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A lattice formulation of 4 dimensional N=2 supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR). Ref) Tomohisa Takimi arXiv:1205.7038 [hep-lat]. 19 th July 2012 Free Meson Seminar. 1. 1. Introduction. Supersymmetric gauge theory One solution of hierarchy problem of SM. - PowerPoint PPT Presentation
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11
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
2
1. Introduction1. Introduction
Supersymmetric gauge theoryOne solution of hierarchy problem of SM. Dark Matter, AdS/CFT correspondence
Important issue for particle physics
2
*Dynamical SUSY breaking. *Study of AdS/CFT
Non-perturbative study is important
3
Lattice: Lattice: A non-perturbative method
lattice construction of SUSY field theory is difficultlattice construction of SUSY field theory is difficult..
Fine-tuning problem
SUSY breaking Difficult
* taking continuum limit* numerical study
4
Lattice: Lattice: A non-perturbative method
lattice construction of SUSY field theory is difficultlattice construction of SUSY field theory is difficult..
Fine-tuning problem
SUSY breaking Difficult
* taking continuum limit* numerical study
5
Lattice: Lattice: A non-perturbative method
lattice construction of SUSY field theory is difficultlattice construction of SUSY field theory is difficult..
Fine-tuning problem
SUSY breaking Difficult
* taking continuum limit* numerical study
6
Lattice: Lattice: A non-perturbative method
lattice construction of SUSY field theory is difficultlattice construction of SUSY field theory is difficult..
Fine-tuning problem
SUSY breaking Difficult
* taking continuum limit* numerical study
7
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-Fine-
tuningtuningof the too many parameters.
(To suppress the breaking term effects)
Whole symmetry must be recovered at the limit
8
Example). N=1 SUSY with matter fields
gaugino mass, scalar mass
fermion massscalar 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
9{ ,Q}=P
_
P
Q
A lattice model of Extended SUSY
preserving a partial SUSY
O.K
Lattice formulations free from fine-tuning
10
We call as BRST charge Q
A lattice model of Extended SUSY
preserving a partial SUSY
: does not include the translation
O.K
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Picking up “BRS” charge from SUSY
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=4There 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.
(E.Witten, Commun. Math. Phys. 117 (1988) 353, N.Marcus, Nucl.
Phys. B431 (1994) 3-77
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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
:bosonic fields :fermionic fields
:derivatives :Some mass
parameters
(1) Let us check the 2-dimensional case
Let us consider the local operators
Mass dimensions
Quantum corrections of the operators are
:bosonic fields :fermionic fields
:derivatives :Some mass
parameters
Mass dimensions 2!
Super-renormalizable
Relevant or marginal operators show up only at 1-loop level.
(1) Let us check the 2-dimensional case
Let us consider the local operators
Mass dimensions
Quantum corrections of the operators are
:bosonic fields :fermionic fields
:derivatives :Some mass
parameters
Mass dimensions 2!
Super-renormalizable
Relevant or marginal operators show up only at 1-loop level.
Irrelevant
(1) Let us check the 2-dimensional case
Let us consider the local operators
Mass dimensions
:bosonic fields :fermionic fields
:derivatives :Some mass
parameters
Mass dimensions 2!
Super-renormalizable
Relevant or marginal operators show up only at 1-loop level.
Only these are relevant operators
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.
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Remaining Task (4 dimensional case)
19
(2) 4 dimensional case,
If
dimensionless !
All order correction can be relevant or marginal remaining at continuum limit.
Operators with
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(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.
21
(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
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
(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.
4d to 2d treatment:
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
(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.
4d to 2d treatment:
31
(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.
4d to 2d treatment:
32
(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.
4d to 2d treatment:
33
(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.
4d to 2d treatment:
34
The treatment with steps (i) ~ (iii)
will not require fine-tunings.
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.
35
Non-perturbative formulation using anisotropy.
Hanada-Matsuura-SuginoProg.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-SuginoProg.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
Full SUSY is recovered in the UV region
Theory on the
37
Non-perturbative formulation using anisotropy.
Hanada-Matsuura-SuginoProg.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
Full SUSY is recovered in the UV region
Theory on the
(2) Moyal plane limit or commutative limit of .
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Non-perturbative formulation using anisotropy.
Hanada-Matsuura-SuginoProg.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
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)
39
Non-perturbative formulation using anisotropy.
Hanada-Matsuura-SuginoProg.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
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 !!
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.
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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.
We introduce a new moduli fixing term with preserving the SUSY on the lattice !!
