Closed Closed k-k-strings in 2+1 dimensional strings in 2+1 dimensional SUSU((NNcc) gauge theories ) gauge theories

Andreas AthenodorouAndreas Athenodorou

St. John’s CollegeSt. John’s College

University of OxfordUniversity of Oxford

GradCATS 2008GradCATS 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)

General Question:General Question: What effective string theory describes What effective string theory describes kk-strings in -strings in SUSU((NNcc) gauge theories?) gauge theories?

I. Introduction: GeneralI. Introduction: General

Two cases:Two cases: Open Open kk-strings-strings Closed Closed kk-strings-strings

During the last decade:During the last decade: 2+1 D2+1 D 3+1 D3+1 D ZZ22, , ZZ44, , UU(1), (1), SUSU((NNcc≤6)≤6)

Questions:Questions: Excitation spectrum (Calculate states with non-trivial quantum numbers)?Excitation spectrum (Calculate states with non-trivial quantum numbers)? Degeneracy pattern?Degeneracy pattern? Do Do kk-strings fall into particular irreducible representations?-strings fall into particular irreducible representations?

I. Introduction: Closed I. Introduction: Closed kk-strings-strings

Open flux tube (Open flux tube (kk=1 string)=1 string) Closed flux tube (Closed flux tube (kk=1 string)=1 string)

Periodic Periodic Boundary Boundary ConditionsConditions

I. Introduction: I. Introduction: kk-strings-strings

Confinement in 3-d Confinement in 3-d SUSU((NNcc) leads to a linear potential between colour ) leads to a linear potential between colour

charges in the fundamental representation.charges in the fundamental representation.

For For SUSU((NNcc ≥ 4) there is a possibility of new stable strings which join ≥ 4) there is a possibility of new stable strings which join

test charges in representations higher than the fundamental!test charges in representations higher than the fundamental!

We can label these by the way the test charge transforms under the We can label these by the way the test charge transforms under the center of the group: center of the group: ψψ((xx) → ) → zzkkψψ((xx), ), z z ∈ ∈ ZZNN,,

The string has The string has NN-ality -ality kk,,

The string tension does not depend on the representation The string tension does not depend on the representation RR but rather but rather on its on its NN-ality -ality kk..

II. Theoretical Expectations: Nambu-GotoII. Theoretical Expectations: Nambu-Goto

22

22 2

24

2

28

pl

πqDNNπσlwE RL

,q,w,NN RL

Described by:Described by: The winding number,The winding number, The winding momentum,The winding momentum, The transverse momentum,The transverse momentum, NNL L andand N NRR connected through the relation: connected through the relation: NNRR-N-NLL=qw,=qw,

Spectrum given by:Spectrum given by:

In 2+1 D: String states are eigenvectors of Parity In 2+1 D: String states are eigenvectors of Parity PP ( (PP==±1),±1),

Motivated by recent results Motivated by recent results (L(Lüüscher&Weisz. 04)scher&Weisz. 04)::

pp

NGl

CEE

22

fit

III. Lattice Calculation: Lattice setupIII. Lattice Calculation: Lattice setup

The lattice represents a mathematical trick: The lattice represents a mathematical trick: It provides a regularisation scheme.It provides a regularisation scheme.

We define our theory on a 3D discretised We define our theory on a 3D discretised periodic Euclidean space-time with periodic Euclidean space-time with LL‖‖LL┴┴LLTT

sites.sites.

Usually in QFT we are interested in calculating quantities like:Usually in QFT we are interested in calculating quantities like:

][

,

1 AS

x

eAxdAZ

A

][

,

1 USL

nL

LeUndUZ

U

)( cNSUxU

a

Usually in LQFT we are interested in calculating quantities like:Usually in LQFT we are interested in calculating quantities like:

a a a

p c

cL U

Nag

NS p2

TrRe1

12

pU

III. Lattice Calculation: Energy CalculationIII. Lattice Calculation: Energy Calculation

Masses of certain states can be calculated using the correlation Masses of certain states can be calculated using the correlation functions of specific operators:functions of specific operators:

0

0

0

00

†

1

†††

tmt

m

tmtm

e

emet m

We use variational technique:We use variational technique: We construct a basis of operators, ΦWe construct a basis of operators, Φ ii : : i i = 1, ..., = 1, ..., NNOO, with transverse , with transverse

deformations described by the quantum numbers of parity deformations described by the quantum numbers of parity PP, winding number , winding number ww, longitudinal momentum , longitudinal momentum pp and transverse momentum and transverse momentum pp⊥⊥ = 0. = 0.

We calculate the correlation function (matrix): ,We calculate the correlation function (matrix): , We diagonalise the matrix: We diagonalise the matrix: CC-1-1(0)(0)CC((aa),), We extract the correlator of each state,We extract the correlator of each state, By fitting the results, we extract the energy (mass) for each state.By fitting the results, we extract the energy (mass) for each state.

0†jiij ttC

III. Lattice Calculation: Energy CalculationIII. Lattice Calculation: Energy Calculation

Example: Closed Example: Closed k k = 1 string= 1 string

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

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

IV. Results: Spectrum of IV. Results: Spectrum of SUSU(3) for(3) for k k=1, =1, qq=0 =0

/E

l

24

1822 nlEn Nambu-Goto predictionNambu-Goto prediction::

PP= +, ̶

IV. Results: Spectrum of IV. Results: Spectrum of SUSU(6) for(6) for k k=1, =1, qq=0 =0

/E

l

24

1822 nlEn Nambu-Goto predictionNambu-Goto prediction::

PP= +, ̶

IV. Results: Spectrum of IV. Results: Spectrum of SUSU(3) for(3) for k k=1, =1, qq≠0≠0 22 /2/ lqE

l

24

1

28/2 222 RL NN

llqE Nambu-Goto predictionNambu-Goto prediction::

P= +, ̶ , q=1, 2

Constraint:Constraint: NNR R ̶ ̶ NNLL=qw=qw

IV. Results: Spectrum of IV. Results: Spectrum of SUSU(4) for(4) for k k=2, =2, qq=0 =0

24

18 2

22

2 nlE kkn Nambu-Goto predictionNambu-Goto prediction::

/E

l

IV. Results: Spectrum of IV. Results: Spectrum of SUSU(4) for(4) for k k=2A, =2A, qq=0 =0

24

18 2

22

2 nlE ARARn Nambu-Goto predictionNambu-Goto prediction::

/E

l

IV. Results: Spectrum of IV. Results: Spectrum of SUSU(4) for(4) for k k=2S, =2S, qq=0 =0

24

18 2

22

2 nlE SRSRn Nambu-Goto predictionNambu-Goto prediction::

/E

l

V. Future: 3+1 DV. Future: 3+1 D

OperatorsOperators::