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
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
In some cases, we can investigate the nonperturbative 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, nonholomorphic quantities or quantities not
restricted by the Chiral properties..)
3
Lattice: A non-perturbative method
lattice construction of SUSY field theory is difficult.
SUSY breaking
Difficult
Fine-tuning problem
* taking continuum limit
* numerical study
4
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
5
Example). N=1 SUSY with matter fields
By standard lattice action.
(Plaquette gauge action + Wilson fermion action)
gaugino mass,
scalar mass
scalar quartic coupling
4 parameters too many
fermion mass
Computation time grows as the power of
the number of the relevant parameters
6
Lattice formulations free from fine-tuning
A lattice model of Extended SUSY
preserving a partial SUSY
: does not include the translation
We call
P
as BRST charge
_
{Q ,Q}=P
7
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
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
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.
(2) 4 dimensional case,
If
dimensionless !
All order correction can be relevant or
marginal remaining at continuum limit.
Operators with
11
(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.
12
(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
13
Fine-tuning of 4 parameters are required.
The formulation has not been useful..
14
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.
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.
How should we manage ?
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 ?
Anisotropic treatment !!
17
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.
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.
(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.
24
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.
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.
The treatment with steps (i) ~ (iii)
will not require fine-tunings.
26
Non-perturbative formulation using anisotropy.
Hanada-Matsuura-Sugino
Prog.Theor.Phys. 126 (2012) 597-611
Hanada
Nucl.Phys. B857 (2012) 335-361
JHEP 1011 (2010) 112
Supersymmetric regularized formulation on
Two-dimensional lattice regularized directions.
27
Non-perturbative formulation using anisotropy.
Hanada-Matsuura-Sugino
Prog.Theor.Phys. 126 (2012) 597-611
Hanada
Nucl.Phys. B857 (2012) 335-361
JHEP 1011 (2010) 112
Supersymmetric regularized formulation on
(1) Taking continuum limit of
Theory on the
Full SUSY is recovered in the UV region
28
Non-perturbative formulation using anisotropy.
Hanada-Matsuura-Sugino
Prog.Theor.Phys. 126 (2012) 597-611
Hanada
Nucl.Phys. B857 (2012) 335-361
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 .
29
Non-perturbative formulation using anisotropy.
Hanada-Matsuura-Sugino
Prog.Theor.Phys. 126 (2012) 597-611
Hanada
Nucl.Phys. B857 (2012) 335-361
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)
30
Non-perturbative formulation using anisotropy.
NoHanada-Matsuura-Sugino
fine-tunings !!
Prog.Theor.Phys. 126 (2012) 597-611
Hanada
Nucl.Phys. B857 (2012) 335-361
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
Our work
32
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.
33
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)
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)
We introduce a new moduli fixing term with
preserving the SUSY on the lattice !!
35
Our Formulation
36
Outline of the way to construct.
37
(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
38
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
39
Detail of how to construct.
40
(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.
41
42
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”
43
The
transformation laws are
44
The important property of
Global
generators
:doublets
:triplet
If
45
The model has
symmetry with following charge assignment
singlet
46
Charge is unchanged under the
(1) Orbifold lattice gauge theory
47
(1) Orbifold lattice gauge theory
Orbifold projection operator
on fields with r-charge
(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
49
Under the projection, matrix model fields become lattice fields
50
SUSY on the orbifold lattice theory
SUSY charges commuting with orbifold
projection will be the SUSY on the lattice
SUSY on the orbifold lattice theory
SUSY charges commuting with orbifold
projection will be the SUSY on the lattice
# of SUSY on the lattice = # of fermions with
= # of site fermions
52
SUSY on the orbifold lattice theory
SUSY charges commuting with orbifold
projection will be the SUSY on the lattice
# of SUSY on the lattice = # of fermions with
= # of site fermions
53
SUSY on the orbifold lattice theory
SUSY charges commuting with orbifold
projection will be the SUSY on the lattice
# of SUSY on the lattice = # of fermions with
= # of site fermions
4 fermions
54
SUSY on the orbifold lattice theory
SUSY charges commuting with orbifold
projection will be the SUSY on the lattice
# of SUSY on the lattice = # of fermions with
= # of site fermions
4 fermions
4SUSY is preserved on the
lattice !!
55
Deconstruction and continuum limit.
