1. Closed k-strings in 2+1 dimensional SU(N c ) gauge theories Andreas Athenodorou St. John’s College University of Oxford GradCATS 2008 Mostly based on: AA, Barak Bringoltz and Mike Teper: arXiv:0709.0693 and arXiv:0709.2981 (k = 1 in 2+1 dimensions) Barak Bringoltz and Mike Teper: arXiv:0708.3447 and arXiv:0802.1490 (k > 1 in 2+1 dimensions) AA, Barak Bringoltz and Mike Teper: Work in progress (excited k-strings, 3+1 dimensions)

2. General Question:  What effective string theory describes k-strings in SU(N c ) gauge theories? I. Introduction: General Two cases:  Open k-strings  Closed k-strings During the last decade:  2+1 D  3+1 D  Z 2, Z 4, U(1), SU(N c ≤6) Questions:  Excitation spectrum (Calculate states with non-trivial quantum numbers)?  Degeneracy pattern?  Do k-strings fall into particular irreducible representations?

3. I. Introduction: Closed k-strings Open flux tube (k=1 string) Closed flux tube (k=1 string) Periodic Boundary Conditions

4. I. Introduction: k-strings Confinement in 3-d SU(N c ) leads to a linear potential between colour charges in the fundamental representation. For SU(N c ≥ 4) 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, The string tension does not depend on the representation R but rather on its N-ality k.

5. II. Theoretical Expectations: Nambu-Goto Described by:  The winding number,  The winding momentum,  The transverse momentum,  N L and N R connected through the relation: N R -N L =qw, Spectrum given by: In 2+1 D: String states are eigenvectors of Parity P (P=±1), Motivated by recent results (Lüscher&Weisz. 04):

6. III. Lattice Calculation: Lattice setup The lattice represents a mathematical trick: It provides a regularisation scheme. We define our theory on a 3D discretised periodic Euclidean space-time with L ‖ L ┴ L T sites. Usually in QFT we are interested in calculating quantities like: a Usually in LQFT we are interested in calculating quantities like: a a a

7. III. Lattice Calculation: Energy Calculation Masses of certain states can be calculated using the correlation functions of specific operators: We use variational technique:  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.  We calculate the correlation function (matrix):,  We diagonalise the matrix: C -1 (0)C(a),  We extract the correlator of each state,  By fitting the results, we extract the energy (mass) for each state.

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

9. III. Lattice Calculation: Operators for P = +, k = 1

10. III. Lattice Calculation: Operators for P = ̶, k = 1

11. IV. Results: Spectrum of SU(3) for k=1, q=0 Nambu-Goto prediction : P P= +, ̶

12. IV. Results: Spectrum of SU(6) for k=1, q=0 Nambu-Goto prediction : P P= +, ̶

13. IV. Results: Spectrum of SU(3) for k=1, q≠0 Nambu-Goto prediction : P= +, ̶, q=1, 2 Constraint: N R ̶ N L =qw

14. IV. Results: Spectrum of SU(4) for k=2, q=0 Nambu-Goto prediction :

15. IV. Results: Spectrum of SU(4) for k=2A, q=0 Nambu-Goto prediction :

16. IV. Results: Spectrum of SU(4) for k=2S, q=0 Nambu-Goto prediction :

17. V. Future: 3+1 D Operators :