Lattice simulation of supersymmetric gauge theories

Transcription

1 Lattice simulation of supersymmetric gauge theories Hiroshi Suzuki Theoretical Physics Laboratory, RIKEN 25 Sept. RIKEN D. Kadoh and H.S., Phys. Lett. B 682 (2010) 466 [arxiv: [hep-lat]]. I. Kanamori and H.S., Nucl. Phys. B 811 (2009) 420 [arxiv: [hep-lat]]. I. Kanamori and H.S., Phys. Lett. B 672 (2009) 307 [arxiv: [hep-lat]]. I. Kanamori, H.S. and F. Sugino, Phys. Rev. D 77 (2008) [arxiv: [hep-lat]]. I. Kanamori, F. Sugino and H.S., Prog. Theor. Phys. 119 (2008) 797 [arxiv: [hep-lat]]. Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 1 / 26

2 SUper SYmmetry (SUSY) SUSY is a theoretical possibility that postulates invariance under exchange of BOSONS and FERMIONS Two strong motivations for SUSY Particle physics beyond the Standard Model Foundation of the string theory and its possible nonperturbative definition via AdS/CFT Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 2 / 26

3 Our real world is not however supersymmetric... Mass spectrum BOSON FERMION mass Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 3 / 26

4 Our real world is not however supersymmetric... Mass spectrum BOSON FERMION mass Higgs? W, Z photon top quark u, d quarks electron neutrino Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 3 / 26

5 Our real world is not however supersymmetric... Mass spectrum BOSON SUSY FERMION mass Higgs? W, Z photon top quark u, d quarks electron neutrino Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 3 / 26

6 Our real world is not however supersymmetric... Mass spectrum BOSON SUSY FERMION mass Higgs? W, Z photon top quark u, d quarks electron neutrino Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 3 / 26

7 Our real world is not however supersymmetric... Mass spectrum BOSON SUSY FERMION mass Higgs?? top quark W, Z? u, d quarks electron neutrino photon? Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 3 / 26

8 Our real world is not however supersymmetric... Mass spectrum BOSON SUSY FERMION mass Higgs??? top quark W, Z?? u, d quarks? electron? neutrino photon? Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 3 / 26

9 Our real world is not however supersymmetric... Mass spectrum BOSON SUSY FERMION mass Higgs? W, Z photon top quark u, d quarks electron neutrino Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 3 / 26

10 Our real world is not however supersymmetric... Mass spectrum BOSON sneutrino? SUSY FERMION photino? mass Higgs? W, Z photon top quark u, d quarks electron neutrino Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 3 / 26

11 Our real world is not however supersymmetric... Mass spectrum BOSON sneutrino? SUSY FERMION photino? mass Higgs? W, Z photon top quark u, d quarks electron neutrino Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 3 / 26

12 Our real world is not however supersymmetric... Mass spectrum BOSON sneutrino? SUSY FERMION photino? mass Higgs? W, Z photon top quark u, d quarks electron neutrino Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 3 / 26

13 Our real world is not however supersymmetric... Mass spectrum BOSON sneutrino? SUSY FERMION photino? mass Higgs? W, Z photon top quark u, d quarks electron neutrino SUSY is spontaneously broken!? and How? Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 3 / 26

14 Nonperturbative formulation of SUSY theories? Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 4 / 26

15 Nonperturbative formulation of SUSY theories? At the current moment, no nonperturbative definition of SUSY theories in general (consistency arguments only... ) We want to investigate nonperturbative phenomena, s.t., confinement, bound states, spontaneous chiral symmetry breaking, quantum tunneling, spontaneous SUSY breaking Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 4 / 26

16 Nonperturbative formulation of SUSY theories? At the current moment, no nonperturbative definition of SUSY theories in general (consistency arguments only... ) We want to investigate nonperturbative phenomena, s.t., confinement, bound states, spontaneous chiral symmetry breaking, quantum tunneling, spontaneous SUSY breaking Lattice formulation can be helpful here? Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 4 / 26

17 SUSY and the lattice SUSY on the lattice would be impossible, because { Q A α, (Q B β ) } = 2δ AB σ m α β P m (cf. Dondi-Nicolai (1977)) Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 5 / 26

18 SUSY and the lattice SUSY on the lattice would be impossible, because { Q A α, (Q B β ) } = 2δ AB σ m α β P m (cf. Dondi-Nicolai (1977)) SUSY will be restored only by fine tuning in the continuum limit Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 5 / 26

