Tomohisa Takimi (TIFR)

1. An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. • Tomohisa Takimi (TIFR) Ref) Tomohisa Takimi arXiv:1205.7038 [hep-lat] 8th June 2012 at (NTU) 1

2. 1. Introduction • 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 2

3. In some cases, we can investigate the non-perturbative quantity in the analytic way, (For example, by utilizing the duality, holomorphy, so on.) But if we want to calculate wider class of general dynamical quantities not relying on such structures, direct numerical calculation would be stronger. (For example, non-holomorphic quantities or quantities not restricted by the Chiral properties..)

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 breaking in the UV region • Many SUSY breaking counter terms appear; • prevents the restoration of the symmetry Fine-tuning of the too many parameters. is required. • (To suppress the breaking term effects) Time for computation becomes huge. Difficult to perform numerical analysis

6. Example). N=1 SUSY with matter fields By standard lattice action. • (Plaquette gauge action + Wilson 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

7. A lattice model of Extended SUSY preserving a partial SUSY Lattice formulations free from fine-tuning : does not include the translation P We call as BRST charge _ { ,Q}=P Q

8. Does the BRST strategy work to solve the fine-tuning ? (1) Let us check the 2-dimensional case Let us consider the local operators Mass dimensions Mass dimensions 2! Super-renormalizable Relevant or marginal operators show up only at 1-loop level. Quantum corrections of the operators are

9. Does the BRST strategy work to solve the fine-tuning ? (1) Let us check the 2-dimensional case Let us consider the local operators Mass dimensions Mass dimensions 2! Super-renormalizable Relevant or marginal operators show up only at 1-loop level. Quantum corrections of the operators are Irrelevant

10. Relevant Only following operators are relevant: No fermionic partner, prohibited by the SUSY on the lattice At all order of perturbation, the absence of the SUSY breaking quantum corrections are guaranteed, requiring no fine-tuning.

11. (2) 4 dimensional case, dimensionless ! If All order correction can be relevant or marginal remaining at continuum limit. Operators with

12. (2) 4 dimensional case, dimensionless ! If All order correction can be relevant or marginal remaining at continuum limit. Prohibited by SUSY and the SU(2)R symmetry on the lattice.

13. (2) 4 dimensional case, dimensionless ! If All order correction can be relevant or marginal remaining at continuum limit. Marginal operators are not prohibited only by the SUSY on the lattice

14. Fine-tuning of 4 parameters are required. The formulation has not been useful..

15. The reason why the four dimensions have been out of reach. (1) UV divergences in four dimensions are too tough to be controlled only by little preserved SUSY on the lattice.

16. The reason why the four dimensions have been out of reach. (1) UV divergences in four dimensions are too tough to be controlled only by little preserved SUSY on the lattice. How should we manage ?

17. The reason why the four dimensions have been out of reach. (1) UV divergences in four dimensions are too tough to be controlled only by little preserved SUSY on the lattice. How should we manage ? Anisotropic treatment !!

18. Anisotropic treatment: (i) We separate the dimensions into several parts in anisotropic way. (ii) We take the continuum limit of only a part of the four directions. During this step, the theory is regarded as a lower dimensional theory, where the UV divergences are much milder than ones in four -dimensions.

19. Anisotropic treatment: (i) We separate the dimensions into several parts in anisotropic way. (ii) We take the continuum limit of only a part of the four directions. During this step, the theory is regarded as a lower dimensional theory, where the UV divergences are much milder than ones in four -dimensions.

20. Anisotropic treatment: (i) We separate the dimensions into several parts in anisotropic way. (ii) We take the continuum limit of only a part of the four directions. During this step, the theory is regarded as a lower dimensional theory, where the UV divergences are much milder than ones in four -dimensions.

21. Anisotropic treatment: (i) We separate the dimensions into several parts in anisotropic way. (ii) We take the continuum limit of only a part of the four directions. During this step, the theory is regarded as a lower dimensional theory, where the UV divergences are much milder than ones in four -dimensions.

22. Anisotropic treatment: (i) We separate the dimensions into several parts in anisotropic way. (ii) We take the continuum limit of only a part of the four directions. During this step, the theory is regarded as a lower dimensional theory, where the UV divergences are much milder than ones in four -dimensions.

23. Anisotropic treatment: (i) We separate the dimensions into several parts in anisotropic way. (ii) We take the continuum limit of only a part of the four directions. During this step, the theory is regarded as a lower dimensional theory, where the UV divergences are much milder than ones in four -dimensions.

24. Anisotropic treatment: (i) We separate the dimensions into several parts in anisotropic way. (ii) We take the continuum limit of only a part of the four directions. During this step, the theory is regarded as a lower dimensional theory, where the UV divergences are much milder than ones in four -dimensions. (1) Even little SUSY on the lattice can manage such mild divergences. (2)A part of broken symmetry can be restored by the first step, to be helpful to suppress the UV divergences in the remaining steps.

