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

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

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

5 Fine-tuning problem Difficult to perform numerical analysis Time for computation becomes huge. To take the desired continuum limit. SUSY breaking in the UV region Many SUSY breaking counter terms appear; is required. prevents the restoration of the symmetry Fine-tuning of the too many parameters. (To suppress the breaking term effects) Whole symmetry must be recovered at the limit

6 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 fermion action) too many4 parameters

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

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 Quantum corrections of the operators are Mass dimensions 2!Super-renormalizable Relevant or marginal operators show up only at 1-loop level.

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 Quantum corrections of the operators are Mass dimensions 2!Super-renormalizable Relevant or marginal operators show up only at 1-loop level. Irrelevant

Only following operators are relevant: 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, If dimensionless ! All order correction can be relevant or marginal remaining at continuum limit. Operators with

12 (2) 4 dimensional case, If dimensionless ! 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, If dimensionless ! 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) Nucl.Phys. B857 (2012) 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) Nucl.Phys. B857 (2012) Hanada JHEP 1011 (2010) 112 Supersymmetric regularized formulation on (1) Taking continuum limit of Full SUSY is recovered in the UV region Theory on the

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

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

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

32 Our work

33 We construct the analogous model toHanada-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 Uplift to 4D by Fuzzy 2-sphere solution Actually all of SUSY are broken but “harmless” (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.

42

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 Globalgenerators :doublets :triplet If

46 The model hassymmetry with following charge assignment singlet Charge is unchanged under the

47 (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

SUSY on the orbifold lattice theory SUSY charges commuting with orbifold projection will be the SUSY on the lattice

52 SUSY on the orbifold lattice theory SUSY charges commuting with orbifold projection will be the SUSY on the lattice = # of site fermions # of SUSY on the lattice = # of fermions with

53 SUSY on the orbifold lattice theory SUSY charges commuting with orbifold projection will be the SUSY on the lattice = # of site fermions # of SUSY on the lattice = # of fermions with

54 SUSY on the orbifold lattice theory SUSY charges commuting with orbifold projection will be the SUSY on the lattice = # of site fermions # of SUSY on the lattice = # of fermions with 4 fermions

55 SUSY on the orbifold lattice theory SUSY charges commuting with orbifold projection will be the SUSY on the lattice = # of site fermions # of SUSY on the lattice = # of fermions with 4 fermions 4SUSY is preserved on the lattice !!

56 Deconstruction and continuum limit. *Orbifodling is just picking up the subsector of matrix model. (No space has appeared.) *No kinetic terms.

57 *Orbifodling is just picking up the subsector of matrix model. (No space has appeared.) *No kinetic terms. To provide the kinetic term and continuum limit, we expand the bosonic link fields around as Deconstruction and continuum limit.

58 Continuum limit. *By taking *If fluctuation around is small, We can obtain the mass deformed 2d SYM with 8SUSY at the continuum limit

59 To provide the proper continuum limit, the fluctuation must be small enough compared with. But in the SUSY gauge theory, there are flat directions which allows huge fluctuation. We need to suppress the fluctuation by adding the moduli fixing terms Moduli fixing terms. These break the SUSY on the lattice eventually. (Softly broken, so UV divergence will not be altered.)

60 Proposed new Moduli fixing terms with keeping SUSY We proposed a new moduli fixing terms without breaking SUSY !!

61 Proposed new Moduli fixing terms with keeping SUSY We proposed a new moduli fixing terms without breaking SUSY !! We utilized the fact 61 If

62 Orbifold lattice action for 2d mass deformed SYM with moduli fixing terms is

63

64

65 (2) Momentum cut-off on the orbifold lattice theory.

66 To perform the numerical simulation, Remaining one continuum direction also must be regularized. We employ the momentum cut-off regularization in Hanada-Nishimura-ShingoTakeuchi Momentum cut-off is truncating the Fourier expansion in the finite-volume

67 Momentum cut-off in gauge theory To justify the momentum cut-off, we need to fix the gauge symmetry by the gauge fixing condition These condition fix the large gauge transformation which allows the momentum to go beyond the cut-off.

68 Momentum cut-off action on (Hybrid regularized theory) after gauge fixing.

69 And so on.. (Remaining parts are really boring, so I will omit the parts…)

70 Notes: (1) About the gauge fixing.

71 Notes: (1) About the gauge fixing. Gauge fixing does not spoil the quantum computation based on the gauge symmetry, because it is just putting the BRS exact term to the action, which does not affect the computation of gauge invariant quantity. Rather we should take this fixing as being required to justify the momentum cut-off to be well defined. Only for this purpose !!

72 Notes: (2) The cut-off might break the gauge symmetry, is it O.K ?

73 Notes: (2) The cut-off might break the gauge symmetry, is it O.K ? O.K !

74 Notes: (2) The cut-off might break the gauge symmetry, is it O.K ? O.K ! If the gauge symmetry is recovered only by taking, completely no problem. I would like to emphasize that what we are interested in is the theory at, not the theory with finite cut-off. There is no concern whether the regularized theory break the gauge sym. or not, since it is just a regularization.