*Orbifodling is just picking up the subsector of
matrix model. (No space has appeared.)
*No kinetic terms.
56
Deconstruction and continuum limit.
*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
57
Continuum limit.
*By taking
*If fluctuation around
is small,
We can obtain the mass deformed 2d SYM with 8SUSY at the
continuum limit
58
Moduli fixing terms.
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
These break the SUSY on the lattice eventually.
(Softly broken, so UV divergence will not be
altered.)
59
Proposed new Moduli fixing terms with keeping SUSY
We proposed a new moduli fixing terms without
breaking SUSY !!
60
Proposed new Moduli fixing terms with keeping SUSY
We proposed a new moduli fixing terms without
breaking SUSY !!
We utilized the fact
If
61
61
Orbifold lattice action for 2d mass deformed SYM
with moduli fixing terms is
62
63
64
(2) Momentum cut-off on the orbifold lattice theory.
65
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
66
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.
67
Momentum cut-off action on
(Hybrid regularized theory) after gauge fixing.
68
And so on.. (Remaining parts are really boring,
so I will omit the parts…)
69
Notes:
(1) About the gauge fixing.
70
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 !!
71
Notes:
(2) The cut-off might break the gauge symmetry, is it O.K ?
72
Notes:
(2) The cut-off might break the gauge symmetry, is it O.K ?
O.K !
73
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.
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.
I will explain it later by including the quantum
effects
75
(3) Uplifting to 4d by Fuzzy 2-sphere solution
76
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!
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!
Derivative
operators along
fuzzy S2
78
We expand the fields in the spherical harmonics:
79
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.
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.
Fuzzy Sphere solution does not break 8 SUSY at all !!
81
By this uplifting, we have completed the construction of
non-perturbative formulation for N=2 4d noncommutative SYM theories.
82
How to take the target continuum theory
83
In our formulation, 4-dimensions are divided into 3-parrts.
Regularized by momentum cut-off
sites
parameters
84
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 ?
85
Criteria.
In early lower dimensional stage, it is easier to handle the
crude regularization breaking much symmetries.
86
Criteria.
In early lower dimensional stage, it is easier to handle the
crude regularization breaking much symmetries.
We should undertake the crude regularization first !
87
誰是壞人？
Regularized by momentum cut-off
sites
parameters
88
就是你！！
Regularized by momentum cut-off
sites
parameters
You are so rude !!
89
On the other hand,
Regularized by momentum cut-off
sites
parameters
BPS state, SUSY is well protected.
90
Then order of taking the limit becomes
91
We start from momentum cut-off directions.
92
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.
93
We start from momentum cut-off directions.
94
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
Mass dimensions 2!
Super-renormalizable
Relevant or marginal operators show up only at 1-loop level.
Quantum corrections of the operators are
Irrelevant
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.
Relevant
Only following operator is relevant:
In this
step, partner,
the full 8 SUSY is restored !!
No
fermionic
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.
We start from momentum cut-off directions.
98
In this step, since the full SUSY is preserved, we do not
need to mind any quantum correction
99
In this step, since the full SUSY is preserved, we do not
need to mind any quantum correction
No fine-tuning !!
100
Notes:
In the case of N=4 theory, we can continuously
connect to the commutative theory in
101
Notes:
OurIn
theory
is a non-perturbative
for the nonthe case
of N=4 theory, formulation
we can continuously
commutative
theory, but it istheory
usefulin
enough to
connect togauge
the commutative
investigate the non-perturbative aspects of gauge
theories.
On the other hand, N=2 theory, it is expected
not to be continuously connectted to the
commutative theory in
102
Summary
We provide a simple non-perturbative
formulation for N=2 4-dimensional theories,
which is easy to put on computer.
103
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)
104
Anisotropic treatment is useful for controlling the UV
divergences.
105
End
わんたんら
106
Precise discussion
107
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
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
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
110
Momentum integration of the odd function
Bosonic tadpole
with fermionic loop
Bosonic 2-point function
with fermionic loop
=0
Bosonic 2-point function
with bosonic loop and derivative
coupling
=0
111
Momentum integration of the odd function
Bosonic tadpole
with fermionic loop
Bosonic 2-point function
with fermionic loop
No quantum correction !!
=0
Bosonic 2-point function
with bosonic loop and derivative
coupling
=0
112
113
It becomes the theory on
114
115
116
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