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%&'# ( 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