%&'# ( Yang-Mills )*"$ !

K. Itoh, M. Kato�, H. So

N. Ukita, and H. Sawanaka

Niigata Univ.

Univ. of Tokyo, Komaba�

2002. 3. 19 at KEK

� Nucl.Phys.Proc.Suppl.106:947-949,2002(hep-lat/0110082)

� hep-lat/0112052

� $"% 6:8

!

x1. 147'SUSY (SYM),(#+=;&39

x2. 5<(-/0!.

x3. Our Formalism

x3.0 Real Staggered Fermion: �0 = �0 and �0 = �1

x3.1 One Cell Model

x3.2 Interacting Cells Model

x4. )&*&2>

1

x1. �ÁÓG SUSY (SYM)WH;T!�D·þ

0. E+7B

� ú� � � � �ßúõF'±ÝõF( ôÇ�

� �� � � � Û勵

1. 4Å© Super Yang-Mills (SYM) !)

L=1

4tr F2�� +

1

2tr TC�1( �D�)

|||||||||

C: Charge Conjugation Matrix, 4Å©HÕ¶#

CT = �C; C�1 � = (C�1 �)T

|||||||||

SUSY ��

�A� = �TC�1 �

� = F�� ���

� O7 Bianchi Id. D[�F��] = 0 D Leibnizã/

ÝS"BJ O( ) ´I ��HÅ©Câ¼

� O( 3) ´I D= 3, 4,( 6,) 10 Å©Câ¼

2

2. �ÁÓH·þ

P� "in�nitesimal Translation Generator

� Breaking of Leibniz Rule

@(ABC) 6= (@A)BC +A(@B)C +AB(@C)

�(Breaking of) Bianchi Identity

D�F��+ cyclic 6= 0?

� Degrees of Freedom,

Euclid Space and Minkowski Space

Majorana Condition ( c = ???)

Degrees of Fermionic Freedom

Two Keys:

Gauge Inv. and \Majorana" or \Real" Fermion

3

x2. Ë HYr{&h

D=4 Case

� An Approach by Wilson Fermion

Curci-Veneziano ('87), Taniguchi ('00),

Montvay et al ('01)

Yang-Mills on Lattice: One-Plaquette Action

Gaugino on Lattice! 'Majorana' Wilson Fermion

C-V Proposal

Axial and SUSY Ward-Takahashi Ids.

(Fermion Mass) Single Tuning �M

Wilson ´

! induces O(1=a) Chiral and SUSY Breaking

! Mass Counter Terms, �MA and �MS

�MA = �MS

Around SUSY Fixed Point,

Fermion Mass Term is a Unique Relevant

(Gauge Invariant, SUSY Breaking) Operator

4

� Domain Wall Fermion: Kaplan-Schmaltz ('00)

From C-V Proposal, What we need to consider

SUSY on Lattice is to impose the Masslessness

of Fermion.

�È$p[yt]|H=NG Domain Wall Fermion

WÀ+!

0−π π

−M

M

θ

5m(x )

−π/2 π/2

0−π π θ

b f00

zero mode component

−π/2 π/2

Mass Function, Left- and Right-Handed Zero

mode Components

5-Dim. Majorana (Non Local) Condition

(x�; x5) = (C�14 )T �T (x�;�x5)

5

x3. Our Formalism

x3.0 Real Staggered Fermion

Ë HYr{&h>D

�ÁÓC Exact Fermionic SymmetryIF*!

�ÁÓHYang-Mills!)W±Ý9TH/þ7*íÎF!�

� Naiive Lattice Fermion ! Doubling

� Introduction of Fermion 'Mass'

(Wilson Term or Domain Wall Mass Term)

! Gauge Field 'Mass'

(Gauge Symmetry Breaking) by SUSY

Otherwise, Lattice Fermion Doubling

! generates Gauge Field Doubling??

� Problem of Degrees of Freedom

Another Way is Staggered Fermion (K-S Fermion)

6

DÅ©^xp\&kéÛD�ø �³'"

f �; �g = ��� 12[D2 ]�2

[D2 ]

C�1 �C = �0 T�

CT = �0C

'v&^xik ' �ø �Ö§(Majorana Ö§)"

� = TC�1

p[yt]|�ú$´"

� �@� = TC�1 �@� = 1+�0�0

2 TC�1 �@�

ALS �0�0 = 1HÄG p[yt]|H»�/Ï3T!

