I. Introduction: General

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1 xford hysics TM Particle Theory Journal Club k-strings in S U(N) gauge theories Andreas Athenodorou St. John s College University of Oxford Mostly based on: AA, Barak Brigoltz and Mike Teper: arxiv: and arxiv: (k = 1 in +1 dimensions) Barak Bringoltz and Mike Teper: arxiv: and arxiv: (k > 1 in +1 dimensions) AA, Barak Bringoltz and Mike Teper: Work in progress (excited k-strings, +1 dimensions) 1

2 I. Introduction: General General question: What effective string theory describes k-strings in SU(N) gauge theories? Two cases: Open k-strings Closed k-strings During the last decade: D, D with Z, Z,U(1), SU(N 6) (Caselle and collaborators, Gliozzi and collaborators, Kuti and collaborators, Lüscher&Weisz, Majumdar and collaborators, Teper and collaborators, Meyer) Questions to be studied in D = + 1 dimensional SU(N) theories: Calculation of excited states, and states with p 0 and P = ± with k = 1, What is the degeneracy pattern of these states? Do k-strings fall into specific irreducible representations?

3 Open flux tube (k = 1 string) I. Introduction: General Closed flux tube (k = 1 string) l periodic b.c l t x t x Φ(l, t) = φ (0, t)u(0, l; t)φ(l, t) Φ(l, t) = TrU(l; t)

4 I. Introduction: General General question: What effective string theory describes k-strings in SU(N) gauge theories? Two cases: Open k-strings Closed k-strings During the last decade: D, D with Z, Z,U(1), SU(N 6) (Caselle and collaborators, Gliozzi and collaborators, Kuti and collaborators, Lüscher&Weisz, Majumdar and collaborators, Teper and collaborators, Meyer) Questions to be studied in D = + 1 dimensional SU(N) theories: Calculation of excited states, and states with p 0 and P = ± with k = 1,. What is the degeneracy pattern of these states? Do k-strings fall into specific irreducible representations?

I. Introduction: k- strings What is a k-string? Confinement in -d SU(N) leads to a linear potential between colour charges in the fundamental representation.

5 I. Introduction: k- strings What is a k-string? Confinement in -d SU(N) leads to a linear potential between colour charges in the fundamental representation. For SU(N ) there is a possibility of new stable strings which join test charges in representations higher than the fundamental! We can label these by the way the test charge transforms under the center of the group: ψ(x) z k ψ(x), z Z N. The string has N-ality k,(k = : = ) ( fund. fund. = symm. anti. ) 5

6 I. Introduction: k- strings The string tension does not depend on the representation R but rather on its N-ality k. The fundamental string has: N-ality k = 1, with string tension σ f. A k-string can be thought of as a bound state of k fundamental strings. Operators: φ = Tr{U k j }Tr{U} j where U is a Polyakov loop and j = 0,.., k 1. For k = 1, φ Tr{U} For k =, φ 1 Tr{U },φ Tr{U}Tr{U} Predictions: Casimir Scaling: σ R = σ f CR C f => σ k σ f = k(n k) (N 1) MQCD: σ k σ f = sin kπ N sin π N We expect: σ k N kσ f 6

7 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: EN L,N R,q,w = (σlw) + 8πσ ( N L +N R D ) ( + πq ) l + p. Described by: 1. The winding number w (w=1),. The winding momentum p = πq/l with q = 0, ±1, ±,. The transverse momentum p (p = 0),. N L and N R connected through the relation: N R N L = qw. N L = n L (k)k and N R = n R (k )k k>0 n L (k)>0 k >0 n R (k )>0 String states are eigenvectors of P (In D = + 1) with eigenvalues: P = ( 1) m i=1 n L (k i )+ m j=1 n R(k j ) 7

8 II. Theoretical expectations A. The seven lowest NG energy levels for the w = 1 closed string level N R N L q P =+ P = α α 1 ᾱ α 1 α 1 0 α α ᾱ 1 0 α 1 α 1 ᾱ α ᾱ 0, α 1 α 1 ᾱ 1 ᾱ 1 0 α ᾱ 1 ᾱ 1 0, α 1 α 1 ᾱ α ᾱ 1 0, α 1 α 1 α 1 ᾱ 1 0 α α 1 ᾱ 1 0 8

