- Email: [email protected]
Pion mass and the PCAC relation in the overlap fermion formalism: Gauged Gross–Neveu model on a lattice
5 August 1999
Physics Letters B 460 Ž1999. 164–169
Pion mass and the PCAC relation in the overlap fermion formalism: Gauged Gross–Neveu model on a lattice Ikuo Ichinose 1, Keiichi Nagao
2
Institute of Physics, UniÕersity of Tokyo, Komaba, Tokyo, 153-8902 Japan Received 12 May 1999 Editor: H. Georgi
Abstract We investigate chiral properties of the overlap lattice fermion by using solvable model in two dimensions, the gauged Gross–Neveu model. In this model, the chiral symmetry is spontaneously broken in the presence of small but finite fermion mass. We calculate the quasi-Nambu–Goldstone ŽNG. boson mass as a function of the bare fermion mass and two parameters in the overlap formula. We find that the quasi-NG boson mass has desired properties as a result of the extended chiral symmetry found by Luscher. We also examine the PCAC relation and find that it is satisfied in the continuum limit. ¨ Comparison between the overlap and Wilson lattice fermions is made. q 1999 Published by Elsevier Science B.V. All rights reserved.
Species doubling is a long standing problem in the lattice fermion formulation. Wilson fermion is the most suitable formulation w1x and it is used in most of the numerical studies of lattice gauge theory. However in order to reach the desired continuum limit, fine tuning must be done with respect to the ‘‘bare fermion mass’’ and the Wilson parameter. Recently a very promising formulation of lattice fermion named overlap fermion was proposed by Narayanan and Neuberger w2x. In that formula the Ginsparg and Wilson ŽGW. relation w3x plays a very
1 2
E-mail: [email protected] E-mail: [email protected]
important role, and because of that there exists an ‘‘extended’’ Žinfinitesimal. chiral symmetry. In this paper we shall study or test the overlap fermion by using the gauged Gross–Neveu model in two dimensions. This is a solvable model which has similar chiral properties with QCD4 , i.e., chiral symmetry is spontaneously broken with a small but finite bare fermion mass and pion appears as quasiNambu–Goldstone boson 3. Actually a closely related model was studied on a lattice in order to test properties of the Wilson fermion in the continuum limit w5x. Therefore advantage of the overlap fermion becomes clear by the investigation in this paper.
3 Overlap fermion with a finite fermion mass was recently studied in Ref. w4x
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 7 4 9 - 2
I. Ichinose, K. Nagaor Physics Letters B 460 (1999) 164–169
The model is defined by the following action on the two-dimensional square lattice with the lattice spacing a, N
Ss
2
Ý Ł Um Ž n . q a2 Ý c Ž m . D Ž m,n . c Ž n . n, m
pl
q a2 MB Ý cc Ž n . y n
a2
'N Ý n
f i Ž n . Ž ct ic .
c Ž n . ™ c Ž n . q c Ž m . t ku kg 5 dn m y 12 aD Ž n,m . 4 f i Ž n . ™ f i Ž n . q d i k ju kf 5j Ž n . ,
a2 q 2 gÕ
Ý
i
i
f Ž n. f Ž n.
q f 5i
Ž n.
f 5i
Ž n. ,
f 5i Ž n . ™ f 5i Ž n . y d i k ju kf j Ž n . ,
n
Ž 2.
Ž 3.
As in the continuum model, we expect that the field f 0 acquires a nonvanishing vacuum expectation value ŽVEV.,
where UmŽ n. is UŽ1. gauge field defined on links, cal Ž a s 1, . . . , N,l s 1, . . . , L. are fermion fields with flavour index l, and the matrix t i Ž i s 0, . . . , L2 y 1. acting on the flavour index is normalized as Tr Ž t it k . s d i k and 1
i j i jk k 'L , t ,t 4 s d t ,
where d i jk ’s are the structure constants of SUŽ L.. Fields f i and f 5i are scalar and pseudo-scalar bosons,respectively. The covariant derivative in Eq. Ž1. is defined by the overlap formula 1 a
ž
1qX
1
'X † X
/
,
1 2a
² f 0 : s 'NL M s , and we define subtracted fields,
w 0 s f 0 y 'NL M s ,
Ss
N 2
dmq m , nUm Ž m . y dm , nq m Um† Ž n . , M0 a
r q 2a
Ý
2 dn , m
Ž i / 0 . , w 5i s f5i . Ž 9.
n, m
pl 2
'N Ý n
w i Ž n . Ž ct ic . Ž n .
qw 5i Ž n . Ž ct ig 5 c . Ž n .
m
ydmq m , nUm Ž m . y dm , nq m Um† Ž n . ,
Ž 4.
where r and M0 are dimensionless nonvanishing free parameters of the overlap lattice fermion formalism w2,6x. The overlap Dirac operator D does not have the ordinary chiral invariance but satisfies the GW relation instead, Dg 5 q g 5 D s aDg 5 D.
w isf i
Ý Ł Um Ž n . q a2 Ý c Ž m . DM Ž m,n . c Ž n . a
y
B Ž n,m . s y
Ž 8.
