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Super Lattice Brothers

Super Lattice Brothers. Super Lattice gauge theories. Tomohisa Takimi (NCTU). 14 th May 2008 at (NCU). 1. Contents. 1 .Motivation of the supersymmetric lattice gauge theory (SLGT) and the general difficulty. 2 .The studies of the SLGT. 2-1. Simulation in the theory free from difficulty.

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Super Lattice Brothers

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  1. Super Lattice Brothers Super Lattice gauge theories Tomohisa Takimi (NCTU) 14th May 2008 at (NCU) 1

  2. Contents 1.Motivation of the supersymmetric lattice gauge theory (SLGT)and the general difficulty 2.The studies of the SLGT 2-1. Simulation in the theory free from difficulty 2-2. Overcoming the difficulty Actually they are not sufficient at all !!

  3. 1. General Motivation & Difficulty • Supersymmetric gauge theory • One solution of hierarchy problem • Dark Matter, AdS/CFT correspondence • Important issue for particle physics *Dynamical SUSY breaking. *Study of AdS/CFT Non-perturbative study is important 3

  4. Lattice: A non-perturbative method lattice construction of SUSY field theory is difficult. SUSY breaking Fine-tuning problem * taking continuum limit Difficult * numerical study

  5. Fine-tuning problem To take the desired continuum limit. Whole symmetry must be recovered at the limit SUSY case • in the standard action. • (Plaquette gauge action + Wilson or Overlap fermion action) • Violation is too hard to repair the symmetry at the limit. • Many SUSY breaking counter terms appear; • prevents the restoration of the symmetry Fine-tuning is required. • (To suppress the breaking term effects) Tuning of the too many parameters. Time for computation becomes huge. Difficult to perform numerical analysis

  6. How is the situation terrible ? Let us compare with the Lorentz symmetry case. (1)Lorentz symmetry in 4-d theory Lorentz symmetry is also broken on the lattice Subgroup (90o rotation) is still preserved - Symmetry breaking term Relevant counter terms are forbidden by the subgroup !

  7. (2)SUSY case No preserved subgroup Example). N=1 SUSY with matter fields By standard lattice action. • (Plaquette gauge action + Wilson or Overlap fermion action) gaugino mass, scalar mass fermion mass scalar quartic coupling 4 parameters too many Computation time grows as the power of the number of the relevant parameters

  8. 2. What Should We do under This Situation ? The studies of SLGT

  9. Only N=1 pure super Yang-Mills is not difficult. 2-1. Studying only the theory free from the difficulty 2-2. Paying effort to overcome Theory with scalar field (But N> 1)

  10. 2-1 Study free from difficulty • Only in the N= 1 pure Super Yang Mills, (Without scalar) the problem is not serious. Gaugino mass only! Only the fine-tuning of this parameter are necessary Numerical simulation might be doable !?

  11. How they Calculated • Gaugino mass  prohibited by Chiral sym It can be prohibited even when the SUSY is broken How about to suppress by the Chiral symmetry? We will not suffer from the fine-tuning problem

  12. G-W Fermion method Problem • Exact Chiral Symmery  Doubling problem (Nielsen-Ninomiya’s theorem ) G-W fermion formalism  Gives us “Chiral Symmetry (modified)” without doubling (Chiral anomaly is also realizable in this method) Let us use G-W formulation to avoid gaugino mass

  13. Domain Wall Fermion D.B Kaplan Phys.Lett.B288 (1992) 342 One of the G-W fermion method The solution of the 5 dimensional Dirac eq. with heavy mass 0 0 5-d is finite Left chirality Right chirality 5-d direction G-W fermion

  14. Domain wall works Proposed by D.B Kaplan Phys.Lett.B288 (1992) 342 Kaplan, Schmaltz Chin.J.Phys.38 (200)543 J.Nishimura Phys.Lett. B406 (1997) 215 N.Maru, J.Nishimura, Int. J. Mod. Phys. A13 (1998) 2841 T.Hotta et al Nucl. Phys. Proc. Suppl. 63 (1998) 685 T.Fleming, J.B.Kogut, P.M.Vranas, Phys.Rev.D64 (2001)034510

  15. What they calculated ? Gaugino condensation In N=1 SYM, it is expected that U(1) R-symmetry breaks down by gaugino condensation Anomaly Further symmetry breaking by Gaugino condensation Infinite volume :Spontaneous Breaking Finite volume: Fractional instanton They tried to watch this directly from the direct numerical calculation on the lattice.

  16. Calculation • Gaugino condensation :Magnitude of gaugino condensation :5-d length :Inverse of lattice spacing Continuum limit They observe the gaugino condensation numericaly. ½ fractional instanton contributes

  17. Next Task If we include the scalar fields..

  18. Scalar fields make situation so serious. 2-2 Overcoming the difficulty gaugino mass, scalar mass fermion mass scalar quartic coupling Difficult to suppress the scalar mass effect etc by the usual bosonic symmetry So many fine-tuning parmaeter Main difficulty of SUSY lattice

  19. Looking for the methods prohibiting Scalar mass effect Preserving the Fermionic symmetry i.E SUSY! On the lattice

  20. How should we preserve the SUSY A lattice model of Extended SUSY preserving a partial SUSY : does not include the translation P We call as BRST charge _ { ,Q}=P Q 20