9 Effective string theory II. Theoretical expectations B. First prediction for w = 1 and q = 0 (Lüscher, Symanzik&Weisz. 80): E n = σl + π ( n D ) + O ( 1/l ). l Lüscher&Weisz effective string action (Lüscher&Weisz. 0): For any D the O ( 1/l (1/l) ) (Boundary term) is absent from E n (E n ) Spectrum in D = + 1 (Drummond 0, Dass and Matlock 06 for any D.): E n = σl + π ( n 1 ) ( 8π n 1 ) + O ( 1/l ) l σl Equivalently: E n = (σl) + 8πσ ( n 1 ) + O ( 1/l ), 9

10 Effective string theory II. Theoretical expectations B. First prediction for w = 1 and q = 0 (Lüscher, Symanzik&Weisz. 80): E n = σl + π ( n D ) + O ( 1/l ). l Lüscher&Weisz effective string action (Lüscher&Weisz. 0): For any D the O ( 1/l (1/l) ) (Boundary term) is absent from E n (E n) Spectrum in D = + 1 (Drummond 0, Dass and Matlock 06 for any D.): E n = σl + π ( n 1 ) ( 8π n 1 ) + O ( 1/l ) l σl Equivalently: En = ENG + O ( 1/l ) Fit : E fit = E NG σ ( ) p (p ) l σ C p 10

11 III. Lattice Calculation: Lattice setup Usually we are interested in calculating quantities like: Ψ(A) = 1 S [A] da μ ( x)ψ(a)e Z x,μ The lattice represents a mathematical trick: It provides a regularisation scheme. We define our gauge theory on a D discretized periodic Euclidean space-time with L L L T sites. 11

12 III. Lattice Calculation: Lattice setup Usually we are interested in calculating quantities like: Ψ(A) = 1 S [A] da μ ( x)ψ(a)e Z x,μ The lattice represents a mathematical trick: It provides a regularisation scheme. U μ ( x) SU(N) (μ = 1,, ) a sites, x We define our gauge theory on a D discretized periodic Euclidean space-time with L L L T sites. 1

13 III. Lattice Calculation: On the Lattice Using link variables, the EFPI is written as: Ψ L (U) = 1 du μ (n)ψ L (U)e S L[U] Z n,μ (1) Where: S L = β {1 1 ReTrU p } () N c p β = N c (D = + 1) () ag U p = U μ ( x)u ν ( x + aˆμ)u μ( x + aˆν)u μ( x) () ν μ 1

14 III. Lattice Calculation: Monte - Carlo Simulations Why Monte - Carlo?: We want to calculate the expectation value of colour singlet operators. High multidimensionality of these integrals makes traditional mesh techniques impractical. Monte Carlo Methods. We need to generate n c different field configurations with probability distribution: du l e β p{1 Nc 1 ReT ru p} l (5) Then the expectation value of Ψ L (U) will be just the average over these fields. Ψ L (U) = 1 n c n c I=1 Ψ L (U I ) ±O( 1 nc ) (6) 1

15 III. Lattice Calculation: Energy Calculation Masses of certain states can be calculated using the correlation functions of specific operators: Φ (t)φ(0) = Ω Φ 0 e tm 0 + Ω Φ m e tm m Let us define the effective mass: m 1 t Ω Φ 0 e tm 0 (7) am ef f (t) = ln Φ (t)φ(0) Φ (t a)φ(0) (8) The mass will be equal to: am k am ef f (t 0 ) (9) where t 0 is the lowest value of t for which m ef f (t 0 ) = am ef f (t > t 0 ) 15

16 III. Lattice Calculation: Energy Calculation Example: Closed k = 1 string: Φ Φ 16

17 III. Lattice Calculation: Energy Calculation We expect to calculate masses for some excited states, so we need to use operators which merely project onto the excited states. We construct a basis of operators, Φ i : i = 1,..., N O, with transverse deformations described by the quantum numbers of parity P, winding number w, longitudinal momentum p and transverse momentum p = 0. For example: k = 1: Φ p,p ± = 1 L L x,x {φ u ± φ d } e ip x +ip x φ u = Tr { } and φ d = Tr { } 17