In terms of the above fields,
X n m s gm Cm Ž n,m . q B Ž n,m . , Cm s
Ž 6.
where u i is an infinitesimal transformation parameter. The above symmetry Ž6. coincides with the ordinary chiral symmetry up to O Ž a.. From the action Ž1., it is obvious that f i and f 5i are composite fields of the fermions, gÕ gÕ i fis ct ic , f 5i s Ž 7. 'N 'N cg 5t c .
Ž 1.
Ds
From Ž1. it is obvious that the systematic 1rN expansion is possible and we shall employ it. The action Ž1. contains the bare fermion mass MB which explicitly breaks the chiral symmetry. This bare mass also breaks the following infinitesimal w7x transformation, which was discovered by Luscher ¨ and we call ‘‘extended chiral symmetry’’,
c Ž n . ™ c Ž n . q t ku kg 5 dn m y 12 aD Ž n,m . 4 c Ž m . ,
= Ž n . q f 5i Ž n . Ž ct ig 5 c . Ž n .
t 0s
165
Ž 5.
a2 q 2 gÕ qw 5i
Ý
w i Ž n . w i Ž n . q 2'NL M s w 0 Ž n .
n
Ž n . w5i Ž n . ,
Ž 10 .
where DM s D y M ,
M s MB q M s .
Obviously, M is the dynamical fermion mass.
Ž 11 .
I. Ichinose, K. Nagaor Physics Letters B 460 (1999) 164–169
166
From the chiral symmetry and Ž8., we can expect that quasi-Nambu–Goldstone ŽNG. bosons appear as a result of the spontaneous breaking of the chiral symmetry. They are nothing but w 5i . The VEV M s is determined by the tadpole cancellation condition of w 0 . In order to perform an explicit calculation of the 1rN-expansion, it is useful to employ the momentum representation, and also we introduce the gauge potential lmŽ n. in the usual way, i.e., UŽ n, m . s w ia Ž .. expŽ 'N x lm n . By using weak-coupling expansion by Kikukawa and Yamada w8x, Dn m s
yi aŽ q nyp m.
e HH p q
D Ž p,q . ,
The vertex function is explicitly given as
ž
V1 m p q
D Ž p,q . s D 0 Ž p . Ž 2p . d Ž p y q . q
a
E s
V Ž p,q . ,
2
q
1
v Ž p. q v Ž q.
y
X0 Ž p .
a 2v Ž p .
5
X 1† Ž p,q .
v Ž p.
,
Ž 14 .
'
i a
Am Ž p . gm q B Ž p .
v Ž q.
1
Ý Ž 1 y cos apm . y a M0 , a
X 1 Ž q, p . s Ž 2p . d Ž q y p y k .
ž
k 2
/
1
'N
,
lm Ž k . V1 m
Ms
Hk
eyS eff s
H
Ž 17 .
Dc Dc eyS .
2
sin2 Ž apm . q r Ý Ž 1 y cos Ž apm . . y M0
ž
m
ab Ž p . s r Ý Ž 1 y cos Ž apm . . y M0 . m
Ž 21 .
/
Ž 22 .
Especially we are interested in w 5 and lm part of the effective action, because w 5 is the quasi-NG bosonŽpion. and its coupling with the gauge boson is related with anomaly. We define
Hk
1 2
w 5i Ž yk . Gi 5j Ž k 2 . w 5j Ž k .
Ž 23 .
where
Gi 5j Ž k 2 . s d i j
)
Ž 20 .
sy Tr Ž Dy1 M Ž 0. Ž k . .
Ž2. w w5 x s Seff
av Ž p. s
.
Effective action of w i , w 5i and the gauge field Ž lm n. is obtained by integrating out the fermions,
Ž 16 .
= pq
Ž 19 .
2
Hk v Ž k . 1qŽ1y Ma.2 4 q2 b Ž k . Ž1y Ma. .