TD+1C�1 = (�)

D(D�1)2 C�1 D+1 for D even

L= D=2, 10CI Weyl and Majorana /��%

v&^xikÅ©(D)~æÜ"

D (mod 8) 1 2 3 4 5 6 7 8

�0 + + � � + � + + � � + ��0 + + � � � � � � + + + +

D= 1 2 3 4 8 (mod 8) ! OK

7

5B 'Real' Fermion H staggered �I$

n = Vn�n ;

� n = TnC�1 = �TnV

Tn C

�1 = �TnC�1V y

n(�0)jnj

44C

Vn = n11

n22 � � �

ndd

V yn �Vn+�̂ = ��(n) = (�)

P�<� n�

uT0C�1u0H=N

�n =

8><>:�nu0; if �0 = 1

�1nu10+ �2nu

20; if �0 = �1

44C ui0I constant spinorC)S

�in/ staggered fermionC)T!

8

lZ&qp[yt]|WÏ0�,B

Xn;�

� n � n+�̂ � n��̂

2

= uT0C�1u0

Pn;� ��(n)�n�n+�̂

for �0 = �0 = 1, D=1,2,8 (mod 8)

Xn;�

� n � n+�̂ � n��̂

2

= u1T0 C�1u20Pn;� �

ij(�)jnj��(n)�in�jn+�̂

for �0 = �0 = �1, D=2,3,4 (mod 8)

44C �n )T*I �in /

(spinorless) staggered fermion!

9

x3.1 One Cell Model

Minimal Model (One Cell Model)W²,T

(a) D=2 (b) D=3

�Fundamental Lattice

Coordinates r� = 0 or 1 (a= 1)

�Gauge Action

Sg = ��

2

Xn=r(����)0<�<�

tr (Un;�� + Un;��)

r(����) � (r1; r2; � � � ; r� = 0; � � � ; r� = 0; � � �)

Sg = ��

2

Xn

X0<�<�

tr (-6

n(�)n��

(�)n�� + (�$ �))

10

�Fermion Action �0 = �0 = 1 Case

Sf =X

n=r(��)0<�

b�(n) tr�nUn;��n+�̂Uyn;�

=X

n=r(��)0<�

b�(n) tr �n��n =

Xn=r(��)0<�

b�(n) tr ���n+�̂�n+�̂

=1

2

Xn

X��

b�[n](n) trh

-�

n � i

44C r(��) � (r1; � � � ; r��1; r� = 0; r�+1; � � �)

p[yt]|»�H¶£Û b�(n) = uT0C�1u0 ��(n)

��nI �nW �� �°G�³�ú5;=�¿Á#

��n = Un;��n+�̂U

yn;�, �

��n = U

yn��̂;��n��̂Un��̂;�

11

�Pre-SUSY Transformation for Gauge Fields

�Un;� = (� � �)n;�Un;�+ Un;�(� � �)n+�̂;�

44C (� � �)n;� �P��

�[n]n;� �

�[n]n , �[n] � (�1)n��

�

-

n

��

n+��̂

!= �

X��

[ 6?

-n

�� + 6?

-n

�� ]

(å!)CI �A� = �TC�1 �

�Pre-SUSY Transformation for Fermi Fields

��n =12

P0<�;� C

(��)[n]n

�Un;(��)[n] � Un;(��)[n]

�

44C (��)[n] � �[n]�[n] = (�1)n�� (�1)n��

� (n) =

X0<�<�

"-

6

�(�)n��

(�)n�� �-

6

�

(�)n��

(�)n��

#

(å!)CI � = ���F��

12

!)HèÒÜHh[i^H�#

S = Sg + Sf ;

(1) �USf = 0 O(�3) ´/Ñ,TÖ§

(2) �S = �USg + ��Sf = 0 O(�1) ´/Ñ,T

Action Invariance

(3) Path Integral Measure

(4) Pre-SUSY éÛ

13

(1) �USf = 0 O(�3) Term Vanishing

tr

264 6 -

r

��

375 = O(�3) � �USf

(�)r���[r]r;�

b�(r)+ (�)r�

��[r]r;�

b�(r)= 0 ! �

�[r]r;�[r]