18 III. Lattice Calculation: Energy Calculation k = : 1 st set of operators: φ u = Tr { }, φ d = Tr { } nd set of operators: φ u = Tr { }, φ d = Tr { } rd set of operators: φ u = Tr { }, φ d = Tr { } Projection onto the Antisymmetric representation: φ u = [Tr { } Tr { }], φ d = [Tr { } Tr { }] Projection onto the Symmetric representation: φ u = [Tr { } + Tr { }], φ d = [Tr { } + Tr { }] We calculate the correlation function (Matrix): C ij (t) = Φ i (t)φ j(0) We diagonalize the matrix: C 1 (0)C(a). We extract the correlator for each state. By fitting the results, we extract the mass (energy) for each state. 18

19 III. Lattice Calculation: Large Basis of Operators Using this large basis of operators: We extract masses of excited states. (up to 15 states for k = 1) It increases the Overlaps (using single exponential fits): Ground state %, First excited state % ( with just 5bl), Second excited state 95 99% ( 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) 19

20 III. Lattice Calculation: Operators for P = + and k = 1 5 bl 5 bl 5 bl + 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 5 bl 0

21 III. Lattice Calculation: Operators for P = and k = 1 5 bl 5 bl 5 bl + bl 5 bl 5 bl 5 bl 5 bl 5 bl 5 bl + 5 bl 5 bl 5 bl 5 bl 5 bl 1

22 IV. Results: Spectrum of SU() and β = 1.0 for k = 1 Group: SU(), a 0.08 fm, Quantum Numbers: P = +, and q = 0 1 E/ σ n = n = n = 1 n = σl NG Prediction: E n = (σl) + 8πσ ( n 1 ), where n = NL = N R since q = 0.

23 IV. Results: Spectrum of SU() and β = 0.0 for k = 1 Group: SU(), a 0.0 fm, Quantum Numbers: P = +, and q = 0 1 E/ σ n = n = n = 1 n = σl NG Prediction: E n = (σl) + 8πσ ( n 1 ), where n = NL = N R since q = 0.

24 IV. Results: Spectrum of SU(6) and β = 90.0 for k = 1 Group: SU(6), a 0.08 fm, Quantum Numbers: P = +, and q = 0 1 E/ σ n = n = n = 1 n = σl NG Prediction: E n = (σl) + 8πσ ( n 1 ), where n = NL = N R since q = 0.

25 IV. Results: Spectrum of SU(N) for k = 1 Groups: SU() and SU(6), a 0.0 fmand 0.08 fm, Quantum Numbers: P = ± and q = n = 7 6 E 1 = σl + π l ( 1 1 ) n = 1 E/ σ 5 E 1 = σl + π l ( 1 1 ) 8π σl ( 1 1 ) n = 0 1 NG Prediction: En = (σl) + 8πσ ( n 1 ) σl 5

26 IV. Results: Non-zero winding momentum for k = 1. Group: SU(), a 0.08 fm, Quantum Numbers: P = +,, q = 1, and w = 1 E /σ (πq/ σl) q = 1, N R =, N L = q =, N R =, N L = 1 q = 1, N R =, N L = 1 q =, N R =, N L = 0 q = 1, N R = 1, N L = l σ NG Prediction: E (πq/l) = (σlw) + 8πσ ( ) N R +N L D. Constraint: N R N L = qw 6

27 IV. Results: Spectrum of SU() for P =+and k =. Group: SU(), a 0.06 fm, Quantum Numbers: P = +, q = 0 and k = E/(σl) σl Basis: TrU (w=1)+ + TrU (w=1),tr(u (w=1)+) + Tr(U (w=1) ), TrU (w=)+ + TrU (w=) 7

28 IV. Results: Spectrum of SU() for P =+and k =. Group: SU(), a 0.06 fm, Quantum Numbers: P = +, q = 0 and k = E/(σl) σl Basis: TrU (w=1)+ + TrU (w=1),tr(u (w=1)+) + Tr(U (w=1) ) 8

29 IV. Results: Spectrum of SU() for P =+and k = A. Group: SU(), a 0.06 fm, Quantum Numbers: P = +, q = 0 and k = E/(σl) σl Basis: [TrU (w=1)+ Tr(U (w=1)+) ] + [TrU (w=1) Tr(U (w=1) )] 9

30 IV. Results: Spectrum of S U() for P = + and k = S. Group: SU(), a 0.06 fm, Quantum Numbers: P = +, q = 0 and k = E/(σl) σl Basis: [TrU (w=1)+ + Tr(U (w=1)+) ] + [TrU (w=1) + Tr(U (w=1) )] 0