X0 Ž q .
r
4
.
sy
m
Hk
/
/
2 a b Ž k . q Ž 1y Ma . v Ž k . 4
X 1 Ž p,q .
Ž 15 .
gm sin apm q
2
a b Ž p . q Ž 1 y Ma . v Ž p . 4 y igm sin Ž apm .
J Ž p.
gÕ
q ... X0 Ž p . s
km
From Ž10. and Ž13.,
gm isin apm
av Ž p.
½
2
/
v Ž p . 1 q Ž 1 y Ma . 4 q 2 b Ž p . Ž 1 y Ma . 4
d pra where Hp s Hy p r a Ž2p . and
V Ž p,q . s
ž
k
2
Dy1 M Ž0.
2p
D0 Ž p . s
E pm
X0 p q
km
From Ž14., the tree level propagator is obtained as
Ž 13 . bŽ p. q v Ž p.
ž
s igm cos a pm q
ž
s 1
2
/
q r sin a pm q
Ž 12 .
2
k
1 gÕ
q Tr g 5 ² c Ž k y p .
Hk
,
Ž 18 .
=c Ž k y p . :g 5 ² c Ž k . c Ž k . : s d i j e q 2 k 2 M02 A Ž k 2 ; M . .
Ž 24 .
I. Ichinose, K. Nagaor Physics Letters B 460 (1999) 164–169
Parameter e in Gi 5j Ž24. is proportional to the pion mass and measures the derivation from the limit of the exact chiral symmetry. In the leading order of the 1rN, e sy
2
It is also straightforward to calculate AŽ k 2 ; M . in Ž24., 1 AŽ k 2 ; M . s 2 4p k Ž k 2 q m 2 .
(
a2
=H
k
(Ž k q 2 m . y 4m =ln k q 2 m y (Ž k q 2 m . y 4m k 2 q 2 m2 q
1
p
M s w v Ž k r a . 1q Ž 1y Ma . 2 4 q2 b Ž k r a . Ž 1y Ma . x
2
2
=w a b Ž k r a. q Ž1y Ma. v Ž k r a. 4 a. 4 v Ž k r a.
™
2
=Ž1q Ž1y Ma. . q2 b Ž k r a.Ž1y Ma. 4 q M s sin2 km q a 2 Ž b Ž k r a . q Ž 1y Ma . v Ž k r a .. 2 4 x s
MB M02 Ms
167
w yln Ž M0 M 2 a . qconst. x q O Ž a. ,
Ž 25 .
1 4pm2
2
2
2 2
4
2
2 2
4
q Ž k2 . ,
Ž 28 .
where m s M0 M and therefore the pion mass is given as mp2 s 2p M 2e . There exists a mixing term of the gauge boson lm and the pion w 50 ,
2
d k p where Hkp s Hy p Ž2p . and we took the continuum limit to obtain the last line of Ž25.. From Ž25., e A MB q O Ž a. and therefore the limit MB ™ 0 is considered as the chiral limit. It is instructive to compare the above result with that in the continuum theory and the lattice model with the Wilson fermions. The corresponding expression of e in the continuum theory is given as 2
L2
2 MB 1
w e x cont s
M s 4p
ln
ž / M2
q O Ž 1rL . ,
Ž 26 .
where L is the momentum cutoff. It is obvious that e in the overlap formalism has a very close resemblance to that in the continuum theory. On the other hand, the corresponding expression in the Wilson fermion formalism was obtained in Ref. w5x as follows,
w e xW s y
4 rW Ms a
L Ž rW . q
2 MB Ms
I Ž k . s Ý sin2 km q y2 r W Ý sin2
ž
m p
L Ž rW . s
Hk
Ý sin2 Ž kmr2. IŽ k.
1
p
Hk
IŽ k. km 2
Ž2. Seff lm , w 50 s y2'L M02 M
Hk Ý l Ž yk . m
=emn kn A Ž k 2 ; M . w 50 Ž k . ,
Ž 29 .
which is identical with the continuum calculation. This mixing term is related to the discussion of the UŽ1. problem in QCD4 and the above result suggests that the correct anomaly appears in the Ward– Takahashi identity of the axial-vector current. We shall examine the PCAC relation. By changing variables as follows in the path-integral representation of the partition function 4 ,
c Ž n. ™ c Ž n. q t ku k Ž n . g 5 dn m y aD Ž n,m . 4 c Ž m . ,
c Ž n . ™ c Ž n . 1 q t ku k Ž n . g 5 4 , f i Ž n . ™ f i Ž n . q d i k ju kf 5j Ž n . , f 5i Ž n . ™ f 5i Ž n . y d i k ju kf j Ž n . , Ž 30 . we obtain the Ward–Takahashi ŽWT. identity,
, 2
² Em j5,k m Ž n . y 2 M Ž ct kg 5 c . Ž n . q
/
q Ma ,
2'N gÕ
M s w 5k Ž n .
