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