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

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 3 *Dynamical SUSY breaking. *Study of AdS/CFT Non-perturbative study is important

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

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

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

Example). N=1 SUSY with matter fields gaugino mass,scalar massfermion mass scalar quartic coupling Computation time grows as the power of the number of the relevant parameters By standard lattice action. (Plaquette gauge action + Wilson or Overlap fermion action) too many4 parameters (2) SUSY case No preserved subgroup

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

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

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 !?

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

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

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

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)

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

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

Next Task If we include the scalar fields..

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

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

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

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

23 CKKU models (Cohen-Kaplan-Katz-Unsal) 2-d N=(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) ） Geometrical approach Catterall (JHEP 11 (2004) 006, JHEP 06 (2005) 031) (Relationship between them: SUSY lattice gauge models with the 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) They might be applicable to the numerical simulation. Sugino ( JHEP 01 (2004) 015, JHEP 03 (2004) 067, JHEP 01 (2005) 016 Phys.Lett. B635 (2006) ）

The simulation using these method Study of the SSB in N=(2,2) 2-d theory by the numerical simulation (Kanamori-Sugino-Suzuki, arXiv: ,arXiv: ) 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 simulation without fundamental matter (N = 0 flavor))

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

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

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.

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

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

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

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

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

Let us compare the BRST cohomology In Continuum VS on Lattice

Continuum

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

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

Lattice

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

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

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

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

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

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!

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