q DAk Ž n . y d k 0 N'L aTr g 5 D Ž n,n . : s 0, ,
Ž 27 .
where r W is the Wilson parameter. Therefore it is obvious that the fine tuning of the ‘‘bare mass’’ MB and the Wilson parameter r W is required in order to reach the chiral limit. In this sense, the overlap fermion is better than the Wilson fermion.
Ž 31 . where the last term comes from the measure of the path integral, and the explicit form of the current 4
We employ this form of change of variables instead of that is given by Ž6.. This is merely for technical reason here.
I. Ichinose, K. Nagaor Physics Letters B 460 (1999) 164–169
168
operator j5,k m is obtained by Kikukawa and Yamada w9x as follows, j5,k m Ž n . s t k Ý c Ž l . K n5m Ž l,m . c Ž m . ,
Ž 32 .
D5A Ž p . 4
b
s y'N aM w 5k Ž p . Tr ² c Ž p q q .
Hq
lm
½
K n5m Ž l,m . s K n m
H
'H 2
5
=c Ž p q q . :g 5 ² c Ž q . c Ž q . :g 5 D 0 Ž p q q .
Ž l,m . ,
Ž 33 . s y'N aMw 5k Ž p .
H s yg 5 X ,
Ž 34 .
= yMsin aqm sin a Ž q y p . m
aK n m Ž l,m . s g5
=
½
2
Hq J Ž q . J Ž q y p .
`
dt
Ž t 2 Wn m y HWn m H . Ž t2qH 2 .
1 2
2
Ž t qH . Wn m Ž l,m . s g
y Ž 2 y Ma . Ž v Ž q . q b Ž q . . a b Ž q y p .
1
Hy` p
5
,
q Ž 1 y Ma . v Ž q y p . 4 .
Ž 35 .
lm
From Ž25. and Ž38., we obtain D5A Ž p . s y2'N M s ew 5k Ž p . .
1 5 2
Ž gm y 1. dn l dnq mˆ , m Un m
† q 12 Ž gm q 1 . d l , nq mˆ dn m Unq mˆ , m 4 ,
Ž 36 .
a
b
DmA s DmA Ž p . 4 q DmA Ž p . 4 ,
DAk Ž n . s aM Ý c Ž n . t kg 5 D Ž n,m . c Ž m .
DmA Ž p . 4
m
a
a
By integrating out the fermions, the above WT identity is expressed in terms of the pions and the gauge field. Matrix element D5A s ² w 5k < DAk <0: has the contribution from the following two terms,
Ž p. s
a
Ž p. 4 q
a
D5A Ž p . 4 s'N aH Tr q
D5A
Hq
Ž p. 4 ,
² c Ž q . c Ž q . : D0 Ž q .
2 Ž 2 y Ma . Ž v Ž q . q b Ž q . . JŽ q.
X
5 D0
DmA Ž p . 4
b
s yNM'L
HqTr ² c Ž q . c Ž q . :
=g 5 d k 0 V Ž p q q,q . .
Ž 40 .
It is not so difficult to show that the above two terms cancel with each other and DmA Ž p . s 0. On the other hand, the last term of Ž31. is evaluated by a similar method for the four-dimensional case by Kikukawa and Yamada w8x, and we obtain
b
=w 5k Ž p . s'N a w 5k Ž p . =
X
Hqq Tr ² c Ž q . c Ž q . :g
=V Ž p q qX ,q . d k 0 ,
Ž 37 .
m
D5A
s NM'L
= Ž p q qX . ² c Ž p q qX . c Ž p q qX . :
i i 'N c Ž n . Ž w Ž n . g 5 y w5 Ž n . .
=t it k Ý D Ž n,m . c Ž m . .
D5A
Ž 39 .
In a similar way, matrix element DmA s ² lm < DA0 <0: is evaluated as,
and the operator DAk Ž n. is given by
q
Ž 38 .
aTr g 5 D Ž n,n . s ,
i
Ý emn En lm . p'N mn
Ž 41 .
Eq. Ž41. is nothing but the chiral anomaly in two dimensions.
