11 super lattice brothers tomohisa takimi (nctu) 14 th may 2008 at (ncu) super lattice gauge...
TRANSCRIPT
11
Super Lattice Brothers
Tomohisa Takimi (NCTU)
14th May 2008 at (NCU)
Super Lattice gauge theories
ConteContentsnts 1.Motivation of the supersymmetric lattice gauge theory (SLGT) and the general difficulty
2.The studies of the SLGT2-1. Simulation in the theory free
from difficulty2-2. Overcoming the difficulty
Actually they are not sufficient at all !!
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1. General Motivation & 1. General Motivation & DifficultyDifficulty Supersymmetric gauge theory
One solution of hierarchy problem Dark Matter, AdS/CFT correspondence
Important issue for particle physics
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*Dynamical SUSY breaking. *Study of AdS/CFT
Non-perturbative study is important
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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
Fine-tuning problem
Difficult to perform numerical analysisTime for computation becomes huge.
To take the desired continuum limit.
SUSY SUSY casecaseViolation is too hard to repair the symmetry at the limit.
in the standard action. (Plaquette gauge action + Wilson or Overlap fermion action)
Many SUSY breaking counter terms appear;
is required.
prevents the restoration of the symmetry
Fine-Fine-tuningtuningTuning of the too many parameters.
(To suppress the breaking term effects)
Whole symmetry must be recovered at the limit
(1) Lorentz symmetry in 4-d theoryLorentz symmetry is also broken on
the lattice
Relevant counter terms are forbidden by the subgroup !
Subgroup (90o
rotation) is still preserved -
Symmetry breaking term
How is the situation terrible ?
Let us compare with the Lorentz symmetry case.
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 or Overlap fermion action)
too many4 parameters
(2) SUSY case
No preserved subgroup
2. What Should We do under This Situation ?
The studies of SLGT
2-1. Studying only the theory free from the difficulty
2-2. Paying effort to overcome
Only N=1 pure super Yang-Millsis not difficult.
Theory with scalar field (But N> 1)
2-1 Study free from difficulty
Only in the N= 1 pure Super Yang Mills, (Without scalar) the problem is not serious.
Gaugino mass only!
Only the fine-tuning of this parameter are necessary
Numerical simulation might be doable !?
How they CalculatedGaugino mass prohibited by Chiral sym
How about to suppress by the Chiral symmetry?
We will not suffer from the fine-tuning problem
It can be prohibited even when the SUSY
is broken
G-W Fermion method
Exact Chiral Symmery Doubling problem(Nielsen-Ninomiya’s theorem )
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Problem
Let us use G-W formulation to avoid gaugino mass
G-W fermion formalism
Gives us “Chiral Symmetry
(modified)” without
doubling(Chiral anomaly is also realizable in this
method)
Domain Wall Fermion
One of the G-W fermion method
The solution of the 5 dimensional Dirac eq. with heavy mass
D.B Kaplan Phys.Lett.B288 (1992) 342
0 0
G-W fermion 5-d direction
Left chirality
Right chirality
5-d is finite
Domain wall works
Proposed by D.B Kaplan Phys.Lett.B288 (1992) 342
Kaplan, Schmaltz Chin.J.Phys.38 (200)543
J.Nishimura Phys.Lett. B406 (1997) 215N.Maru, J.Nishimura, Int. J. Mod. Phys. A13 (1998)
2841T.Hotta et al Nucl. Phys. Proc. Suppl. 63 (1998) 685
T.Fleming, J.B.Kogut, P.M.Vranas, Phys.Rev.D64 (2001)034510
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Gaugino condensation In N=1 SYM, it is expected that U(1) R-symmetry breaks down by gaugino condensation
They tried to watch this directly from the direct numerical calculation on the lattice.
Anomaly Further symmetry
breaking by
Gaugino condensation
Infinite volume
:Spontaneous Breaking Finite
volume: Fractional instanton
What they calculated ?
CalculationGaugino condensation
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They observe the gaugino condensation numericaly.
:Inverse of lattice
spacing
:Magnitude of gaugino
condensation :5-d length
Continuum
limit
½ fractional instanton contributes
Next Task
If we include the scalar fields..
2-2 Overcoming the difficulty
Scalar fields make situation so serious.
Difficult to suppress the scalar mass effect etc by the usual bosonic symmetry So many fine-tuning
parmaeter Main difficulty of SUSY
lattice
gaugino mass, scalar mass
fermion massscalar quartic coupling
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Looking for the methods prohibiting Scalar mass effect
Preserving the Fermionic symmetry i.E SUSY! On the lattice
2020
How should we preserve the SUSY
We call as BRST charge
{ ,Q}=P_
P
Q
A lattice model of Extended SUSY
preserving a partial SUSY
: does not include the translation
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Twist in the Extended SUSY
Redefine the Lorentz algebra
.
(E.Witten, Commun. Math. Phys. 117 (1988) 353, N.Marcus, Nucl.
