CONFINEMENT WITHOUT
A CENTER:
THE EXCEPTIONAL
GAUGE GROUP G(2)
MICHELE
PE PE
University of Bern
(S w i t z e r l a n d)
OUTLINE
• Overview of the deconfinement transition in YM
theory with a general gauge group and motivations
• The group G(2): generalities
• G(2) gauge theories
• Numerical results
• Conclusions
YM
YM + Higgs
• What is the role of the center of the gauge group in the
deconfinement transition of Yang-Mills theory?
gauge theory
Svetitsky-Yaffe
conjecture
scalar theory
complicated, local, effective action for the Polyakov loop
order of the deconfinement
phase transition
K.Holland, M.P., U.J. Wiese
SU(N)
Sp(N)
Nucl.Phys.B694 (2004) 35
M.P., Nucl.Phys.B PS
141 (2005) 238
SO(N)
Otah and Wingate
potential mechanism of
confinement in YM theory
Spin(N)
Lucini, Wenger, and Teper
G(2), F(4), E(8)
exceptional
E(6)
groups
E(7)
Greensite and Lautrup
Tomboulis
Datta, Gavai et al.
De Forcrand and Jahn
Burgio, Muller-Preussker et al.
(N)
(2)
N odd
(2)
N=4k+2
(4)
(2) (2) N=4k
trivial center
(3)
(2)
• Sp(N): increase the size of the group keeping the center (2) fixed
generalization of SU(2)=Sp(1); pseudo-real representation
K.Holland, M.P., U.J. Wiese
Nucl.Phys.B694 (2004) 35
Sp(2)10
Sp(3) 21
center: no information about the order of the deconfinement transition
confined phase
(colorless states)
conjecture
deconfined phase
(gluon plasma)
size of the group determines
the order of the p.t.
• (3+1)-d: only Sp(1)=SU(2) YM theory has a 2nd order deconfinement p.t.
What about confinement in YM theory with a gauge group with trivial center?
K.Holland, P. Minkowski, M.P., U.J. Wiese
Nucl.Phys.B668 (2003) 207
• Potential relevance of topological objects in the
mechanism of confinement in non-Abelian gauge
theories. Possible candidates: ’t Hooft flux vortices.
1( G / center(G) )  {}
• Gauge theories without ’t Hooft flux vortices: study
how confinement shows up.
What about confinement in YM theory with a gauge group with trivial center?
• G(2): simplest group such that
1( G(2) / {} ) = {}
G(2)
SU(3)
G(2): generalities
• G(2)  SO(7)
[ rank = 3; generators = 21]
det  = 1 ;
ab = a´b´ a a´ b b´
Ta b c = Ta´ b´ c´ a a´ b b´ c c´ ; T is antisymmetric
14 generators; real representations (fundamental 77)
G(2)-"quarks" ~ G(2)-"antiquarks"
• G(2) has rank 2
• G(2)  SU(3) in a real
{7}
rep.
{3}{3}{1}
SU(3)
a 0 0
a = 0 -a* 0
0 0
0
a = Gell-Mann matrices
• 14 generators: adjoint representation is {14}
{14}
SU(3)
{8}{3}{3}
14 G(2)-"gluons" SU(3) 8 gluons +
"vector quark" + "vector antiquark"
•
rank 2
G(2): its own universal
covering group
center(G(2)) = {}
G(2)  SU(3)
1( G(2) / {} ) = {}
"3-ality" : all reps mix together in the tensor product decomp.
string breaking without dynamical G(2)-"quark"
{7}{14}{14}{14} = {1} …
• Interesting homotopy groups
3( G(2) ) = 
instantons
2( G(2)/U(1)2) =  

1( G(2) / {} ) =
{}
monopoles
like SU(3)
no ’t Hooft unlike SU(3)
flux vortices
G(2) Yang-Mills
• Pure gauge: 14 G(2)-"gluons"
{14}
SU(3)
{8}{3}{3}
6 G(2)-"gluons" explicitly break (3)  center(G(2)) = {}
quarks for SU(3)
V(r)
• G(2)-YM is asymptotically free
at low energies:
- confinement
- string breaking: =0 (QCD)
~ 6 G(2)-"gluons"
• G(2)-"laboratory": confinement similar to QCD without
complications related to fermions.
• Wilson loop
perimeter law
r
• Fredenhagen-Marcu order parameter: confining/Higgs or
Coulomb phase
R
T/2
(R,T) =
0
Confining/Higgs
=0
Coulomb
R,T 
1/2
T
R
ab
cd
ef
= (Ux Tabc) Uxy (Uy Tdef)
U =
no counterpart when the gauge group has a non-trivial center
In strong coupling we are in the confining/Higgs phase
• Finite temperature: different behaviour than SU(3)-YM
P
zP
 P0 P*r r  
2
(3) unbroken
 P   e-Fq/T
0
P
(3) broken
 P  = 0, 0
 P   0,  =
• In SU(3)-YM there is a global symmetry that breaks down.
In G(2)-YM no symmetry  no 2nd order phase transition
1st order
or
crossover ?
Conjecture: Sp(2) has 10 generators and it has 1st deconfinement p.t.
We expect G(2) YM to have also a 1st deconfinement p.t.
dynamical issue: numerical simulations
2436
Tr U/7

High temperature effective potential
N. Weiss,
Phys. Rev. D24 (1981) 475
• 1-loop expansion of the effective potential for the Polyakov loop
P=
diag(ei(1+2),
ei(-1+2),
~
P (1, 2) = ( P, P*,1)
e-2i2)
2
2
SU(3)
G(2)
1
1
G(2) Yang-Mills + Higgs {7}
• Higgs {7}: G(2)
SU(3)
{14}
SU(3)
  = v
{8}{3}{3}
6 G(2)-"gluons" pick up a mass MG  v
• For MG  QCD the 6 massive G(2)-"gluons" participate in
the dynamics; for MG  QCD they decouple  SU(3)
Higgs {7}: handle for G(2)
SU(3)
• confinement G(2)
SU(3). 6 massive G(2)-"gluons"
are {3} and {3}  quarks  string breaking
V(r)
V(r)
MG
r
=0
r
0
SHYM = SYM - 
  (x) U
+
x,
(x)
^
(x+)
Nt=6
SU(3)-YM

G(2)-YM
1/(7g2)
=
1.3
=
1.3
=
1.3
Nt=6
SU(3)-YM

G(2)-YM
1/(7g2)
=
1.3
=
1.3
=
1.3
=
1.5
=
1.5
=
1.5
Nt=6
SU(3)-YM

G(2)-YM
1/(7g2)
=
1.3
=
1.3
=
1.3
=
1.5
=
1.5
=
1.5
=
2.5
=
2.5
=
2.5
Conclusions
• Confinement is difficult problem: not only SU(N) but all Lie groups!
• Conjecture: the size of the group determines the order of the deconfinement p.t.
The center is relevant only if the transition is 2nd order: G(2)14 YM 1st order
(3+1)-d only Sp(1)=SU(2)3 YM has a 2nd order deconfinement p.t.
SU(3)8 YM weak 1st order, no known universality class available
YM with all other gauge groups have 1st order
(2+1)-d SU(2)3, SU(3)8, Sp(2)10 YM has a 2nd order deconfinement p.t.,
SU(4)15 YM: weak 1st or 2nd ?, G(2)14 YM: not known
YM with all other gauge groups have 1st order
Outlook
• Finite temperature behaviour of G(2) YM in (2+1)-d
• Static quark-quark potential and string breaking
• Study of the Fredenhagen-Marcu order parameter