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CONFINEMENT WITHOUT A CENTER: THE EXCEPTIONAL GAUGE GROUP G(2)

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). in collaboration with. UWE-JENS WIESE 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). YM. YM + Higgs.

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CONFINEMENT WITHOUT A CENTER: THE EXCEPTIONAL GAUGE GROUP G(2)

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  1. 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) in collaboration with UWE-JENS WIESE 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)

  2. YM YM + Higgs 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

  3. gauge theory Svetitsky-Yaffe conjecture scalar theory complicated, local, effective action for the Polyakov loop order of the deconfinement phase transition potential mechanism of confinement in YM theory SU(N) (N) K.Holland, M.P., U.J. Wiese Nucl.Phys.B694 (2004) 35 Sp(N) (2) M.P., Nucl.Phys.B PS 141 (2005) 238 (2) N odd Spin(N) SO(N) (4) N=4k+2 Otah and Wingate Lucini, Wenger, and Teper (2) (2) N=4k G(2), F(4), E(8) trivial center Greensite and Lautrup exceptional Tomboulis E(6) (3) Datta, Gavai et al. groups De Forcrand and Jahn E(7) (2) Burgio, Muller-Preussker et al. • What is the role of the center of the gauge group in the deconfinement transition of Yang-Mills theory?

  4. K.Holland, M.P., U.J. Wiese Nucl.Phys.B694 (2004) 35 • 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? Sp(2)10 Sp(3) 21 center: no information about the order of the deconfinement transition conjecture confined phase deconfined phase (colorless states) (gluon plasma) size of the group determines the order ofthe p.t. K.Holland, P. Minkowski, M.P., U.J. Wiese Nucl.Phys.B668 (2003) 207

  5. G(2) SU(3) • 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) / {} ) = {}

  6. {7} {3}{3}{1} SU(3) • a 0 0 0 -a* 0 0 0 0 a = 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 rep. a= Gell-Mann matrices

  7. C(K) D*(K) K … … … = D(K) C*(K) K* + T -K -K (K) U 0 0 CU D*U* K {14} {8}{3}{3} 0 U* 0 DU C*U* K* SU(3) 0 0 1 + T -K U -K U*  • 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} 14 G(2)-"gluons" 8 gluons + "vector quark" + "vector antiquark" SU(3)

  8. G(2): its own univ. covering group rank 2 1( G(2) / {} ) = {} center(G(2)) = {} G(2)  SU(3) string breaking without dynamical G(2)-"quark" {7}{14}{14}{14} = {1} … • Interesting homotopy groups "N-ality" : all reps mix together in the tensor product decomp. 3( G(2) ) =  instantons like SU(3) 2( G(2)/U2(1) ) =    monopoles no center 1( G(2) / {} ) = {} unlike SU(3) vortices

  9. {14} {8}{3}{3} 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 V(r) ~6 G(2)-"gluons" r

  10. 1/2 ab cd ef = (UxTabc) Uxy (UyTdef) U = R T/2  0 Confining/Higgs • Fredenhagen-Marcu order parameter: confining/Higgs or Coulomb phase (R,T) = R,T  = 0 Coulomb T R no counterpart when the gauge group has a non-trivial center In strong coupling we are in the confining/Higgs phase

  11. P z P  P0 P*r   P2 r   • 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

  12. Tr U/7  2436

  13. ~ P = diag(ei(1+2), ei(-1+2), e-2i2) P (1, 2) = ( P, P*,1) 2 2 SU(3) G(2) 1 1 N. Weiss, Phys. Rev. D24 (1981) 475 High temperature effective potential • 1-loop expansion of the effective potential for the Polyakov loop

  14. {14} {8}{3}{3} SU(3) 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 V(r) V(r) MG r r  = 0 0

  15. ^ x, Nt=6 SU(3)-YM SHYM = SYM -   +(x) U(x) (x+)  G(2)-YM 1/(7g2)

  16. =1.3 =1.3 =1.3

  17. Nt=6 SU(3)-YM  G(2)-YM 1/(7g2)

  18. =1.5 =1.3 =1.3 =1.5 =1.5 =1.3

  19. Nt=6 SU(3)-YM  G(2)-YM 1/(7g2)

  20. =1.5 =2.5 =1.3 =1.3 =1.5 =2.5 =1.5 =2.5 =1.3

  21. 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 YMweak 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

  22. A NEW EFFICIENT CLUSTER ALGORITHM FOR THE ISING MODEL 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) in collaboration with UWE-JENS WIESE and MATTHIAS NYFELER 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)

  23. 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

  24. X,  • 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 3x+ Quantum partition function  = Tr e -  • Due to the trace-structure of , we can perform a unitary transformation and rotate to a basis with 1in   = - 1x 1x+ X,  X, 

  25. x even x odd • Checker-boarddecomposition in 1d =1 + 2 1 = - 1x 1x+ 2 = - 1x 1x+  = Tr e -  = Tr (e - 1e - 2)M = = s1|e- 1|s2s2|e- 2|s3.….s2M-1|e- 1|s2M s2M|e- 2|s1 = =  s1|e-  1|s2s2|e-  2|s1 the interactions are on the shaded plaquettes = M 

  26. 0 0 0 1 0 1 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 0 0 0 • Transfer matrix of a single plaquette  = 1x 1x+ =  =  = e - = • From this matrix we can see that there are only 8 physically allowed plaquette configurations Ch() 0 0 Sh() 0 Ch() Sh() 0 0 Sh() Ch() 0 Sh() 0 0 Ch() Ch() Ch() Sh() Sh()

  27. 0 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 • 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 A/(A+B) B1 B2 B2 B/(A+B) A B2 A A B1 B2 A A

  28. 1 0 1 1 0  • Correlation function: 101x = Tr(101xe - ) The matrices 1 can be viewed as violations 1 = A B2 A A B1 A A A

  29. 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

  30. (x) (1) (x) (2) (0) (0) (x-1) (1) • 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1x = = …… 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)

  31. Diagonal correlation function in 2 dimensions Lattice size L = 80; couplings  = 0.01

  32. 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!)

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