= 0

# of � = D2 ! D(D�1)=2 = # of C

(2) �S = �USg + ��Sf = 0

Pre-SUSYGQT Sg H��

�Sg = �2�X

n=r(����);0<�<�

tr�(�)n�(� � �)n;� � (�)n�(� � �)n;�

�� (Un;�� � Un;��)

Pre-SUSYGQT Sf H��

�Sf = 2X

n=r(����);0<�<�;0<�

b�(n(�)) tr [C��(n)n ��n

� (Un;�� � Un;��)]

Action W��G9T=NHÖ§Wü=%

14

-6

� �

�

(a) � 6= �; � (b) � = � or �

(a)

b�(r)C(��)[r]r = �[(�)r���[r]

r;� � (�)r���[r]r;� ]

(b)

b�(r)C(��)[r]r + b�(rd)C

(��)[rd]rd

= ��((�)r���[r]r;� + (�)r���[rd]

rd;� )

|||||||||||Combining (1)'s Result

C(��)[r]r = �(�)r�

��[r]r;�

b�(r)= ��(�)r�

��[r]r;�

b�(r)

|||||||||||Extra Constraints

C(��)[r]r + C(��)[r]

r + C(��)[r]r = 0

|||||||||||

Numbers of Transformation Parameter

D(D � 1)=2 ! D � 1 per Site

15

(3) Invariance of Path Integral Measure

ÿî�C²,T;

Step A

U0

n;� = Un;�

�0

n = �n+ C��n (Un;�� � Un;��)

Step B

U00

n;� = e�n;���0

nU0

n;�e�n+�̂;���

0

n+�̂

�00

n = �0

n

�0�n = U

0

n;�f�0

n+�̂�C��n+�̂(U

0

n+�̂;���U0

n+�̂;��)gU0yn;�

Step AC Jacobian = 1I�!

Step BCI, O7��n;� = 0FR Jacobian I 1

�®FR Un;�H��C©Hx|^GÞ×9T��Ü

! ��n;�H� = 0

4UI (1) O(�3) = 0 Ö§D consistent!

16

�­��COêÔ�.$

O(�2; C2; �C)GÖ§/CT!'<U}ÓIÍF*!(

��G �¨F]sz&f&I

6

�

-

C rw`ij/©Hx|^GÞ×9TDPJ*%

(��[n]n;� C

��[n];�n+�[n]

� ��[n]n+�̂;�C

��[n];��n+�[n]+�)( 6

n �

�[n] � 6n

)

= 0

Ë@B ÅHÖ§/ÍT#

��[n]n;� C

��[n];�n+�[n]

� ��[n]n+�̂;�C

��[n];��n+�[n]+�

= 0:

17

(4) Pre-SUSYéÛHh[i^

a&cÕWÿ��� �2�1Un;�

6n �

�[n] - 6n

4H_wpI ÉIÓHmeasure��ÜCH�¨Fop-

eratorDû8%

O?VX £ÛHnwu&f&I�+'ÿAI ý"(

[�2; �1]Un;� = [�2; �1](1 + iagA�(x) + � � �)

= (�1;�n;�C2;���n+�̂ � �2;�n;�C

1;���n+�̂ )

�f(1 + iagA�(x))2(1� iagA�(x+ �̂))

�(1 + iagA�(x+ �̂)) +O(�2) + � � �g

� �2iag�+� A�(x) +O(�2) + � � �

O(�2)´I lZ&q¡­CI aHµÅ´L= A�

H��°¸�}�I � � �G)T!

p[yt]|H�O¸�D7BÍ T/ staggeredH

;*C½%$

18

x3. Our Formalism

x3.2 Interacting Cell Model

� Cell ModelWlZ&qGá¢�G�òC0F*

íÎF!�

= Pre-SUSYI O(a) SymmetryGÝS�/T!

�AHbZjGAF/@= 4Ê&Hrw`ijI

Opposite Circulations WÃA!

-

6

� (< �)

�

6

? ?

6

Pre-SUSYW�9D 4URWáB û8¶Cä56T

-,F*!

19

Pre-SUSY éÛH�)C

O7 CellCF.@=R rw`ijH��/��°G

Oæº7 ¶¥CO(a2�2�A�)DFT%

� Problem of Degrees of Freedom

Too Many Plaquettes

! Reduction of Plaquettes

Ichimatsu Pattern Lattice

20

L: á¢�WÅH¾AHUnitG�3T!