I. Ichinose, K. Nagaor Physics Letters B 460 (1999) 164–169
Then the final form of the WT identity is given by
Em j5,k m s i d k 0
sid
'NL p
'NL k0 p
(
2N
q
p
169
fermion is quite useful for numerical studies of lattice QCD4 and other realistic theories.
Ý emn En lm q 2 Me'N w5k mn
Acknowledgements K.N., one of the authors, would like to thank Prof. T. Yoneya for useful discussions.
Ý emn En lm mn
mp2 =
w 5k
'2p M 2
.
Ž 42 .
Then it is obvious that the PCAC relation is satisfied in the overlap fermion formalism. In this paper, we studied the overlap fermion formalism by using the two-dimensional gauged Gross–Neveu model in the large-N limit, and showed that the pion mass is automatically proportional to the bare quark mass Ži.e., the current quark mass. without any fine tuning and that the PCAC relation is satisfied. This result means that the chiral limit of the overlap fermion formalism is reached by MB ™ 0 w10x. This is in sharp contrast to the Wilson fermion formalism in which fine tuning of the Wilson parameter is required, and it is expected that the overlap
References w1x K.G. Wilson, in: A. Zichichi ŽEd.., New Phenomena in Subnuclear Physics, Plenum New York 1977. w2x R. Narayanan, H. Neuberger, Nucl. Phys. B 412 Ž1994. 574; Nucl. Phys. B 443 Ž1995. 305. w3x P.H. Ginsparg, K.G. Wilson, Phys. Rev. D 25 Ž1982. 2649. w4x T.W. Chiu, hep-latr9810052; S. Chandrasekharan, heplatr9805015; Phys. Rev. D 59 Ž1999. 094502; W. Bietenholz, I. Hip, hep-latr9902019. w5x I. Ichinose, Phys. Lett. B 110 Ž1982. 284; Ann. Phys. 152 Ž1984. 451. w6x H. Neuberger, Phys. Lett. B 417 Ž1998. 141; heplatr9801031; Phys. Rev. D 57 Ž1998. 5417. w7x M. Luscher, Phys. Lett. B 428 Ž1998. 342. ¨ w8x Y. Kikukawa, A. Yamada, Phys. Lett. B 448 Ž1999. 265. w9x Y. Kikukawa, A. Yamada, hep-latr9808026. w10x P. Hasenfratz, Nucl. Phys. B 525 Ž1998. 401.
Physics Letters B 460 Ž1999. 164–169
Pion mass and the PCAC relation in the overlap fermion formalism: Gauged Gross–Neveu model on a lattice Ikuo Ichinose 1, Keiichi Nagao
2
Institute of Physics, UniÕersity of Tokyo, Komaba, Tokyo, 153-8902 Japan Received 12 May 1999 Editor: H. Georgi
Abstract We investigate chiral properties of the overlap lattice fermion by using solvable model in two dimensions, the gauged Gross–Neveu model. In this model, the chiral symmetry is spontaneously broken in the presence of small but finite fermion mass. We calculate the quasi-Nambu–Goldstone ŽNG. boson mass as a function of the bare fermion mass and two parameters in the overlap formula. We find that the quasi-NG boson mass has desired properties as a result of the extended chiral symmetry found by Luscher. We also examine the PCAC relation and find that it is satisfied in the continuum limit. ¨ Comparison between the overlap and Wilson lattice fermions is made. q 1999 Published by Elsevier Science B.V. All rights reserved.
Species doubling is a long standing problem in the lattice fermion formulation. Wilson fermion is the most suitable formulation w1x and it is used in most of the numerical studies of lattice gauge theory. However in order to reach the desired continuum limit, fine tuning must be done with respect to the ‘‘bare fermion mass’’ and the Wilson parameter. Recently a very promising formulation of lattice fermion named overlap fermion was proposed by Narayanan and Neuberger w2x. In that formula the Ginsparg and Wilson ŽGW. relation w3x plays a very
1 2
E-mail: [email protected] E-mail: [email protected]
important role, and because of that there exists an ‘‘extended’’ Žinfinitesimal. chiral symmetry. In this paper we shall study or test the overlap fermion by using the gauged Gross–Neveu model in two dimensions. This is a solvable model which has similar chiral properties with QCD4 , i.e., chiral symmetry is spontaneously broken with a small but finite bare fermion mass and pion appears as quasiNambu–Goldstone boson 3. Actually a closely related model was studied on a lattice in order to test properties of the Wilson fermion in the continuum limit w5x. Therefore advantage of the overlap fermion becomes clear by the investigation in this paper.