Phys. B431 (1994) 3-77
by a diagonal subgroup of (Lorentz) (R-symmetry)
Ex) d=2, N=2
d=4, N=4
they do not include in their algebra
Scalar supercharges under , BRST
charge
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Extended Supersymmetric gauge theory action
Topological Field
Theory action Supersymmetric Lattice Gauge
Theory action latticeregularization
Twisting
BRST charge is extracted from spinor
charges
is preserved
equivalent
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CKKU models (Cohen-Kaplan-Katz-Unsal)
2-d N=(4,4),3-d N=4, 4-d N=4 etc. super Yang-Mills theories
( JHEP 08 (2003) 024, JHEP 12 (2003) 031, JHEP 09 (2005) 042)
Sugino models (JHEP 01 (2004) 015, JHEP 03 (2004) 067, JHEP 01
(2005) 016 Phys.Lett. B635 (2006) 218-224 ) Geometrical approach
Catterall (JHEP 11 (2004) 006, JHEP 06 (2005) 031)
(Relationship between them:
SUSY lattice gauge models with the
T.T (JHEP 07 (2007) 010)) Damgaard, Matsuura
(JHEP 08(2007)087)
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Do they really solve fine-tuning problem?
Perturbative investigation solved CKKU JHEP 08 (2003) 024, JHEP 12 (2003)
031, Onogi, T.T Phys.Rev. D72 (2005) 074504
They might be applicable to the numerical simulation.
Sugino (JHEP 01 (2004) 015, JHEP 03 (2004) 067, JHEP 01 (2005) 016 Phys.Lett. B635 (2006)
218-224 )
The simulation using these method
Study of the SSB in N=(2,2) 2-d theory
by the numerical simulation(Kanamori-Sugino-Suzuki,
arXiv:0711.2099,arXiv:0711.2132) They calculated the VEV of Hamiltonian
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Recent analytic study of 2-d N=(2,2) SUSY gauge by Hori -Tong
Few number of flavor spontaneous SUSY breaking?Try to confirm it in the numerical simulationwithout fundamental matter (N = 0 flavor))
They calculated the VEV of Hamiltonian
VEV of Hamiltonian becomes the order parameter of the SUSY breaking.
Numerical simulation
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Numerical result
2-d N=(2,2) SUSY gauge theory is not spontaneously broken
Vertical: Hamiltonian Horizon: lattice spacing
Continuum limit
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Material they did not do
*Simulation with fundamental matterHori-tong’s analysisincludes the fundamental
representationFormulation with fundamental rep.does not exist
yet.
2-2-1 Insufficient things in present formulations with scalar fields.
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2-2-1 Insufficient things in present formulations with scalar fields.(1)Fundamental Matter
(2) Non-perturbative confirmation whether Fine-tuning problem is solved or not.
TFT is basically based on adjoint representation fields. There is not still.
K.Ohta, T.T Prog.Theor. Phys. 117 (2007) No2
Non-perturbative investigation
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Extended Supersymmetric gauge theory action
Topological Field
Theory action Supersymmetric Lattice Gauge
Theory action
limit a 0continuum
latticeregularization
3333
Topological fieldtheory
Must be realized
Non-perturbative
quantity
How to perform the Non-perturbative investigation
Lattice
Target continuum theory
BRST-cohomol
ogy
For 2-d N=(4,4) CKKU models
2-d N=(4,4)
CKKU
Forbidden
Imply
The target continuum theory includes a topological field theory as a subsector.
Judge
3434
what is BRST cohomology? (action
)
BRST cohomology (BPS state)
We can obtain this value non-perturbatively in the semi-classical limit.
these are independent of gauge coupling
Because
Hilbert space of topological field theory:
Not BRST exact
Let us compare the BRST cohomology
In Continuum VS on Lattice
Continuum
BRST cohomology in the continuum In the continuum theory, the BRST cohomology are
satisfies so-called
descent relation
BRST-cohomology
1-homology cycle
383838
not BRST exact !
not gauge invariant
formally BRST exact
BRST exact (gauge invariant quantity)
In the continuum theory
Lattice
404040
BRST exact !
not
really BRST exact
On the Lattice
gauge invariant
Even in continuum limit, these are BRST exactThis situation is independent of lattice spacing
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Why they are BRST exact ?
Gauge parameters on the lattice are defined on each sites as the independent parameters.
Vn Vn+i
Source of No go
gauge invariant
not
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The realization is difficult due to the independence of gauge parameters
BRST cohomology
Topological quantity
(Singular gauge transformation)Admissibility condition etc. would be needed
Vn Vn+i
(Intersection number)= 1
There are so nice trial to the SUSY lattice formulation with scalar fields,
But,If we consider non-perturbatively and
seriously,they would not solve the fine-tuning
problem.
Further study is required !
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3. Conclusion
*It has been important issue to make the SUSY lattice formulation applicable to numerical simulation
* Recently there are great progress in this direction. (Formulation with preserved SUSY on the lattice)* But at present stage, only limited theories could be calculated
Material already done
*Theory without scalar fieldsSimulation is not difficult
There is no epoch making result*Theory with scalar fields
?
?
Really correct ?
Among the adjoint rep.
Remaining Future work
*Fundamental representation*N=1 with scalar
*Formulation familiar with topology(How about the combination of G-W ferminon method and exact SUSY on the lattice)
So many remaining further study is necessary!
Far fromGame Clear!New advanced game (study) is continuing..