19 SUSY and the lattice SUSY on the lattice would be impossible, because { Q A α, (Q B β ) } = 2δ AB σ m α β P m (cf. Dondi-Nicolai (1977)) SUSY will be restored only by fine tuning in the continuum limit lim a 1 a 0 a = 1 Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 5 / 26

20 SUSY and the lattice SUSY on the lattice would be impossible, because { Q A α, (Q B β ) } = 2δ AB σ m α β P m (cf. Dondi-Nicolai (1977)) SUSY will be restored only by fine tuning in the continuum limit lim a 1 a 0 a = 1 tune lim a 0 a 0 a = 0 Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 5 / 26

21 SUSY and the lattice SUSY on the lattice would be impossible, because { Q A α, (Q B β ) } = 2δ AB σ m α β P m (cf. Dondi-Nicolai (1977)) SUSY will be restored only by fine tuning in the continuum limit lim a 1 a 0 a = 1 tune lim a 0 a 0 a = 0 In lucky cases, other lattice symmetries SUSY in a 0 Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 5 / 26

22 SUSY and the lattice SUSY on the lattice would be impossible, because { Q A α, (Q B β ) } = 2δ AB σ m α β P m (cf. Dondi-Nicolai (1977)) SUSY will be restored only by fine tuning in the continuum limit lim a 1 a 0 a = 1 tune lim a 0 a 0 a = 0 In lucky cases, other lattice symmetries SUSY in a 0 A nilpotent sub-algebra of SUSY, Q: {Q, Q} = 2Q 2 = 0, S = QX QS = Q 2 X = 0 Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 5 / 26

23 SUSY and the lattice SUSY on the lattice would be impossible, because { Q A α, (Q B β ) } = 2δ AB σ m α β P m (cf. Dondi-Nicolai (1977)) SUSY will be restored only by fine tuning in the continuum limit lim a 1 a 0 a = 1 tune lim a 0 a 0 a = 0 In lucky cases, other lattice symmetries SUSY in a 0 A nilpotent sub-algebra of SUSY, Q: {Q, Q} = 2Q 2 = 0, S = QX QS = Q 2 X = 0 2D N = (2, 2) SYM Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 5 / 26

24 SUSY and the lattice SUSY on the lattice would be impossible, because { Q A α, (Q B β ) } = 2δ AB σ m α β P m (cf. Dondi-Nicolai (1977)) SUSY will be restored only by fine tuning in the continuum limit lim a 1 a 0 a = 1 tune lim a 0 a 0 a = 0 In lucky cases, other lattice symmetries SUSY in a 0 A nilpotent sub-algebra of SUSY, Q: {Q, Q} = 2Q 2 = 0, S = QX QS = Q 2 X = 0 Chiral symmetry 2D N = (2, 2) SYM Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 5 / 26

25 SUSY and the lattice SUSY on the lattice would be impossible, because { Q A α, (Q B β ) } = 2δ AB σ m α β P m (cf. Dondi-Nicolai (1977)) SUSY will be restored only by fine tuning in the continuum limit lim a 1 a 0 a = 1 tune lim a 0 a 0 a = 0 In lucky cases, other lattice symmetries SUSY in a 0 A nilpotent sub-algebra of SUSY, Q: {Q, Q} = 2Q 2 = 0, S = QX QS = Q 2 X = 0 Chiral symmetry 4D N = 1 SYM 2D N = (2, 2) SYM Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 5 / 26

26 2Dim. N = (2, 2) Supersymmetric Yang-Mills theory Action (dimensional reduction of 4D N = 1 SYM to 2D) S 2DSYM = 1 [ ] 1 g 2 d 2 x tr 2 F MNF MN + Ψ T CΓ M D M Ψ + H 2 SUSY δa M = iɛ T CΓ M Ψ, δψ = i 2 F MNΓ M Γ N ɛ + i HΓ 5 ɛ We set and decompose δ H = iɛ T CΓ 5 Γ M D M Ψ Ψ ψ 0 ψ 1 χ η/2, ɛ ε (0) ε (1) ε ε δ ε (0) Q (0) + ε (1) Q (1) + ε Q + εq Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 6 / 26

