INT Summer School on "Lattice QCD and its applications"
The Closed String Spectrum of S U(N) gauge Theories in2+1 Dimensions
Andreas Athenodorou
St. John’s College
Oxford University
with Barak Bringoltz and Mike Teper
xford
hysicsTM
1
I. Introduction.
General question:
→What effective string theory describes flux tubes in S U(N) gauge theories?
Two cases:
→ Open string
→ Closed string
During the last decade:
→ 3D, 4D with Z2, Z4,U(1), S U(N ≤ 6) (Caselle and collaborators, Gliozzi and collaborators,
Kuti and collaborators, Luscher&Weisz, Majumdar and collaborators, Teper and collaborators, Meyer)
Questions to be studied in D = 2 + 1 dimensional S U(N) theories:
→ Calculation of excited states, and states with p‖ � 0 and P = ±→ What is the degeneracy pattern of these states?
→ What is the leading correction in E2n?
2
I. Introduction.Open flux tube Closed flux tube
x
l
t
periodic b.c−→l t
x
Φ(l, t) = φ†(0, t)U(0, l; t)φ(l, t) Φ(l, t) = TrU(l; t)
3
I. Introduction.
General question:
→What effective string theory describes flux tubes in S U(N) gauge theories?
Two cases:
→ Open string
→ Closed string ← in D = 2 + 1
During the last decade:
→ 3D, 4D with Z2, Z4,U(1), S U(N ≤ 6) (Caselle and collaborators, Gliozzi and collaborators,
Kuti and collaborators, Luscher&Weisz, Majumdar and collaborators, Teper and collaborators, Meyer)
Questions to be studied in D = 2 + 1 dimensional S U(N) theories:
→ Calculation of excited states, and states with p‖ � 0 and P = ±→ What is the degeneracy pattern of these states?
→ What is the leading correction in E2n?
4
II. Theoretical expectations A.
The Spectrum of the Nambu-Goto (NG) String Model
◦ Action of Nambu-Goto free bosonic string leads to:
→ Spectrum given by:
E2NL ,NR ,q,w
= (σlw)2 + 8πσ(
NL+NR2 − D−2
24
)+
(2πq
l
)2+ p2
⊥.
→ Described by:
1. The winding number w (w=1),
2. The winding momentum p‖ = 2πq/l with q = 0,±1,±2,
3. The transverse momentum p⊥ (p⊥ = 0),
4. NL and NR connected through the relation: NR − NL = qw.
NL =∑k>0
∑nL(k)>0
nL(k)k and NR =∑k′>0
∑nR(k′)>0
nR(k′)k′
→ String states are eigenvectors of P (In D = 2 + 1) with eigenvalues:
P = (−1)∑m
i=1 nL(ki)+∑m′
j=1 nR(k′j)
5
II. Theoretical expectations B.Effective string theory
◦ First prediction for w = 1 and q = 0 (Luscher, Symanzik&Weisz. 80):
En = σl +4πl
(n − D − 2
24
)+ O
(1/l2
).
◦ Luscher&Weisz effective string action (Luscher&Weisz. 04):
→ Using open-closed string duality:
∗ For any D the O(1/l2 (1/l)
)(Boundary term) is absent from En(E2
n)
→ Spectrum in D = 2 + 1 (Drummond ’04, Dass and Matlock ’06 for any D.):
En = σl +4πl
(n − 1
24
)− 8π2
σl3
(n − 1
24
)2
+ O(1/l4
)
→ Equivalently:
E2n = (σl)2 + 8πσ
(n − 1
24
)+ O
(1/l3
),
6
II. Theoretical expectations B.Effective string theory
◦ First prediction for w = 1 and q = 0 (Luscher, Symanzik&Weisz. 80):
En = σl +4πl
(n − D − 2
24
)+ O
(1/l2
).
◦ Luscher&Weisz effective string action (Luscher&Weisz. 04):
→ Using open-closed string duality:
∗ For any D the O(1/l2 (1/l)
)(Boundary term) is absent from En(E2
n)
→ Spectrum in D = 2 + 1 (Drummond ’04, Dass and Matlock ’06 for any D.):
En = σl +4πl
(n − 1
24
)− 8π2
σl3
(n − 1
24
)2
+ O(1/l4
)
→ Equivalently:
E2n = E2
NG + O(1/l3
)−→ Fit : E2
fit = E2NG − σ
Cp(l√σ)p (p ≥ 3)
7
III. Lattice CalculationOur approach:
• Construct a large basis of operators (80 − 200) with transverse deformations.
• Calculate the correlation matrix Ci j,p,± (t) = 〈Φ†i,p,± (t)Φ j,p,± (0)〉).• Use the variational technique to extract correlators of different states.
• Fit our results using single cosh fits, and look at large t mass plateaus.
