(3)
Luscher
Weisz
SU(2)
Caselle
Pepe
Rago
classical +
quantum
Z(2) D=3
Juge et al.
also Majumdar
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 = 5.8 and 6.0 163 32 lattice
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 = 0.60 483 lattice
Y shape configuration is favored by Casimir energy
Holland, JK
β = 0.59 1003 lattice
β = 0.56 1003 lattice
D=3 Z(3) gauge model in dual potts spin representation
work in progress: space-time picture, baryon string excitations
wetting at transition point
β = 0.554 1003 lattice
β = 0.5512 1003 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
Summary of main results on the
spectrum of the fixed end Z(2) string
first NG term
full PS
end distortions
in field theory
first subleading
x= σR
dimensionless scale variable
EN / σ = x 1− D − 22 π + 2π2N
12 x
x
NG
Expand energy gaps for large x
R( EN − E0 )
π
=N+
a ( N ) b( N )
+ 4 +…
x2
x
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
Majumdar
?
full PS
Juge et al.
Juge et al.
universal
subleading
Juge et al.
Juge et al.
Surprise: data point approach the PS prediction
from above
Juge et al.
universal subleading term suggested
the approach from below
data follows the full PS curve more
closely than expected
Juge et al.
SU(3) D=4
Opposite fixed color
source (antiquark)
Three exact quantum numbers
characterize gluon excitations:
R
Λ+ −
Λ Æ angular momentum
projected along
quark-antiquark axis
+−
+−
Fixed color source
(quark)
S states (Λ =0)
Σg
P states (Λ =1)
Π
D states (Λ =2)
∆
.
.
.
+−
CP
Angular momentum
with chirality
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
Multipole states adiabatically evolve into
string states with expected level ordering
in large R limit
Σu+ , Π u' , ∆ u
Σu−
Short distance multipole expansion:
LW
C~1
string?
ΠU
Bali an Pineda
H Cb = H gauge +
Q⋅Q ={
g 2Q ⋅ Q
− g 3 (Q × Q)a ∫ d 3 rAa (r,t)J(r,t)
4π | r 1 − r 2 |
4
singlet
3
1
+ octet
6
−
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
universal
subleading
Juge et al.
full PS
no end effects
universal
subleading
Juge et al.
full PS
no end effects
full PS
F. Maresca and M. Peardon
Trinity, 2004
Typical effective mass plots
show good plateaus
universal
subleading
15 basic torelon operators translated
and fuzzed in large correlation matrices
Perfect string degeneracies
E12 = E02 + 4πσ + p32
p3 =
E12 = E02 + 8πσ
2π
L
Expected string behavior
more lattice systematics is needed
required technology is in place
E = E + 8πσ + p
2
1
2
0
4π
p3 =
L
2
3
E12 = E02 + 12πσ + p32
p3 =
2π
L
Conclusions
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