27 Nilpotent Q-transformation Q-transformation (H H + if 01 ) QA µ = ψ µ, Qψ µ = id µ φ Qφ = 0 Q φ = η, Qη = [ φ, φ ] Qχ = H, QH = [φ, χ] is nilpotent on gauge invariant combinations Q 2 = δ φ 0 Moreover, the continuum action is Q-exact, S = QX S 2DSYM = Q 1 [ g 2 d 2 x tr 2iχF 01 + χh η [ φ, φ ] ] iψ µ D µ φ Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 7 / 26

28 Lattice formulation (Sugino (2004); cf. Cohen-Kaplan-Katz-Ünsal (2003)) Lattice Q-transformation (U µ (x) SU(N): link variables) QU µ (x) = iψ µ (x)u µ (x) Qψ µ (x) = iψ µ (x)ψ µ (x) i (φ(x) U µ (x)φ(x + aˆµ)u µ (x) 1) Qφ(x) = 0 Q φ(x) = η(x), Qη(x) = [ φ(x), φ(x) ] Qχ(x) = H(x), QH(x) = [φ(x), χ(x)] is nilpotent on gauge invariant combinations on the lattice Q 2 = δ φ 0 Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 8 / 26

29 Lattice formulation (Sugino (2004)) Lattice action S LAT 2DSYM = Q 1 a 2 g 2 x Λ [ tr iχ(x)ˆφ(x) + χ(x)h(x) η(x) [ φ(x), φ(x) ] i 1 µ=0 ( ψ µ (x) U µ (x) φ(x + aˆµ)u µ (x) 1 φ(x) ) ] where ˆΦ(x) ( 2F 01 ) is (basically) the plaquette Q is a manifest symmetry of S LAT 2DSYM, QSLAT 2DSYM = 0 U(1) A is another manifest symmetry Ψ(x) exp (αγ 2 Γ 3 ) Ψ(x), φ(x) exp (2iα) φ(x), φ(x) exp ( 2iα) φ(x) Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 9 / 26

30 Scalar mass term Only with S2DSYM LAT (and SU(2)), no thermalization was archived, owing to the flat directions trajectory Figure: Monte Carlo evolution of a 2 tr[ φ(x)φ(x)] , ag = , antiperiodic BC To avoid this, we added a SUSY breaking scalar mass term Smass LAT = µ2 g 2 tr [ ] φ(x)φ(x) x Λ Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 10 / 26

31 How to see SUSY (symmetry) restoration? Noether s theorem (Continuous) symmetry Conservation law More specifically, conservation law is expressed by the Ward-Takahashi (WT) identity δ µ J µ (x)o(y) = i δɛ(x) O(y), J µ (x): Noether current In the present system, we would expect where µ s µ (x) O(y) = µ2 f (x) O(y) i g2 s µ (x): supercurrent, δ δɛ(x) O(y), f (x): explicit breaking owing to the scalar mass Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 11 / 26

32 On the lattice, however, Identity on the lattice µ s µ (x) O(y) = µ2 g 2 f (x) O(y) + 1 g 2 ao S(x) O(y) δ i δɛ(x) O(y), where s µ (x) is a lattice supercurrent and, f (x) = tr[φ(x)η(x)/2], O S(x) = 0 Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 12 / 26

33 On the lattice, however, Identity on the lattice µ s µ (x) O(y) = µ2 g 2 f (x) O(y) + 1 g 2 ao S(x) O(y) δ i δɛ(x) O(y), where s µ (x) is a lattice supercurrent and, f (x) = tr[φ(x)η(x)/2], O S(x) = 0 Q symmetry! Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 12 / 26

34 Restoration of full SUSY (cf. Bochicchio-Maiani-Martinelli-Rossi-Testa (1985)) For simplicity, assume x y (no contact term) Dimensional counting, gauge symmetry, U(1) A symmetry (and assuming no SUSY anomaly), since [g] = M 1, O R S (x) = O S(x) + a 1 Z f g 2 f (x) + g 2 j Z (j) 7/2 O(j)R 7/2 (x), which means µ s µ (x) O(y) = µ2 Z f g 2 f (x) O(y) + O(a) However, since O S (x) = (,,, 0) T, we conclude Z f = 0 and g 2 lim a 0 µ s µ (x) O(y) = µ2 g 2 lim f (x) O(y) a 0 Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 13 / 26