3 Overlap fermion with a finite fermion mass was recently studied in Ref. w4x
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 7 4 9 - 2
I. Ichinose, K. Nagaor Physics Letters B 460 (1999) 164–169
The model is defined by the following action on the two-dimensional square lattice with the lattice spacing a, N
Ss
2
Ý Ł Um Ž n . q a2 Ý c Ž m . D Ž m,n . c Ž n . n, m
pl
q a2 MB Ý cc Ž n . y n
a2
'N Ý n
f i Ž n . Ž ct ic .
c Ž n . ™ c Ž n . q c Ž m . t ku kg 5 dn m y 12 aD Ž n,m . 4 f i Ž n . ™ f i Ž n . q d i k ju kf 5j Ž n . ,
a2 q 2 gÕ
Ý
i
i
f Ž n. f Ž n.
q f 5i
Ž n.
f 5i
Ž n. ,
f 5i Ž n . ™ f 5i Ž n . y d i k ju kf j Ž n . ,
n
Ž 2.
Ž 3.
As in the continuum model, we expect that the field f 0 acquires a nonvanishing vacuum expectation value ŽVEV.,
where UmŽ n. is UŽ1. gauge field defined on links, cal Ž a s 1, . . . , N,l s 1, . . . , L. are fermion fields with flavour index l, and the matrix t i Ž i s 0, . . . , L2 y 1. acting on the flavour index is normalized as Tr Ž t it k . s d i k and 1
i j i jk k 'L , t ,t 4 s d t ,
where d i jk ’s are the structure constants of SUŽ L.. Fields f i and f 5i are scalar and pseudo-scalar bosons,respectively. The covariant derivative in Eq. Ž1. is defined by the overlap formula 1 a
ž
1qX
1
'X † X
/
,
1 2a
² f 0 : s 'NL M s , and we define subtracted fields,
w 0 s f 0 y 'NL M s ,
Ss
N 2
dmq m , nUm Ž m . y dm , nq m Um† Ž n . , M0 a
r q 2a
Ý
2 dn , m
Ž i / 0 . , w 5i s f5i . Ž 9.
n, m
pl 2
'N Ý n
w i Ž n . Ž ct ic . Ž n .
qw 5i Ž n . Ž ct ig 5 c . Ž n .
m
ydmq m , nUm Ž m . y dm , nq m Um† Ž n . ,
Ž 4.
where r and M0 are dimensionless nonvanishing free parameters of the overlap lattice fermion formalism w2,6x. The overlap Dirac operator D does not have the ordinary chiral invariance but satisfies the GW relation instead, Dg 5 q g 5 D s aDg 5 D.
w isf i
Ý Ł Um Ž n . q a2 Ý c Ž m . DM Ž m,n . c Ž n . a
y
B Ž n,m . s y
Ž 8.
In terms of the above fields,
X n m s gm Cm Ž n,m . q B Ž n,m . , Cm s
Ž 6.
where u i is an infinitesimal transformation parameter. The above symmetry Ž6. coincides with the ordinary chiral symmetry up to O Ž a.. From the action Ž1., it is obvious that f i and f 5i are composite fields of the fermions, gÕ gÕ i fis ct ic , f 5i s Ž 7. 'N 'N cg 5t c .
Ž 1.
Ds
From Ž1. it is obvious that the systematic 1rN expansion is possible and we shall employ it. The action Ž1. contains the bare fermion mass MB which explicitly breaks the chiral symmetry. This bare mass also breaks the following infinitesimal w7x transformation, which was discovered by Luscher ¨ and we call ‘‘extended chiral symmetry’’,
c Ž n . ™ c Ž n . q t ku kg 5 dn m y 12 aD Ž n,m . 4 c Ž m . ,
= Ž n . q f 5i Ž n . Ž ct ig 5 c . Ž n .
t 0s
165
Ž 5.
a2 q 2 gÕ qw 5i
Ý
w i Ž n . w i Ž n . q 2'NL M s w 0 Ž n .
n
Ž n . w5i Ž n . ,
Ž 10 .
where DM s D y M ,
M s MB q M s .
Obviously, M is the dynamical fermion mass.
Ž 11 .