Example:
t
T − t
Φi,p,± Φ†j,p,±
φL = Tr{ }
and φR = Tr{ }
Φp,± = 1L‖L⊥
∑x‖ ,x⊥ {φL ± φR} eipx‖
8
III. Lattice Calculation
Monte-Carlo:
• We define our gauge theory on a 3D discretized periodic Euclidean space-time
with L × L⊥ × LT sites.
• We choose to use the standard Wilson action:
S W = β∑
P
[1 − 1
NReTrUP
]
with: β = 2Nag2 .
• Simulation:
– We use a mixture of Kennedy-Pendelton heat bath and over-relaxation steps
for updating S U(2) subgroups of S U(N).
– We use Cabibbo-Marinary for updating S U(N).
9
III. Lattice Calculation: Operators for P = +
×5 bl ×5 bl ×5 bl ×5 bl ×5 bl
×5 bl ×5 bl ×5 bl ×5 bl f (L‖, L⊥, bl)
×5 bl ×5 bl ×5 bl ×5 bl ×5 bl
10
III. Lattice Calculation: Operators for P = −
×5 bl ×5 bl ×5 bl ×5 bl ×5 bl
×5 bl ×5 bl ×5 bl
×5 bl ×5 bl ×5 bl ×5 bl ×5 bl
11
III. Lattice Calculation: Large Basis of Operators
Using this large basis of operators:
• We extract masses of excited states.(up to 15 states)
• It increases the Overlaps (using single exponential fits):
– Ground state ∼ 99 − 100%,
– First excited state ∼ 98 − 100% (∼ 90 − 95 with just ×5bl),
– Second excited state ∼ 95 − 99% (∼ 85 − 90 with just ×5bl),
• We can extract energies of non-zero winding momentum states.
• We can extract energies of P = − states.
• It increases computational time moderately.(ex. ×6 for L = 16a)
12
IV. Results: Spectrum of S U(3) and β = 21.0
Group: S U(3), a � 0.08 f m, Quantum Numbers: P = +,− and q = 0
√σl
E/√σ
654321
12
10
8
6
4
2
0
n = 0
n = 1
n = 2
n = 3
NG Prediction: E2n = (σl)2 + 8πσ
(n − 1
24
), where n = NL = NR since q = 0.
13
IV. Results: Spectrum of S U(3) and β = 40.0
Group: S U(3), a � 0.04 f m, Quantum Numbers: P = +,− and q = 0
√σl
E/√σ
654321
12
10
8
6
4
2
0
n = 0
n = 1
n = 2
n = 3
NG Prediction: E2n = (σl)2 + 8πσ
(n − 1
24
), where n = NL = NR since q = 0.
14
IV. Results: Spectrum of S U(6) and β = 90.0
Group: S U(6), a � 0.08 f m, Quantum Numbers: P = +,− and q = 0
√σl
E/√σ
654321
12
10
8
6
4
2
0
n = 0
n = 1
n = 2
n = 3
NG Prediction: E2n = (σl)2 + 8πσ
(n − 1
24
), where n = NL = NR since q = 0.
15
IV. Results: Spectrum of S U(N)
Groups: S U(3) and S U(6), a � 0.04 f m and 0.08 f m,
Quantum Numbers: P = ± and q = 0
√σl
E/√σ
654321
9
8
7
6
5
4
3
2
1
0
n = 0
n = 1
n = 2
E1 = σl + 4πl
(1 − 1
24
)− 8π2
σl3
(1 − 1
24
)2
E1 = σl + 4πl
(1 − 1
24
)
NG Prediction: E2n = (σl)2 + 8πσ
(n − 1
24
)
16
IV. Results: Non-zero winding momentum.Group: S U(3), a � 0.08 f m, Quantum Numbers: P = +,−, q = 1, 2 and w = 1
l√σ
√ E2/σ−(
2πq/√ σ
l)2
4.543.532.52
9
8
7
6
5
4
q = 1, NR = 1, NL = 0
q = 2, NR = 2, NL = 0
q = 1, NR = 2, NL = 1
q = 2, NR = 3, NL = 1
q = 1, NR = 3, NL = 2
NG Prediction: E2 − (2πq/l)2 = (σlw)2 + 8πσ(
NR+NL2 − D−2
24
).
Constraint: NR − NL = qw
17
IV. Results: Fits.
Cp
p
0
-50
-100
-150
-200
8765432
�
S U(3)β = 21.00
S U(6)β = 90.00
S U(3)β = 41.00
Ansatz: E2fit = E2
NG − σ Cp
(l√σ)p ,
String tension: Calculated using the ground state, fixing p = 3
First excited states exclude p = 1 (Boundary term)!