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)
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
From this regularized space we want to take the continuum limit without any fine-tuning as
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The way to construct(schematic explanation)
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(0) Starting from the Mass deformed 1 dimensional matrix model with 8SUSY
(Analogous to BMN matrix model)
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(0) Starting from the Mass deformed 1 dimensional matrix model with 8SUSY
Performing Orbifolding
54
Performing deconstuction
55
Performing deconstuction
Additional 1 dimension emerges
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Performing deconstuction
But this dimension is unstable, fluctuating,
and it can crush
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To stabilize the space, we introduce Moduli fixing term
58
To stabilize the space, we introduce Moduli fixing term Then the space would be stabilized
59
To stabilize the space, we introduce Moduli fixing term Then the space would be stabilized(I introduce the moduli fixing term
without breaking SUSY on the lattice !)
60
(1) Then we obtain the orbifold lattice theory on
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For the numerical study we need to regularize
(1) Then we obtain the orbifold lattice theory on
62
We will take momentum cut-off regularization
(1) Then we obtain the orbifold lattice theory on
For the numerical study we need to regularize
63
We will take momentum cut-off regularization
(1) Then we obtain the orbifold lattice theory on
For the numerical study we need to regularize
(2)This is the hybrid regularized theory on
64
(2)This is the hybrid regularized theory on
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This is still 2 dimensional theory. Additional 2 dimensions must be emerged.
(2)This is the hybrid regularized theory on
66
Taking Fuzzy Sphere solution.
(2)This is the hybrid regularized theory on
Taking Fuzzy Sphere solution.
67
(3) Finally we obtain the non-perturbative formulation for the 4-d N=2 SYM on
68
69
How to take the continuum limit (schematic explanation)
7070
7171
We manage the momentum cut-off first !
72
73
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Continuum limit of the orbifold lattice gauge theory.
Continuum limit of the orbifold lattice gauge theory.
Moyal plane limit
Moyal plane limit
Moyal plane limit
Until the limit We do not need Fine-tunings !!
But from
But from
But from
This limit is expected not smoothly connected..
83
Although our formulation might not be a formulation for the commutative gauge theory,
It can be used for the non-commutative theories.
84
Detailed explanation
85
(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
86
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
87
Detail of how to construct.
88
(0) Starting from the Mass deformed 1 dimensional matrix model with 8SUSY
(Analogous to BMN matrix model)
88
89
(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.
90
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”
91
The transformation laws are
92
The important property of
Global
generators
:doublets:triplet
If
93
The model has
symmetry with following charge assignment
singlet
Charge is unchanged under the
94
(1) Orbifold lattice gauge theory
959595
9696
(1) Orbifold lattice gauge theoryOrbifold projection operator on fields with
r-charge
98
(1) Orbifold lattice gauge theoryOrbifold projection operator on fields with
r-charge
Orbifold projection:Discarding the mN ×mN components except
the ones with
mN ×mN indices
99
Example in N=3,
100
Example in N=3,
From the gauge transformation law of the above under U(M)3
101
Example in N=3,
From the gauge transformation law of the above under U(M)3
102
Example in N=3,
From the gauge transformation law of the above under U(M)3
Site
Link
103
Under the projection, matrix model fields become lattice fields
SUSY on the orbifold lattice theory
SUSY charges invariant under orbifold projection will be the SUSY on the lattice
105
SUSY on the orbifold lattice theory
SUSY charges invariant under orbifold projection will be the SUSY on the lattice
= # of site fermions
# of SUSY on the lattice = # of SUSY with
106
SUSY on the orbifold lattice theory
SUSY charges invariant under orbifold projection will be the SUSY on the lattice
= # of site fermions
# of SUSY on the lattice = # of SUSY with
107
SUSY on the orbifold lattice theory
SUSY charges invariant under orbifold projection will be the SUSY on the lattice
= # of site fermions
# of SUSY on the lattice = # of SUSY with
4 fermions
108
SUSY on the orbifold lattice theory
SUSY charges invariant under 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 !!
109
I have explained Orbifolding
109
110
Next is Deconstruction
111
Next is Deconstruction
111
112
Deconstruction and continuum limit.
*Orbifodling is just picking up the subsector of matrix model. (No space has appeared.)*No kinetic terms.
113
*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.
114
Continuum limit.
*By taking *If fluctuation around is
small, We can obtain the mass deformed 2d SYM with
8SUSY at the continuum limit
115
Next we need to stabilize the lattice !!
115
116
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.)
117
Proposed new Moduli fixing terms with keeping SUSYWe proposed a new moduli fixing
terms without breaking SUSY !!
118
Proposed new Moduli fixing terms with keeping SUSYWe proposed a new moduli fixing
terms without breaking SUSY !!
We utilized the fact
118
If
119
By a new moduli fixing term, the lattice becomes stabilized !!