35 Monte Carlo verification (Kanamori-H.S. (2008)) For a lowest-dimensional gauge invariant operator O(y) f 0 (y) i a 5/2 g 2 Γ ( 0 Γ tr [φψ(y)] + Γ tr [ ]) φψ(y) and an appropriate supercurrent s µ(x) = s µ (x) + O(a), and examined lim a 0 (s) µ (s µ ) i (x) (O) i (y) = µ2 g 2 lim (f ) i(x) (O) i (y)? for x y, a 0 or equivalently µ (s) (s µ ) i (x) (O) i (y) (f ) i (x) (O) i (y) a 0 µ2 g 2? for x y Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 14 / 26

36 Dynamical simulation with N f = 1/2 Majorana spinor Partition function Z = N [d(fields)] e S = N [d(bosonic fields)] e S B Pf{D} Pseudo-fermion Pf{D} = e i Arg Pf{D} (det D D) 1/4 = e i Arg Pf{D} [dϕ] [dϕ] e ϕ(d D) 1/4 ϕ Rational Hybrid Monte Carlo (RHMC) x 1/4 α 0 + N i=1 α i x + β i Remez algorithm, multi-shift solver,... Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 15 / 26

37 SUSY is restored in a 0! (Kanamori-H.S. (2008)) Continuum limit of the ratio (s µ ) i (x) (O) i (y) (s) µ (f ) i (x) (O) i (y) a 0 µ2 g 2? for x y µ 2 /g 2 Figure: i = 1 (+), i = 2 ( ), i = 3 ( ), i = 4 ( ), antiperiodic BC Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 16 / 26

38 Physics (Kanamori-H.S. (2008)) Absence of mass gap (Witten (1993)) ( t Hooft anomaly matching) in bosonic and fermionic sectors Screening of charges in the fundamental representation (Gross-Klebanov-Matytsin-Smilga (1995), Armoni-Frishman-Sonnenschein (1998)) Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 17 / 26

39 Physics: Proper definition of the hamiltonian density Lattice identity in the massless limit µ 2 0: µ δ (s µ ) i (x) O(y) = i δɛ i (x) O(y) + 1 g 2 (ao S) i (x) O(y) Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 18 / 26

40 Physics: Proper definition of the hamiltonian density Lattice identity in the massless limit µ 2 0: µ δ (s µ ) i (x) O(y) = i δɛ i (x) O(y) + 1 g 2 (ao S) i (x) O(y) set i = 4 Q-transformation and (ao S ) 4 (x) = 0 Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 18 / 26

41 Physics: Proper definition of the hamiltonian density Lattice identity in the massless limit µ 2 0: µ δ (s µ ) i (x) O(y) = i δɛ i (x) O(y) set i = 4 Q-transformation and (ao S ) 4 (x) = 0 and O(y) = (s 0 ) 1(y) Q (0) -transformation, Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 18 / 26

42 Physics: Proper definition of the hamiltonian density Lattice identity in the massless limit µ 2 0: µ δ (s µ ) i (x) O(y) = i δɛ i (x) O(y) set i = 4 Q-transformation and (ao S ) 4 (x) = 0 and O(y) = (s 0 ) 1(y) Q (0) -transformation, we have (for periodic BC) 0 ia 2 µ (sµ ) 4 (x) (s 0 ) 1(y) = Q (s 0 ) 1(y) x Λ Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 18 / 26

43 Physics: Proper definition of the hamiltonian density Lattice identity in the massless limit µ 2 0: µ δ (s µ ) i (x) O(y) = i δɛ i (x) O(y) set i = 4 Q-transformation and (ao S ) 4 (x) = 0 and O(y) = (s 0 ) 1(y) Q (0) -transformation, we have (for periodic BC) 0 ia 2 µ (sµ ) 4 (x) (s 0 ) 1(y) = Q (s 0 ) 1(y) x Λ The RHS can be identified with the hamiltonian density Q (s 0 ) 1 (y) 2 H(y) {Q, Q (0) } = 2i 0 (Kanamori-Sugino-H.S. (2007)) Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 18 / 26

44 Physics: SUSY is spontaneously broken in this system? (Kanamori (2009)) Zero temperature extrapolation of the thermal (i.e., with antiperiodic BC) average of H(x) Vacuum energy density E 0 /g 2 = 0.09 ± 0.09(sys) (stat) (energy density) g a 0 (!g) -a 1 +a !g Dynamical spontaneous SUSY breaking in this system (Hori-Tong (2006)) seems unlikely... Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 19 / 26