  21. Twist in the Extended SUSY (E.Witten, Commun. Math. Phys. 117 (1988) 353, N.Marcus, Nucl. Phys. B431 (1994) 3-77 • Redefine the Lorentz algebra by a diagonal subgroup of (Lorentz) (R-symmetry) Ex) d=2, N=2 d=4, N=4 Scalar supercharges under , . BRST charge they do not include in their algebra

  22. Extended Supersymmetric gauge theory action BRST charge is extracted from spinor charges Twisting equivalent Topological Field Theory action Supersymmetric Lattice Gauge Theory action lattice regularization is preserved

  23. SUSY lattice gauge models with the • CKKU models(Cohen-Kaplan-Katz-Unsal) • 2-dN=(4,4),3-d N=4, 4-d N=4 etc. super Yang-Mills theories • ( JHEP 08 (2003) 024, JHEP 12 (2003) 031, JHEP 09 (2005) 042) • Sugino models • (JHEP 01 (2004) 015, JHEP 03 (2004) 067, JHEP 01 (2005) 016 Phys.Lett. B635 (2006) 218-224） • Geometrical approach • Catterall (JHEP 11 (2004) 006, JHEP 06 (2005) 031) (Relationship between them: T.T (JHEP 07 (2007) 010)) Damgaard, Matsuura (JHEP 08(2007)087)

  24. Do they really solve fine-tuning problem? • Perturbative investigation • solved • CKKU JHEP 08 (2003) 024, JHEP 12 (2003) 031, • Onogi, T.T Phys.Rev. D72 (2005) 074504 Sugino (JHEP 01 (2004) 015, JHEP 03 (2004) 067, JHEP 01 (2005) 016 Phys.Lett. B635 (2006) 218-224） They might be applicable to the numerical simulation.

  25. The simulation using these method • Study of the SSB in N=(2,2) 2-d theory by the numerical simulation (Kanamori-Sugino-Suzuki, arXiv:0711.2099,arXiv:0711.2132) They calculated the VEV of Hamiltonian

  26. Recent analytic study of 2-d N=(2,2) SUSY gauge by Hori -Tong Few number of flavor  spontaneous SUSY breaking? Try to confirm it in the numerical simulationwithout fundamental matter (N = 0 flavor))

  27. Numerical simulation They calculated the VEV of Hamiltonian VEV of Hamiltonianbecomes the order parameter of the SUSY breaking.

  28. Numerical result Vertical: Hamiltonian Horizon: lattice spacing Continuum limit 2-d N=(2,2) SUSY gauge theory is not spontaneously broken

  29. Material they did not do *Simulation with fundamental matter Hori-tong’s analysis includes the fundamental representation Formulation with fundamental rep. does not exist yet.

  30. 2-2-1 Insufficient things in present formulations with scalar fields.

  31. 2-2-1 Insufficient things in present formulations with scalar fields. (1)Fundamental Matter TFT is basically based on adjoint representation fields. There is not still. (2)Non-perturbative confirmation whether Fine-tuning problem is solved or not. K.Ohta, T.T Prog.Theor. Phys. 117 (2007) No2

  32. Non-perturbative investigation Extended Supersymmetric gauge theory action Topological Field Theory action Supersymmetric Lattice Gauge Theory action continuum limit a 0 lattice regularization

  33. How to perform the Non-perturbative investigation The target continuum theory includes a topological field theory as a subsector. For 2-d N=(4,4) CKKU models BRST-cohomology Imply Target continuum theory 2-d N=(4,4) Judge CKKU Lattice Topological fieldtheory Must be realized Forbidden Non-perturbative quantity 33

  34. what is BRST cohomology? (action ) Hilbert space of topological field theory: BRST cohomology (BPS state) Not BRST exact these are independent of gauge coupling Because • We can obtain this valuenon-perturbatively • in the semi-classical limit. 34

  35. Let us compare the BRST cohomology In Continuum VS on Lattice

  36. Continuum

  37. BRST cohomology in the continuum In the continuum theory, the BRST cohomology are satisfies so-called descent relation BRST-cohomology 1-homology cycle

  38. In the continuum theory formally BRST exact not BRST exact ! not gauge invariant BRST exact (gauge invariant quantity) 38 38

  39. Lattice

  40. On the Lattice really BRST exact BRST exact ! gauge invariant not This situation is independent of lattice spacing Even in continuum limit, these are BRST exact 40 40

  41. Why they are BRST exact ? Source of No go gauge invariant not Gauge parameters on the lattice are defined on each sites as the independent parameters. Vn Vn+i

  42. BRST cohomology Topological quantity (Intersection number)= 1 The realization is difficult due to the independence of gauge parameters Vn Vn+i (Singular gauge transformation) Admissibility condition etc. would be needed 42

  43. There are so nice trial to the SUSY lattice formulation with scalar fields, But, If we consider non-perturbatively and seriously, they would not solve the fine-tuning problem. Further study is required !

  44. 3. Conclusion *It has been important issue to make the SUSY lattice formulation applicable to numerical simulation * Recently there are great progress in this direction. (Formulation with preserved SUSY on the lattice) *But at present stage, only limited theories could be calculated

  45. Material already done *Theory without scalar fields Simulation is not difficult There is no epoch making result *Theory withscalar fields Really correct ? ? Among the adjoint rep. ?

  46. Remaining Future work *Fundamental representation *N=1 with scalar *Formulation familiar with topology (How about the combination of G-W ferminon method and exact SUSY on the lattice) So many remaining further study is necessary!

  47. Far from Game Clear! New advanced game (study) is continuing..

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