75 Notes: (2) The cut-off might break the gauge symmetry, is it O.K ? O.K ! If the gauge symmetry is recovered only by taking, completely no problem. I would like to emphasize that what we are interested in is the theory at, not the theory with finite cut-off. There is no concern whether the regularized theory break the gauge sym. or not, since it is just a regularization. I will explain it later by including the quantum effects

76 (3) Uplifting to 4d by Fuzzy 2-sphere solution

77 Until here, the theory is still in the 2 dimensions. We need to uplift the theory to 4 dimensions. We will use the Fuzzy Sphere solutions!

78 Until here, the theory is still in the 2 dimensions. We need to uplift the theory to 4 dimensions. We will use the Fuzzy Sphere solutions! Derivative operators along fuzzy S2

79 We expand the fields in the spherical harmonics:

80 We expand the fields in the spherical harmonics: field on 2d spherical harmonics(kind of Fourier basis) on Fuzzy S2 field variable on target 4d space.

81 We expand the fields in the spherical harmonics: field on 2d spherical harmonics(kind of Fourier basis) on Fuzzy S2 field variable on target 4d space. Fuzzy Sphere solution does not break 8 SUSY at all !!

82 By this uplifting, we have completed the construction of non-perturbative formulation for N=2 4d non- commutative SYM theories.

83 How to take the target continuum theory

84 In our formulation, 4-dimensions are divided into 3-parrts. Regularized by momentum cut-off sites parameters

85 In our formulation, 4-dimensions are divided into 3-parrts. Regularized by momentum cut-off sites parameters Task. Which direction should we deal with first ?

86 Criteria. In early lower dimensional stage, it is easier to handle the crude regularization breaking much symmetries.

87 Criteria. In early lower dimensional stage, it is easier to handle the crude regularization breaking much symmetries. We should undertake the crude regularization first !

88 Regularized by momentum cut-off sites parameters 誰是壞人？

89 Regularized by momentum cut-off sites parameters 就是你！！ You are so rude !!

90 Regularized by momentum cut-off sites parameters On the other hand, BPS state, SUSY is well protected.

91 Then order of taking the limit becomes

92 We start from momentum cut-off directions.

93 In finite the theory is one-dimensional theory. There is no UV divergences. There is no quantum correction breaking 2 SUSY and gauge symmetry. only by taking, orbifold lattice theory is recovered.

94 We start from momentum cut-off directions.

Repeating the renormalization discussion in the early stage of this talk…. Renormalization in the 2-dimensional case Let us consider the local operators Mass dimensions Quantum corrections of the operators are Mass dimensions 2!Super-renormalizable Relevant or marginal operators show up only at 1-loop level. Irrelevant

Only following operator is relevant: 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.

Only following operator is relevant: 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. In this step, the full 8 SUSY is restored !!

98 We start from momentum cut-off directions.

99 In this step, since the full SUSY is preserved, we do not need to mind any quantum correction

100 In this step, since the full SUSY is preserved, we do not need to mind any quantum correction No fine-tuning !!

101 Notes: In the case of N=4 theory, we can continuously connect to the commutative theory in

102 Notes: In the case of N=4 theory, we can continuously connect to the commutative theory in On the other hand, N=2 theory, it is expected not to be continuously connectted to the commutative theory in Our theory is a non-perturbative formulation for the non- commutative gauge theory, but it is useful enough to investigate the non-perturbative aspects of gauge theories.

103 Summary We provide a simple non-perturbative formulation for N=2 4-dimensional theories, which is easy to put on computer.

104 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)

105 Anisotropic treatment is useful for controlling the UV divergences.

106 End わんたんら

107 Precise discussion

108 Only following diagrams can provide quantum corrections Bosonic tadpole with fermionic loop Bosonic 2-point function with fermionic loop Bosonic 2-point function with bosonic loop and derivative coupling

109 Only following diagrams can provide quantum corrections Bosonic tadpole with fermionic loop Bosonic 2-point function with fermionic loop Bosonic 2-point function with bosonic loop and derivative coupling

110 Momentum integration of the odd function Bosonic tadpole with fermionic loop Bosonic 2-point function with fermionic loop Bosonic 2-point function with bosonic loop and derivative coupling

111 Momentum integration of the odd function Bosonic tadpole with fermionic loop Bosonic 2-point function with fermionic loop Bosonic 2-point function with bosonic loop and derivative coupling = 0

112 Momentum integration of the odd function Bosonic tadpole with fermionic loop Bosonic 2-point function with fermionic loop Bosonic 2-point function with bosonic loop and derivative coupling = 0 No quantum correction !!

113

114 It becomes the theory on

115

116