CONFINEMENT WITHOUT A CENTER: THE EXCEPTIONAL GAUGE GROUP G(2) M I C H E L E P E P E U 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 theoryscalar theory Svetitsky-Yaffe conjecture complicated, local, effective action for the Polyakov loop order of the deconfinement phase transition potential 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. Wiese Nucl.Phys.B694 (2004) 35 M.P., Nucl.Phys.B PS 141 (2005) 238 E(7) E(6) exceptional groups 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 2 nd 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 conjecture confined phasedeconfined phase (colorless states)(gluon plasma) size of the group determines the 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´ T a b c = T a´ b´ c´  a a´  b b´  c c´ ; T is antisymmetric 14 generators; real representations (fundamental 7  7) G(2)-"quarks" ~ G(2)-"antiquarks" l a a *  a = G(2)  SU(3) in a real rep. G(2) has rank 2 a = Gell-Mann matrices SU(3) {7} {3}  {3}  {1}

14 generators: adjoint representation is {14} {14} {8}  {3}  {3} SU(3) 14 G(2)- " gluons " 8 gluons + "vector quark " + "vector antiquark " SU(3) G(2): its own universal covering group rank 2 G(2)  SU(3) center(G(2)) = {  }  1 ( G(2) / {  } ) = {  }

string breaking without dynamical G(2)-"quark" {7}  {14}  {14}  {14} = {1}  … Interesting homotopy groups "3-ality" : all reps mix together in the tensor product decomp.  3 ( G(2) ) =   2 ( G(2)/U(1) 2 ) =     1 ( G(2) / {  } ) = {  } instantons monopoles no ’t Hooft flux 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  = (U  x T abc ) U  xy (U  y T def ) ab cdef R R T/2 T no counterpart when the gauge group has a non-trivial center U =U =

Finite temperature: different behaviour than SU(3)-YM  (3) unbroken  (3) broken  P   e -F q /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 2 nd order phase transition 1 st order or crossover ? Conjecture: Sp(2) has 10 generators and it has 1 st deconfinement p.t. We expect G(2) YM to have also a 1 st deconfinement p.t. dynamical issue: numerical simulations P z P  P 0 P * r   P  2 r  

 Tr U   /7  24 3  6

High temperature effective potential 1-loop expansion of the effective potential for the Polyakov loop ~ 11 G(2) P (  1,  2 ) = ( P, P *,1) 22 11 22 SU(3) P = diag(e i(  1 +  2 ), e i(-  1 +  2 ), e -2i  2 ) 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 M G  v For M G   QCD the 6 massive G(2)-"gluons" participate in the dynamics; for M G   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) rr MGMG  0  = 0

S HYM = S YM -    + (x) U  (x)  (x+  ) x,  ^ 1/(7g 2 )  N t =6 SU(3)-YM G(2)-YM

 = 1.3

 SU(3)-YM G(2)-YM N t =6 1/(7g 2 )

 = 1.3  = 1.5

 SU(3)-YM G(2)-YM N t =6 1/(7g 2 )

 = 1.3  = 1.5  = 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 2 nd order: G(2) 14 YM 1 st order (3+1)-d only Sp(1)=SU(2) 3 YM has a 2 nd order deconfinement p.t. SU(3) 8 YM weak 1 st order, no known universality class available YM with all other gauge groups have 1 st order (2+1)-d SU(2) 3, SU(3) 8, Sp(2) 10 YM has a 2 nd order deconfinement p.t., SU(4) 15 YM: weak 1 st or 2 nd ?, G(2) 14 YM: not known YM with all other gauge groups have 1 st 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