I. Ichinose, K. Nagaor Physics Letters B 460 (1999) 164–169
166
From the chiral symmetry and Ž8., we can expect that quasi-Nambu–Goldstone ŽNG. bosons appear as a result of the spontaneous breaking of the chiral symmetry. They are nothing but w 5i . The VEV M s is determined by the tadpole cancellation condition of w 0 . In order to perform an explicit calculation of the 1rN-expansion, it is useful to employ the momentum representation, and also we introduce the gauge potential lmŽ n. in the usual way, i.e., UŽ n, m . s w ia Ž .. expŽ 'N x lm n . By using weak-coupling expansion by Kikukawa and Yamada w8x, Dn m s
yi aŽ q nyp m.
e HH p q
D Ž p,q . ,
The vertex function is explicitly given as
ž
V1 m p q
D Ž p,q . s D 0 Ž p . Ž 2p . d Ž p y q . q
a
E s
V Ž p,q . ,
2
q
1
v Ž p. q v Ž q.
y
X0 Ž p .
a 2v Ž p .
5
X 1† Ž p,q .
v Ž p.
,
Ž 14 .
'
i a
Am Ž p . gm q B Ž p .
v Ž q.
1
Ý Ž 1 y cos apm . y a M0 , a
X 1 Ž q, p . s Ž 2p . d Ž q y p y k .
ž
k 2
/
1
'N
,
lm Ž k . V1 m
Ms
Hk
eyS eff s
H
Ž 17 .
Dc Dc eyS .
2
sin2 Ž apm . q r Ý Ž 1 y cos Ž apm . . y M0
ž
m
ab Ž p . s r Ý Ž 1 y cos Ž apm . . y M0 . m
Ž 21 .
/
Ž 22 .
Especially we are interested in w 5 and lm part of the effective action, because w 5 is the quasi-NG bosonŽpion. and its coupling with the gauge boson is related with anomaly. We define
Hk
1 2
w 5i Ž yk . Gi 5j Ž k 2 . w 5j Ž k .
Ž 23 .
where
Gi 5j Ž k 2 . s d i j
)
Ž 20 .
sy Tr Ž Dy1 M Ž 0. Ž k . .
Ž2. w w5 x s Seff
av Ž p. s
.
Effective action of w i , w 5i and the gauge field Ž lm n. is obtained by integrating out the fermions,
Ž 16 .
= pq
Ž 19 .
2
Hk v Ž k . 1qŽ1y Ma.2 4 q2 b Ž k . Ž1y Ma. .
X0 Ž q .
r
4
.
sy
m
Hk
/
/
2 a b Ž k . q Ž 1y Ma . v Ž k . 4
X 1 Ž p,q .
Ž 15 .
gm sin apm q
2
a b Ž p . q Ž 1 y Ma . v Ž p . 4 y igm sin Ž apm .
J Ž p.
gÕ
q ... X0 Ž p . s
km
From Ž10. and Ž13.,
gm isin apm
av Ž p.
½
2
/
v Ž p . 1 q Ž 1 y Ma . 4 q 2 b Ž p . Ž 1 y Ma . 4
d pra where Hp s Hy p r a Ž2p . and
V Ž p,q . s
ž
k
2
Dy1 M Ž0.
2p
D0 Ž p . s
E pm
X0 p q
km
From Ž14., the tree level propagator is obtained as
Ž 13 . bŽ p. q v Ž p.
ž
s igm cos a pm q
ž
s 1
2
/
q r sin a pm q
Ž 12 .
2
k
1 gÕ
q Tr g 5 ² c Ž k y p .
Hk
,
Ž 18 .
=c Ž k y p . :g 5 ² c Ž k . c Ž k . : s d i j e q 2 k 2 M02 A Ž k 2 ; M . .
Ž 24 .
I. Ichinose, K. Nagaor Physics Letters B 460 (1999) 164–169
Parameter e in Gi 5j Ž24. is proportional to the pion mass and measures the derivation from the limit of the exact chiral symmetry. In the leading order of the 1rN, e sy
2
It is also straightforward to calculate AŽ k 2 ; M . in Ž24., 1 AŽ k 2 ; M . s 2 4p k Ž k 2 q m 2 .
(
a2
=H
k
(Ž k q 2 m . y 4m =ln k q 2 m y (Ž k q 2 m . y 4m k 2 q 2 m2 q
1
p
M s w v Ž k r a . 1q Ž 1y Ma . 2 4 q2 b Ž k r a . Ž 1y Ma . x
2
2
=w a b Ž k r a. q Ž1y Ma. v Ž k r a. 4 a. 4 v Ž k r a.
