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　 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

3

*Dynamical SUSY breaking. *Study of AdS/CFT

Non-perturbative study is important

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

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

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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

4242

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..