(1) Even Unit I �AH Cell Model

(2) Odd Unit I �H Cell Model

(3) Blank Unit Cell CIF*%

Action = (Sg + Sf)jEven+ (Sg + Sf)jOdd

¬*�,TD

(A) Link �Û (a&cÕ) I

Even Cell . L=I Odd Cell ÓG)T!

Un;�, Un;�

(B) PlaquettesI Even Cell ð. L=I

Odd CellG)T%/ BLANK Unit GI F*%

Un;��, Un;��

(C) Site Variables (p[yt]|) I

Even Cell ÓG)S .A Odd Cell ÓGO)T%

�n � �n and �n

(Dual Property"p[yt]| �nI ÿÌÚ�%)

21

Four Invariance Checks Again!

||||||||||||

Two Expressions for a Site n

n = N + r = N 0+ r0 = n0

where N 0 � N � e + 2r and r0 � e � r with

e � (1;1;1;1; � � �).

Extensions; More Parameters

(� � �)n;� =X�

(��[n]n;� �

�[n]n + ~���[n]

n;� ���[n]n )

��n =P

0<�<�[C(��)[n]n

�Un;(��)[n] � Un;(��)[n]

�]n=N+r

+P

0<�<�[C(��)[n0]n0

�Un0;(��)[n0] � Un0;(��)[n0]

�]n0=N 0+r0

(1), (3) and (4) ! The Same Conditions as 1-Cell!

Straightforward Extension

(2) ! Extra Conditions relating Neighboring Cells;

(�)r�b�(n)~���[r]n;� + (�)r

0

�b�(n0)~���[r0]

n0;� = 0

b�(n)C(��)[r0]n0 = �[(�)r

0

�~���[r0]n0;� � (�)r

0

�~���[r0]n0;� ]

b�(n0)C(��)[r]

n = �[(�)r�~���[r]n;� � (�)r�~���[r]

n;� ]

22

� Number of Transformation Parameters

! 2D � 1 per Site

Independent Parameters;

C(�1)[r]n , C

(�1)[r0]n and ~�

�1[r0]n0;1

|||||||||||||||||-

C(��)[r]n = C

(�1)[r]n � C

(�1)[r]n

C(��)[r0]n0

= C(�1)[r0]n0

� C(�1)[r0]n0

��[r]n;� = (�)r�

b�(n)

�

�C(�1)[r]n � C

(�1)[r]n

�

��[r0]n0;�

= (�)r0�b�(n)

�

�C(�1)[r0]n0

� C(�1)[r0]n0

�

~���[r]n;� =

�b�(n0)

b1(n)(�)r

0

1+r�~��1[r0]

n0;1 + (�)r�b�(n0)

�

�C(�1)[r]n � C(�1)[r0]

n0

�

~���[r0]n 0;� =

b�(n)

b1(n)(�)r

0

1+r0

�~��1[r0]n0;1 + (�)r

0

�b�(n)

�

�C(�1)[r0]n0 � C(�1)[r]

n

�

23

x4. Summary and Discussion

� DÅ©C Pre-SUSYWÃAOne-Cell ModelW

±Ý

� DÅ©C One-CellWÂÐ��GAF*C

Pre-SUSYWÃA�¤Wá¢�G�ò

Discussion

� Our Pre-SUSY is NOT BRS Symmetry

�USg 6= 0

� Spinorization of Majorana Staggered Fermion

�0 = �0 = 1 Case (D=1,2,8 mod 8),

We may take b�(n) = i��(n)

�0 = �0 = �1 Case (D=2,3,4 mod 8),

We may take b�(n) = �ij(�)jnj��(n)

even site D odd siteC �1D�2WØS�3T!

24

� Å©H�ë

dol&�'reconstruction(DûÄG�¦

ñ¯#a&cÕH'Pre-SUSY'��/ SUSY��W

�M=NGI

��n;�H�AHäI ý"Fdol&HÝ�HÛ}Ó

F3UJFRF*!'¹óCO Weyl-Majorana(

D � 2[D=2]�2

DI11}�!D=(2),3,4,8,9,10,(11)

� Pre-SUSY(Local)

! SUSY(global) and D-Dim.

