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

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)
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?
I. Introduction: Closed k-strings
Open flux tube (k=1 string)
Closed flux tube (k=1 string)
Periodic
Boundary
Conditions
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.
II. Theoretical Expectations: Nambu-Goto
Spectrum given by:
 N L  N R D  2   2πq 
2
2
 lw   8πσ 


  p
2
24   l 

2
E N2 L ,N R ,q,w
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):
E E
2
fit
2
NG

C p
l  
p
III. Lattice Calculation: Lattice setup
The lattice represents a mathematical trick:
It provides a regularisation scheme.
Up

We define our theory on a 3D discretised
periodic Euclidean space-time with L‖L┴LT
sites.
U  x  SU ( N c )
a
Usually in LQFT
QFT we
weare
areinterested
interestedinincalculating
calculatingquantities
quantitieslike:
like:

11
 S [ A]  S L [U ]



LA

x

A
e
U  Z   dAdU


n



L U e
Z x,n,

2Nc
SL  2
ag


1
p 1  N Re Tr U p 
c


III. Lattice Calculation: Energy Calculation
Masses of certain states can be calculated using the correlation
functions of specific operators:
 † t 0    † 0 e tm0     † m e tmm
m1

t
   † 0 e tm0
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): Cij t    i † t  j 0 ,
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.
III. Lattice Calculation: Energy Calculation
Example: Closed k = 1 string
III. Lattice Calculation: Operators for P = +, k = 1
III. Lattice Calculation: Operators for P = ̶ , k = 1
IV. Results: Spectrum of SU(3) for k=1, q=0
E/ 
P= +, ̶
l
1 

Nambu-Goto prediction: E  l   8  n 

24 

2
n
2
IV. Results: Spectrum of SU(6) for k=1, q=0
E/ 
P= +, ̶
l


Nambu-Goto prediction: En2  l 2  8  n 
1 

24 
IV. Results: Spectrum of SU(3) for k=1, q≠0

E 2 /   2q /  l

2
P= +, ̶ , q=1, 2
l
NL  NR 1 

Nambu-Goto prediction: E  2q / l   l   8 
 
2
24 

Constraint: NR ̶ NL=qw
2
2
2
IV. Results: Spectrum of SU(4) for k=2, q=0
E/ 
l
1 

Nambu-Goto prediction: E   k  2l   8k  2  n 

24 

2
n
2
IV. Results: Spectrum of SU(4) for k=2A, q=0
E/ 
l
1 

Nambu-Goto prediction: E   R  2 Al   8 R  2 A  n 

24 

2
n
2
IV. Results: Spectrum of SU(4) for k=2S, q=0
E/ 
l
1 

Nambu-Goto prediction: E   R  2 S l   8 R  2 S  n 

24 

2
n
2
V. Future: 3+1 D
Operators: