CONFINEMENT WITHOUT A CENTER:

THE EXCEPTIONAL GAUGE GROUP G(2)

M I C H E L E P E P EU n i v e r s i t y o f B e r n

(S w i t z e r l a n d)

in collaboration with

UWE-JENS WIESEU n i v e r s i t y o f B e r n

(S w i t z e r l a n d)

O U T L I N E• 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?

SU(N) (N)

gauge theory scalar theorySvetitsky-Yaffeconjecture

complicated, local, effective action for the Polyakov looporder of the deconfinement

phase transitionpotential mechanism of

confinement in YM theory

SO(N)(2)(4)(2) (2)

N odd

N=4k+2

N=4k

Spin(N)

K.Holland, M.P., U.J. WieseNucl.Phys.B694 (2004) 35

M.P., Nucl.Phys.B PS141 (2005) 238

E(7)

E(6)exceptionalgroups

G(2), F(4), E(8)

(2)

(3)

trivial center

Sp(N) (2)

Otah and Wingate

Lucini, Wenger, and Teper

Greensite and Lautrup

Tomboulis

Datta, Gavai et al.

De Forcrand and Jahn

Burgio, Muller-Preussker et al.

• Sp(N): increase the size of the group keeping the center (2) fixed generalization of SU(2)=Sp(1); pseudo-real representation

• (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?

center: no information about the order of the deconfinement transition

conjectureconfined phase deconfined phase(colorless states) (gluon plasma)

size of the group determinesthe order of the p.t.

Sp(2)10 Sp(3) 21

K.Holland, P. Minkowski, M.P., U.J. Wiese Nucl.Phys.B668 (2003) 207

K.Holland, M.P., U.J. Wiese

Nucl.Phys.B694 (2004) 35

• 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.

G(2) SU(3)

What about confinement in YM theory with a gauge group with trivial center?

• G(2): simplest group such that

1( G(2) / {} ) = {}

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 77)

G(2)-"quarks" ~ G(2)-"antiquarks"

a 0 0

0 -a* 0

0 0 0

a =

• G(2) SU(3) in a real rep.• G(2) has rank 2

a = Gell-Mann matrices

SU(3){7} {3}{3}{1}

• G(2): form of the matrices

U8: 3×3 complex matrix; K6= 3-comp. complex vector

C(K), D(K) = 3×3 matrices; = number

6 + 8 = 14 • 14 generators: adjoint representation is {14}

C(K) D*(K) K

D(K) C*(K) K*

-K -K (K)+ T

U 0 00 U* 00 0 1

CU D*U* K

DU C*U* K*

-K U -K U* + T=

{14} {8}{3}{3}SU(3)

14 G(2)-"gluons" 8 gluons + "vector quark" + "vector antiquark"

SU(3)

………

•

string breaking without dynamical G(2)-"quark"

{7}{14}{14}{14} = {1} …

• Interesting homotopy groups

G(2): its own univ.covering group

rank 2

G(2) SU(3)center(G(2)) = {}

1( G(2) / {} ) = {}

"N-ality" : all reps mix together in the tensor product decomp.

3( G(2) ) =

2( G(2)/U 2

(1) ) =

1( G(2) / {} ) = {}

instantons

monopoles

no center vortices

like SU(3)

unlike SU(3)

G(2) Yang-Mills• Pure gauge: 14 G(2)-"gluons"

6 G(2)-"gluons" explicitly break (3) center(G(2)) = {} quarks for SU(3)• G(2)-YM is asymptotically free

at low energies: - confinement - string breaking: =0 (QCD)

• G(2)-"laboratory": confinement similar to QCD without complications related to fermions.

• Wilson loop perimeter law

{14} {8}{3}{3}SU(3)

V(r)

r

~ 6 G(2)-"gluons"

• Fredenhagen-Marcu order parameter: confining/Higgs or Coulomb phase

(R,T) = 1/2

0 Confining/Higgs

= 0 Coulomb

In strong coupling we are in the confining/Higgs phase

R,T

= (Ux Tabc) Uxy

(Uy Tdef)

ab cd ef

R

R

T/2

T

no counterpart when the gauge group has a non-trivial center

U =

• Finite temperature: different behaviour than SU(3)-YM

(3) unbroken (3) broken

P e-Fq/T P = 0, 0 P 0, = 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

P z P

P0 P*r P 2

r

Tr U/7

2436

High temperature effective potential • 1-loop expansion of the effective potential for the Polyakov loop

~

1

G(2)

P (1, 2) = ( P, P*,1)

2

1

2

SU(3)

P = diag(ei(1+2), ei(-1+2), e-2i2)

N. Weiss, Phys. Rev. D24 (1981) 475

G(2) Yang-Mills + Higgs {7}• Higgs {7}: G(2) SU(3) = v

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

{14} {8}{3}{3}SU(3)

V(r) V(r)

rr

MG

0 = 0

SHYM = SYM - +(x) U(x) (x+)x,

^

1/(7g2)

Nt=6SU(3)-YM

G(2)-YM

=1.3

=1.3

=1.3

SU(3)-YM

G(2)-YM

Nt=6

1/(7g2)

=1.3

=1.3

=1.3

=1.5

=1.5

=1.5

SU(3)-YM

G(2)-YM

Nt=6

1/(7g2)

=1.3

=1.3

=1.3

=1.5

=1.5

=2.5

=2.5

=1.5 =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

A NEW EFFICIENT CLUSTER ALGORITHM

FOR THE ISING MODEL

M I C H E L E P E P EU n i v e r s i t y o f B e r n

(S w i t z e r l a n d)

in collaboration with

UWE-JENS WIESE and MATTHIAS NYFELERU n i v e r s i t y o f B e r n

(S w i t z e r l a n d)

O U T L I N E

• Ising model in the quantum formulation

• Construction of the cluster algorithm

• Observables: susceptibility and n-point function

• Worm algorithm

• Conclusions

• Classical Ising spin model

H[s] = - sx sx+

Partition function

Z = Ds e - H[s] 1/ = temperature

• Rewrite it as a quantum spin model

= - 3x 3

x+

Quantum partition function

= Tr e -

• Due to the trace-structure of , we can perform a unitary transformation and rotate to a basis with 1

in

= - 1x 1

x+

X,

X,

X,

• Checker-board decomposition in 1d

= 1 + 2

1 = - 1x 1

x+ 2 = - 1x 1

x+

= Tr e - = Tr (e - 1 e - 2)M =

= s1|e- 1|s2s2|e- 2|s3.….s2M-1|e- 1|s2M s2M|e- 2|s1 =

= s1|e- 1|s2s2|e- 2|s1

= M

the interactions are on the

shaded plaquettes

x even x odd

• Transfer matrix of a single plaquette

= 1x 1

x+ = =

= e - =

• From this matrix we can see that there are only 8 physically allowed plaquette configurations

0 1

1 0

0 1

1 0

0 0 0 10 0 1 00 1 0 01 0 0 0

Ch() 0 0 Sh() 0 Ch() Sh() 0 0 Sh() Ch() 0Sh() 0 0 Ch()

Ch() Ch() Sh() Sh()

• We choose A and B breakups in order to form our loop-clusters

= A + B1 + B2

B = B1 =B2 = Sh() and A = Ch() - Sh()

A

B1 B2

1 0 0 00 1 0 00 0 1 00 0 0 1

1 0 0 10 0 0 00 0 0 01 0 0 1

0 0 0 00 1 1 00 1 1 00 0 0 0

A

A

A

AA

B1

B2

B2

B2

A/(A+B)

B/(A+B)

• Correlation function:

10 1

x = Tr (10 1

x e - )

The matrices 1 can be viewed as violations

1 =

1

0 1

1 0

A

A

A

A

A

B1

A

B2

• Diagonal correlation function in 2 dimensions

Lattice size L = 64; couplings = 0.42, 0.44, 0.46

• We have also an improved estimator for the susceptibility

• Worm formulation

Instead of using clusters, we can – with the same breakups –

simply move the two violations. This technique is called the

worm algorithm

10 1

x = = ……

This way we can measure an exponentially suppressed signal with a linear effort! (P. de Forcrand, M. D’Elia and M. Pepe, Phys. Rev Lett. 2000)

(x)

(0)

(1)

(0)

(x)

(x-1)

(2)

(1)

• Diagonal correlation function in 2 dimensions

Lattice size L = 80; couplings = 0.01

Conclusions• Using the quantum formulation of the Ising model a new,

efficient algorithm can be constructed

• We have an improved estimator for the susceptibility

• We can measure n-point correlation functions over a large number of orders of magnitudes

• We would like to apply this approach to other theories, possibly to gauge theories. (bold outlook!)