L> p[yt]|HÆ�ù/ç91T%

� x|^�ÛH�« ! 'É'

��n � (Un;�� � Un;��)

Im trUn;�� = 0

25

� Perturbation (Weak Coupling Expansion)

ey¤�¤CI p[yt]|W switch-o�9TD

�ey/mwmwGFT!

a&cÕH%/�2K÷ìC0F*$

! nZr¤Hæº%

p[yt]|W switch-o�7BO

�eyGmwmwGFR:G .A �õGI

ÂÐ��Wkeep7B*T�¤%

nZr�¤CO Pre-SUSYI æº9T%

26

� Lattice Requirements

(A) Ù��Ü (mod 2)

n� ! n�+2a�̂

mod 2 Ichimatsu Pattern and

Property of b�(n)

Momentum Conservation and

Fourier DecompositionZ �=a

�=adp�(p) exp(ipna)!

Z �=(2a)

��=(2a)dp0�(p0) exp(ip0n02a)

for Fermion and Gauge Field!

(B) '�öèÒÜ'

Symteric under �=2 Rotation around

a Center of a Cell

27

(C) Re ection (O-S) Positivity

Osterwalder-Seiler H�):

Re ection Map � for nD ! �nD

ô! 2.1

If F 2 A+ is Gauge Inv. Operator,

h (�F)F i > 0

ô! 2.1W�=9GI S = f + �f + (�g)g

/��%� Hô�"

��n! ��n

��in! �ij�j�n

ÂÐ��>DàS�7��.AbZjHÛ/4H�Û%

p[yt]|H¶£Û b�HÜÈ.R

O-S Positivity OK!

mod N (N � 3 ) P �Û�Á FEI gu%

28

� Pre-SUSYC FG.**4DIF*H$

� Indication of 'SUSY'

Ward-Takahashi Identities for Pre-SUSY

�htr�ni= C��n htr (Un;�� � Un;��)i= 0

�htr�nUn;��i = C��n htr (Un;�� � Un;��)Un;��i

+ ��n;�h�nUn;��n+�̂Uyn;�Un;��i � � � � = 0

� � � � � �

Gaugino-Gaugino Hâ��ÛD

rw`ij-rw`ij Hâ��ÛGª�F�£

� Wilson Loop, Con�nement

and Gluino Condensation

%&"#!$'

29

References

Our ApproachK. Itoh, M. Kato, H. So, H. Sawanaka and N. Ukita,hep-lat/0110082; NIIG-DP-01-7; in preparation

D Dim. super Yang-Mills TheoryM. B. Green, J. H. Schwartz and E. Witten, Superstringtheory Vol.1 pp244-pp247

Wilson Fermion ApproachG. Curci, G. Veneziano, Nucl.Phys.B292:555,1987

1-Loop Calculation by Wilson FermionY. Taniguchi, Phys.Rev.D63:014502,2001

MC Simulation of Wilson Fermion ApproachDESY-Muenster collaboration, Eur. Phys. J. C11 (1999)507; Nucl. Phys. Proc. Suppl. 94 (2001) 787

Domain Wall Fermion ApproachD. B. Kaplan and M. Schmaltz, Chin. J. Phys. 38

(2000) 543

MC Simulation of Domain Wall Fermion ApproachG. T. Fleming, J. B. Kogut, P. M. Vranas, Phys.Rev. D64 (2001) 034510

30

Another Approach by K-S FermionH. Aratyn, M. Goto and A. H. Zimerman, Nuovo Ci-mento A 84 (1984) 255; Nuovo Cimento A 88 (1985)225;H. Aratyn, P. F. Bessa and A. H. Zimerman, Z. Phys.C27(1985) 535H. Aratyn, and A. H. Zimerman, J. Phys. A: Math.Gen. 18 (1985) L487

Staggered FermionL. Susskind, Phys. Rev. D16 (1977) 3031

Dirac-Kaehler FormalismP. Becher and H. Joos, Z. Phys. C15 (1982) 343

Bianchi Idntity on LatticeG. G. Batrouni, Nucl. Phys. B208 (1982) 467; J. Kiskis,Phys. Rev. D26 (1982) 429

D Dim. Majorana ConditionT. Kugo and P. Townsend, Nucl. Phys. B221 (1983)357

Re ection PositivityK. Osterwalder and E. Seiler, Ann. of Phys. 110 (1978)440

31