45 4Dim. N = 1 Supersymmetric Yang-Mills theory Action SUSY S 4DSYM = ( ) 1 d 4 x tr 2 F µνf µν + λγ µ D µ λ δ ξ A µ = ξγ µ λ, δ ξ λ = 1 2 σ µνξf µν, δ ξ λ = 1 2 ξσ µνf µν U(1) A transformation λ exp (iαγ 5 ) λ, λ λ exp (iαγ 5 ) Physics: Chiral symmetry breaking (G = SU(N c )) U(1) A anomaly & instanton Z 2Nc gaugino condensation Z 2 Physics: Low energy spectrum (Veneziano-Yankielowicz (1982)) Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 20 / 26

46 Emergent SUSY (Kaplan (1983), Curci-Veneziano (1987)) Unique SUSY breaking relevant operator is the gaugino mass term: Chiral symmetry a 0 SUSY U(1) A WT identity (up to O(a) terms) (Karsten-Smit (1980), Bochicchio-Maiani-Martinelli-Rossi-Testa (1985)) Z A µ j 5µ(x)O + Z FF e DFF e E (x)o = 2 M 12 «fi fl δ a 1 Z P P(x)O + i δα(x) O SUSY WT identity (up to O(a) terms) (Curci-Veneziano (1987), Farchioni-Feo-Galla-Gebert-Kirchner-Montvay-Münster-Vladikas (2001)) fi fl Z S µ S µ(x)o + Z T µ T µ(x)o = M a 1 δ Z χ χ(x)o δξ(x) O It has been argued that (M 12 ) ) a 1 Z P (M a 1 Z χ Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 21 / 26

47 Chiral symmetry on the lattice is of course not free! Wilson fermion fine tuning (Curci-Veneziano (1987)) Nonperturbative determination of the SUSY (and chiral) point (Donini-Guagnelli-Hernandez-Vladikas (1997)): For x y, µ S µ (x)o(y) + Z ( T M a 1 ) Z χ µ T µ (x)o(y) = χ(x)o(y) Z S Z S Domain wall L s (Nishimura (1997), Maru-Nishimura (1997), Kaplan-Schmaltz (2000)) Overlap (Neuberger (1997)) Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 22 / 26

48 G = SU(2), Wilson fermion (DESY-Münster collaboration: Demmouche-Farchioni-Ferling-Montvay-Münster-Scholz-Wuilloud (2010)) , β = 1.6 (tree-level Symanzik improved), a 0.18r 0 ( M a 1 ) Z χ 1 m gluino, Z S κ 2(4 + am), κ c Lattice: 24 3 x 48 (unstout) 0,5 0,25 kc WI ~ ,4 k c OZI ~ am gluino 0,2 0,15 0,1 0,05 (am p ) 2 0,3 0,2 0, ,95 5 5,05 1/k 4,95 5 5,05 1/k One-loop calculation (plaquette) (Taniguchi (1999)) 1 = κ c 4π 2 N 2 c β (= ) Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 23 / 26

49 G = SU(2), Wilson fermion (DESY-Münster collaboration: Demmouche-Farchioni-Ferling-Montvay-Münster-Scholz-Wuilloud (2010)) a 0.18r 0 Spectrum of N=1 SU(2) Super-Yang-Mills theory on the lattice a-h Gluino-glue Glueball 0 ++ a-f 0 5 r 0 M L = 1.54 fm L = 2.3 fm L = 2.11 fm (stout) (r 0 m p ) 2 Not appear supersymmetric... Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 24 / 26

50 G = SU(2), domain wall fermion Fleming-Kogut-Vranas (2001) , β = 2.3 Giedt-Brower-Catterall-Fleming-Vranas (2008) , β 2.4, a 0.2r 0, m res r Endres (2009) , β 2.4, a 0.2r 0, m res r It must be very interesting to investigate mass spectrum Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 25 / 26

51 Summary Although SUSY on the lattice is still quite difficult, there has been much progress in the last decade, thanks to New theoretical ideas (algorithm, formulation,... ) Increases in computing power (dynamical fermion more or less becomes accessible) For lucky cases, we can expect much in the decade ahead! The puzzling situation in 4D N = 1 SYM will be resolved! Can go into Physics (gaugino condensation etc.) Still, for unlucky cases, we need further theoretical ideas... Brute force fine tuning? Manifest SUSY in non-commutative (and non-local) field theory and its commutative limit? (matrix model etc.) Hiroshi Suzuki (RIKEN) Lattice simulation of Sept. RIKEN 26 / 26