31 IV. Results: Spectrum of SU() for P = and k =. Group: SU(), a 0.06 fm, Quantum Numbers: P =, q = 0 and k = E/(σl) σl Basis: TrU (w=1)+ + TrU (w=1),tr(u (w=1)+) + Tr(U (w=1) ) 1

32 IV. Results: Spectrum of SU() for P = and k =. Group: SU(), a 0.06 fm, Quantum Numbers: P =, q = 0 and k = E/(σl) σl Basis: TrU (w=1)+ + TrU (w=1),tr(u (w=1)+) + Tr(U (w=1) ), TrU (w=)+ + TrU (w=)

33 IV. Results: Spectrum of SU() for P =, q = 1, k = A. Group: SU(), a 0.06 fm, Quantum Numbers: P =, q = 1 and k = A E/(σl) σl Basis: [TrU (w=1)+ Tr(U (w=1)+) ] + [TrU (w=1) Tr(U (w=1) )]

34 IV. Results: Spectrum of SU() for P =, q =, k = A. Group: SU(), a 0.06 fm, Quantum Numbers: P =, q = and k = A E/(σl) σl Basis: [TrU (w=1)+ Tr(U (w=1)+) ] + [TrU (w=1) Tr(U (w=1) )]

35 IV. Results: Spectrum of SU() for P =, q = 1, k = S. Group: SU(), a 0.06 fm, Quantum Numbers: P =, q = 1 and k = S E/(σl) σl Basis: [TrU (w=1)+ + Tr(U (w=1)+) ] + [TrU (w=1) + Tr(U (w=1) )] 5

36 IV. Results: Spectrum of SU() for P =, q =, k = S. Group: SU(), a 0.06 fm, Quantum Numbers: P =, q = and k = S E/(σl) σl Basis: [TrU (w=1)+ + Tr(U (w=1)+) ] + [TrU (w=1) + Tr(U (w=1) )] 6

37 V. Summary We constructed a large basis of operators characterized by the quantum numbers of parity P, and winding momentum πq/l, We calculated the energies of closed k-strings in D=+1 described by P = ± for: SU() with k = 1, β = 1.0 (a 0.08fm) and q = 0, ±1, ±, SU() with k = 1, β = 0.0 (a 0.0fm) and q = 0, SU() with k = 1,, β = 50.0 (a 0.06fm) and q = 0, ±1, ±, SU(5) with k = 1,, β = 80.0 (a 0.06fm) and q = 0, ±1, ±, SU(6) with k = 1, β = 90.0 (a 0.08fm) and q = 0. We fit our data for the ground state using E fit = E NG σc p/ ( l σ ) p and p =, and extract σ. Using σ we compare our results to Nambu-Goto: Nambu-Goto is VERY good 7

38 V. Summary k-strings know about the full SU(N) gauge group. We observe additional states... 8

39 Example of Operators: VI. Future project: + 1 Dimensions. Operators CP J i i ± 0 i i i i ± 1 ± 9

40 VII. Appendix 1: Large-N c Quantum chromodynamics is the theory of strong interactions based on the gauge group SU(N c = ). Yet some of its essential properties including confinement, are poorly understood. It is useful to find a Neighbouring field theory that one can analyze more simply. Neighbouring Theory SU(N c ). Expansion parameter: 1 N c. 0

41 VII. Appendix : T Hooft s coupling T Hooft s coupling: λ = g N c. T Hooft s double line diagrammatic representation: j k k j j i 1

42 VII. Appendix. Planar Diagram i k i j i j k j g N c (1 closed loop) = λ N c N c = λ

43 VII. Appendix. Non-Planar Diagram i i j i j j g 6 N c (1 closed loop) = λ N c N c = λ N c N c 0

44 VII. Appendix : Contribution of Operators SU(), β = 1.000, and L = 16a 1 st excited state nd excited state 0. u i Operators, i

arxiv: v1 [hep-lat] 24 Mar 2013

arxiv: v1 [hep-lat] 24 Mar 2013 Closed flux tubes in higher representations and their string description in D=+ SU(N) gauge theories arxiv:33.59v [hep-lat] Mar 3 Andreas Athenodorou a and Michael Teper b a Department of Physics, Swansea