25. Anisotropic treatment: (iii) We will take the continuum limit of the remaining regularized directions. In this steps, Symmetries restored in the earlier steps help to suppress tough UV divergences in higher dimensions.

26. Anisotropic treatment: (iii) We will take the continuum limit of the remaining regularized directions. In this steps, Symmetries restored in the earlier steps help to suppress tough UV divergences in higher dimensions. The treatment with steps (i) ~ (iii) will not require fine-tunings.

27. Non-perturbative formulation using anisotropy. Hanada-Matsuura-Sugino Prog.Theor.Phys. 126 (2012) 597-611 Nucl.Phys. B857 (2012) 335-361 Hanada JHEP 1011 (2010) 112 Supersymmetric regularized formulation on Two-dimensional lattice regularized directions.

28. Non-perturbative formulation using anisotropy. Hanada-Matsuura-Sugino Prog.Theor.Phys. 126 (2012) 597-611 Nucl.Phys. B857 (2012) 335-361 Hanada JHEP 1011 (2010) 112 Supersymmetric regularized formulation on (1) Taking continuum limit of Theory on the Full SUSY is recovered in the UV region

29. Non-perturbative formulation using anisotropy. Hanada-Matsuura-Sugino Prog.Theor.Phys. 126 (2012) 597-611 Nucl.Phys. B857 (2012) 335-361 Hanada JHEP 1011 (2010) 112 Supersymmetric regularized formulation on (1) Taking continuum limit of Theory on the Full SUSY is recovered in the UV region (2) Moyal plane limit or commutative limit of .

30. Non-perturbative formulation using anisotropy. Hanada-Matsuura-Sugino Prog.Theor.Phys. 126 (2012) 597-611 Nucl.Phys. B857 (2012) 335-361 Hanada JHEP 1011 (2010) 112 Supersymmetric regularized formulation on (1) Taking continuum limit of Theory on the Full SUSY is recovered in the UV region (2) Moyal plane limit or commutative limit of . Bothering UV divergences are suppressed by fully recovered SUSY in the step (1)

31. No fine-tunings !! Non-perturbative formulation using anisotropy. Hanada-Matsuura-Sugino Prog.Theor.Phys. 126 (2012) 597-611 Nucl.Phys. B857 (2012) 335-361 Hanada JHEP 1011 (2010) 112 Supersymmetric regularized formulation on (1) Taking continuum limit of Theory on the Full SUSY is recovered in the UV region (2) Moyal plane limit or commutative limit of . Bothering UV divergences are suppressed by fully recovered SUSY in the step (1)

32. Our work

33. We construct the analogous model to Hanada-Matsuura-Sugino Advantages of our model: (1) Simpler and easier to put on a computer (2) It can be embedded to the matrix model easily. (Because we use “deconstruction”) Easy to utilize the numerical techniques developed in earlier works.

34. Moreover, we resolve the biggest disadvantage of the deconstruction approach of Kaplan et al. In the approach, to make the well defined lattice theory from the matrix model, we need to introduce SUSY breaking moduli fixing terms, SUSY on the lattice is eventually broken (in IR, still helps to protect from UV divergences)

35. Moreover, we resolve the biggest disadvantage of the deconstruction approach of Kaplan et al. In the approach, to make the well defined lattice theory from the matrix model, we need to introduce SUSY breaking moduli fixing terms, SUSY on the lattice is eventually broken (in IR, still helps to protect from UV divergences) We introduce a new moduli fixing term with preserving the SUSY on the lattice !!

36. Our Formulation

37. Outline of the way to construct.

38. (0) Starting from the Mass deformed 1 dimensional matrix model with 8SUSY (Analogous to BMN matrix model) Orbifolding & deconstruction (1) Orbifold lattice gauge theory on 4 SUSY is kept on the lattice (UV) And moduli fixing terms will preserve 2 SUSY

39. Momentum cut off (2) Orbifold lattice gauge theory with momentum cut-off, (Hybrid regularization theory) Theory on Actually all of SUSY are broken but “harmless” Uplift to 4D by Fuzzy 2-sphere solution (3) Our non-perturbative formulation for 4D N=2 non-commutative SYM theories: Theory on

40. Detail of how to construct.

41. (0) The Mass deformed 1 dimensional matrix model With mN × mN matrices and with 8-SUSY For later use, we will rewrite the model by complexified fields and decomposed spinor components.

43. We also pick up and focus on the specific 2 of 8 SUSY. By using these 2 supercharges and spnior decomposition and complexified fields, we can rewrite the matrix model action by “the BTFT form”

44. The transformation laws are

45. The important property of Global generators :doublets :triplet If

46. The model has symmetry with following charge assignment singlet Charge is unchanged under the

47. (1) Orbifold lattice gauge theory

48. (1) Orbifold lattice gauge theory Orbifold projection operator on fields with r-charge

49. (1) Orbifold lattice gauge theory Orbifold projection operator on fields with r-charge Orbifold projection: Discarding the mN ×mN components except the ones with mN ×mN indices

50. Under the projection, matrix model fields become lattice fields