A lattice formulation of 4 dimensional N=2 supersymmetric Yang-Mills theories

1. A lattice formulation of 4 dimensional N=2 supersymmetric Yang-Mills theories • Tomohisa Takimi (TIFR) Ref) Tomohisa Takimi arXiv:1205.7038 [hep-lat] 19th July 2012 Free Meson Seminar 1

2. 1. Introduction • Supersymmetric gauge theory • One solution of hierarchy problem of SM. • Dark Matter, AdS/CFT correspondence • Important issue for particle physics *Dynamical SUSY breaking. *Study of AdS/CFT Non-perturbative study is important 2

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

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

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

7. 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

8. 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

9. A lattice model of Extended SUSY preserving a partial SUSY Lattice formulations free from fine-tuning P _ O.K { ,Q}=P Q

10. A lattice model of Extended SUSY preserving a partial SUSY Lattice formulations free from fine-tuning : does not include the translation We call as BRST charge Q O.K

11. Picking up “BRS” charge from 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 the Lorentz and the R-symmetry • in the extended SUSY • ex. d=2, N=2 • d=4, N=4 There are some scalar supercharges under this diagonal subgroup. If we pick up the charges, they become nilpotent supersymmetry generator which do not include infinitesimal translation in their algebra.

12. Does the BRST strategy work to solve the fine-tuning ?

13. (1) Let us check the 2-dimensional case Let us consider the local operators Mass dimensions :bosonic fields :derivatives :fermionic fields :Some mass parameters Quantum corrections of the operators are

14. (1) Let us check the 2-dimensional case Let us consider the local operators Mass dimensions :bosonic fields :derivatives :fermionic fields :Some mass parameters Quantum corrections of the operators are Mass dimensions 2! Super-renormalizable Relevant or marginal operators show up only at 1-loop level.

15. (1) Let us check the 2-dimensional case Let us consider the local operators Mass dimensions :bosonic fields :derivatives :fermionic fields :Some mass parameters Quantum corrections of the operators are Irrelevant Mass dimensions 2! Super-renormalizable Relevant or marginal operators show up only at 1-loop level.

16. (1) Let us check the 2-dimensional case Let us consider the local operators Mass dimensions :bosonic fields :derivatives :fermionic fields :Some mass parameters Only these are relevant operators Mass dimensions 2! Super-renormalizable Relevant or marginal operators show up only at 1-loop level.

17. Relevant Only following operator is 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, no fine-tuning.

18. Remaining Task (4 dimensional case)

19. (2) 4 dimensional case, dimensionless ! If All order correction can be relevant or marginal remaining at continuum limit. Operators with

20. (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.

21. (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

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

23. 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.

24. 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 ?

25. 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 ? Can we reduce the 4d system to the 2d system ?

26. 4d to 2d 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.

27. 4d to 2d 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.

28. 4d to 2d (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.

29. 4d to 2d 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.

30. 4d to 2d 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.

31. 4d to 2d 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.

32. 4d to 2d 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.

33. 4d to 2d treatment: (iii) Final step: taking the continuum limit of the remaining directions. Symmetries restored in the earlier steps help to suppress tough UV divergences in higher dimensions.

34. 4d to 2d treatment: (iii) Final step: taking the continuum limit of the remaining directions. 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.

35. 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.

36. 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

37. 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 .

38. 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)

39. 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)

40. Our work

41. 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.

42. Moreover, we resolve the biggest disadvantage of the deconstruction approach of Kaplan et al. In the conventional approach, it is necessary to introduce SUSY breaking moduli fixing terms, SUSY on the lattice is eventually broken (in IR, still helps to protect from UV divergences)

43. Moreover, we resolve the biggest disadvantage of the deconstruction approach of Kaplan et al. In the conventional approach, it is necessary 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 !!

44. Our Formulation

45. Schematic explanation

46. 4 –dimensions are divided into

47. 4 –dimensions are divided into

48. 4 –dimensions are divided into

49. From this regularized space we want to take the continuum limit without any fine-tuning

50. From this regularized space we want to take the continuum limit without any fine-tuning as