Also second excited states exclude p = 1 .
e−12 χ
2= f (p,Cp)
18
V. SummaryWe constructed a large basis of operators characterized by the quantum
numbers of parity P, and winding momentum 2πq/l,
We calculated the energies of closed flux tubes in D=2+1 described by P = ±for:
→ S U(3) with β = 21.0 (a � 0.08fm) and q = 0,±1,±2,
→ S U(3) with β = 40.0 (a � 0.04fm) and q = 0,
→ S U(6) with β = 90.0 (a � 0.08fm) and q = 0,
We fit our data for the ground state using E2fit = E2
NG − σCp/(l√σ)p
and p = 3,
and extract σ.
Using σ we compare our results to Nambu-Goto:
→ Nambu-Goto is VERY good
We fit our data for the 1st and 2nd excited states with E2fit, and extract p
→ 1st and 2nd excited states exclude p = 1 (boundary term Luscher&Weisz. 04)
19
VI. Appendix 1: Large-Nc
• Yet some of the essential properties of QCD including confinement, are poorly
understood.
• It is useful to find a ’Neighbouring’ field theory that one can analyze more
simply.
• ’Neighbouring Theory’ −→ S U(Nc −→ ∞).
• Expansion parameter −→ 1Nc
.
• Planar diagrams contribute in the Large-N limit
• Non-Planar diagrams vanish.
20
VI. Appendix 1: ’t Hooft’s coupling
• ’t Hooft’s coupling: λ = g2Nc
• ’t Hooft’s double line diagrammatic representation:
j
j
j
k
i
k
21
VI. Appendix 1: Planar Diagram
k
j jii
i
j
k
∼ g2 × Nc(1 closed loop) =λ
Nc× Nc = λ
22
VI. Appendix 1: Non-Planar Diagram
j
ij
i
ij
∼ g6 × Nc(1 closed loop) =λ3
N3c
× Nc =λ3
N2c
Nc→∞−→ 0
23
VI. Appendix 2: Contribution of Operators
2nd excited state1st excited state
Operators, i
|ui|2
0.3
0.25
0.2
0.15
0.1
0.05
0�
��
��
��
��
��
��
��
�
S U(3), β = 21.000, and L = 16a
�
24
VI. Appendix 3: Why E2n?
Ground state:
S U(3) and β = 14.7172 S U(5) and β = 80.00
1 1.5 2 2.5 3 3.5 4 4.50.5
1
1.5
2
Ceff(l
)
l√
σ
Ceff from Fit 1Ceff from Fit 2
1 1.5 2 2.5 3 3.5 4 4.50.5
1
1.5
2
Ceff(l
)
l√
σ
Ceff from Fit 1Ceff from Fit 2
Fit 1: E0(l, σ) = σl − π6l ×C(1)
eff (l)
Fit 2: E20(l, σ) = (σl)2 − πσ3 ×C(2)
eff (l).
25
VII. Blocking-Smearing
• One needs operators that extend over physical length scales and are smooth on
such scales.
• Iterative ’fuzzing’ algorithm:
26
VII. Blocking-Smearing
• One needs operators that extend over physical length scales and are smooth on
such scales.
• Iterative ’fuzzing’ algorithm:
smear
27
VII. Blocking-Smearing
• One needs operators that extend over physical length scales and are smooth on
such scales.
• Iterative ’fuzzing’ algorithm:
smear block
28
VII. Blocking-Smearing
• One needs operators that extend over physical length scales and are smooth on
such scales.
• Iterative ’fuzzing’ algorithm:
smear block
Iterate
29
VIII. References
• Large-N
G. ’t Hooft, Nucl. Phys. B72 (1974) 461; B75 (1974) 461.
S. Coleman, 1979 Erice Lectures.
E. Witten, Nucl. Phys. B160 (1979) 57.
A. Manohar, 1997 Les Houches Lectures, hep-ph/9802419.
Y. Makeenko, hep-th/0001047
• Strings
J. Polchinski, Cambridge, UK: Univ. Pr. (1998) 402 p
Lust&Theisen, Springer-Verlag, (1989)
• Lattice QCD and Strings
A. M. Polyakov, Gauge Fields and Strings (Harwood, 1987),
J. Polchinski, hep-th/9210045,
J. Kuti, PoS LAT2005, 001 (2006) [PoS JHW2005, 009 (2006)],
30
• Effective string theory
M. Luscher, K. Symanzik and P.Weisz, Nucl. Phys. B 173, 365 (1980),
M. Luscher and P. Weisz, JHEP 0407, 014 (2004) [arXiv:hep-th/0406205],
N. D. Hari Dass and P. Majumdar, JHEP 0610, 020 (2006)
[arXiv:hep-lat/0608024],
P. Olesen, Phys. Lett. B 160, 144 (1985),
J. Arvis, Phys. Lett. 127B (1983) 106.
31