Closed k- strings in 2+1 dimensional SU ( N c ) gauge theories

1. GradCATS 2008 Closed k-strings in 2+1 dimensional SU(Nc) gauge theories Andreas Athenodorou St. John’s College University of Oxford 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. I. Introduction: General • General Question: • What effective string theory describes k-strings in SU(Nc) gauge theories? • Two cases: • Open k-strings • Closed k-strings • During the last decade: • 2+1 D • 3+1 D • Z2, Z4, U(1), SU(Nc≤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(Nc) leads to a linear potential between colour charges in the fundamental representation. • For SU(Nc≥ 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) → zkψ(x), z ∈ ZN, • 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 • Spectrum given by: • Described by: • The winding number, • The winding momentum, • The transverse momentum, • NL and NR connected through the relation: NR-NL=qw, • 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┴LTsites. a a a a • Usually in QFT we are interested in calculating quantities like: • Usually in LQFT we are interested in calculating quantities like:

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, Φi: i = 1, ..., NO, 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 P= +,̶ Nambu-Goto prediction:

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

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