120
Orbifold lattice action for 2d mass deformed SYM with moduli fixing terms is
121
122
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(2) Momentum cut-off on the orbifold lattice theory.
124124124
125125
126
To perform the numerical simulation, Remaining one continuum direction also
must be regularized.
We employ the momentum cut-off regularization in Hanada-Nishimura-Takeuchi
Momentum cut-off is truncating the Fourier expansion in the finite-volume
127
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.
128
Momentum cut-off action on (Hybrid regularized theory) after
gauge fixing.
129
And so on.. (Remaining parts are really boring, so I will omit the parts…)
130
Notes:
(1) About the gauge fixing.
131
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 !!
132
Notes:
(2) The cut-off might break the gauge symmetry, is it O.K ?
133
Notes:
(2) The cut-off might break the gauge symmetry, is it O.K ?
O.K !
134
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.
135
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
136
(3) Uplifting to 4d by Fuzzy 2-sphere solution
137137
138138
139
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!
140
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
141
We expand the fields in the fuzzy sphere basis which is spherical harmonics truncated at spin j:
142
We expand the fields in the fuzzy sphere basis which is spherical harmonics truncated at spin j:
field on 2d
143
We expand the fields in the Fuzzy sphere basis which is spherical harmonics truncated at spin j:
field on 2d Fuzzy S2 basis
144
We expand the fields in the Fuzzy sphere basis which is spherical harmonics truncated at spin j:
field on 2d Fuzzy S2 basis
Fuzzy S2 basis:
(1) If we truncate the spherical harmonic expansion at spin j ⇒ Fuzzy S2 basis
(2) 2j+1 ×2j+1 matrix (Tensor product is altered by
Matrix product of 2j+1 ×2j+1 matrix
⇒Total spin does not exceed j
We expand the fields in the Fuzzy sphere basis which is spherical harmonics truncated at spin j:
field on 2d Fuzzy S2 basis
field variable on target 4d space.
We expand the fields in the Fuzzy sphere basis which is spherical harmonics truncated at spin j:
field on 2d Fuzzy S2 basis
field variable on target 4d space.
Fuzzy Sphere solution does not break 8 SUSY at all !!
147
By this uplifting, we have completed the construction of non-perturbative formulation for N=2 4d non-commutative SYM theories.
148
(ii) How to take the target continuum theory
149
In our formulation, 4-dimensions are divided into 3-parrts.
Regularized by momentum cut-off
sitesparamete
rs
150
In our formulation, 4-dimensions are divided into 3-parrts.
Regularized by momentum cut-off
sitesparamete
rs
Task.
Which direction should we deal with first ?
151
Criteria.
In early lower dimensional stage, it is easier to handle the crude regularization breaking much symmetries.
152
Criteria.
In early lower dimensional stage, it is easier to handle the crude regularization breaking much symmetries.
We should undertake the crude regularization first !
153
Regularized by momentum cut-off
sitesparamete
rs
154
Regularized by momentum cut-off
sitesparamete
rs
This one !!
155
Regularized by momentum cut-off
sitesparamete
rs
On the other hand,
BPS state, SUSY is well protected.
156
Regularized by momentum cut-off
sitesparamete
rs
Then the order of taking the continuum limit is
(1)
(2)
(3)
157
Then order of taking the limit becomes
158
(1) Momentum cut-off directions.
159159
We manage the momentum cut-off first !
160
161
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.
162
(2) Managing the orbifold lattice directions
163
165
Repeating the renormalization discussion in the early stage of this talk….
(1) Let us check the 2-dimensional case
Let us consider the local operators
Mass dimensions
:bosonic fields :fermionic fields
:derivatives :Some mass
parameters
Mass dimensions 2!
Super-renormalizable
Relevant or marginal operators show up only at 1-loop level.
Only these are relevant operators
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 !!
169
(3) Fuzzy S2 directions.
Moyal plane limit
Moyal plane limit
172
In this step, since the full SUSY is preserved, we do not need to mind any quantum correction
173
In this step, since the full SUSY is preserved, we do not need to mind any quantum correction
No fine-tuning !!
174
Notes:
In the case of N=4 theory, we can continuously connect to the commutative theory in
175
Notes:
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.
But from
This limit is expected not smoothly connected..
177
SummarySummary
We provide a simple non-perturbative formulation for N=2 4-dimensional theories, which is easy to put on computer.
178
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)
179
Anisotropic treatment is useful for controlling the UV divergences.
180
End
181
Precise discussion
182
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
183
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
184
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
185
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
= 0
186
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
= 0
No quantum correction !!
187
188
It becomes the theory on
189
190