™
2
=Ž1q Ž1y Ma. . q2 b Ž k r a.Ž1y Ma. 4 q M s sin2 km q a 2 Ž b Ž k r a . q Ž 1y Ma . v Ž k r a .. 2 4 x s
MB M02 Ms
167
w yln Ž M0 M 2 a . qconst. x q O Ž a. ,
Ž 25 .
1 4pm2
2
2
2 2
4
2
2 2
4
q Ž k2 . ,
Ž 28 .
where m s M0 M and therefore the pion mass is given as mp2 s 2p M 2e . There exists a mixing term of the gauge boson lm and the pion w 50 ,
2
d k p where Hkp s Hy p Ž2p . and we took the continuum limit to obtain the last line of Ž25.. From Ž25., e A MB q O Ž a. and therefore the limit MB ™ 0 is considered as the chiral limit. It is instructive to compare the above result with that in the continuum theory and the lattice model with the Wilson fermions. The corresponding expression of e in the continuum theory is given as 2
L2
2 MB 1
w e x cont s
M s 4p
ln
ž / M2
q O Ž 1rL . ,
Ž 26 .
where L is the momentum cutoff. It is obvious that e in the overlap formalism has a very close resemblance to that in the continuum theory. On the other hand, the corresponding expression in the Wilson fermion formalism was obtained in Ref. w5x as follows,
w e xW s y
4 rW Ms a
L Ž rW . q
2 MB Ms
I Ž k . s Ý sin2 km q y2 r W Ý sin2
ž
m p
L Ž rW . s
Hk
Ý sin2 Ž kmr2. IŽ k.
1
p
Hk
IŽ k. km 2
Ž2. Seff lm , w 50 s y2'L M02 M
Hk Ý l Ž yk . m
=emn kn A Ž k 2 ; M . w 50 Ž k . ,
Ž 29 .
which is identical with the continuum calculation. This mixing term is related to the discussion of the UŽ1. problem in QCD4 and the above result suggests that the correct anomaly appears in the Ward– Takahashi identity of the axial-vector current. We shall examine the PCAC relation. By changing variables as follows in the path-integral representation of the partition function 4 ,
c Ž n. ™ c Ž n. q t ku k Ž n . g 5 dn m y aD Ž n,m . 4 c Ž m . ,
c Ž n . ™ c Ž n . 1 q t ku k Ž n . g 5 4 , f i Ž n . ™ f i Ž n . q d i k ju kf 5j Ž n . , f 5i Ž n . ™ f 5i Ž n . y d i k ju kf j Ž n . , Ž 30 . we obtain the Ward–Takahashi ŽWT. identity,
, 2
² Em j5,k m Ž n . y 2 M Ž ct kg 5 c . Ž n . q
/
q Ma ,
2'N gÕ
M s w 5k Ž n .
q DAk Ž n . y d k 0 N'L aTr g 5 D Ž n,n . : s 0, ,
Ž 27 .
where r W is the Wilson parameter. Therefore it is obvious that the fine tuning of the ‘‘bare mass’’ MB and the Wilson parameter r W is required in order to reach the chiral limit. In this sense, the overlap fermion is better than the Wilson fermion.
Ž 31 . where the last term comes from the measure of the path integral, and the explicit form of the current 4
We employ this form of change of variables instead of that is given by Ž6.. This is merely for technical reason here.
I. Ichinose, K. Nagaor Physics Letters B 460 (1999) 164–169
168
operator j5,k m is obtained by Kikukawa and Yamada w9x as follows, j5,k m Ž n . s t k Ý c Ž l . K n5m Ž l,m . c Ž m . ,
Ž 32 .
D5A Ž p . 4
b
s y'N aM w 5k Ž p . Tr ² c Ž p q q .
Hq
lm
½
K n5m Ž l,m . s K n m
H
'H 2
5
=c Ž p q q . :g 5 ² c Ž q . c Ž q . :g 5 D 0 Ž p q q .
Ž l,m . ,
Ž 33 . s y'N aMw 5k Ž p .
H s yg 5 X ,
Ž 34 .
= yMsin aqm sin a Ž q y p . m
aK n m Ž l,m . s g5
=
½
2
Hq J Ž q . J Ž q y p .
`
dt
Ž t 2 Wn m y HWn m H . Ž t2qH 2 .
1 2
2
Ž t qH . Wn m Ž l,m . s g
y Ž 2 y Ma . Ž v Ž q . q b Ž q . . a b Ž q y p .
1
Hy` p
5
,
q Ž 1 y Ma . v Ž q y p . 4 .
Ž 35 .
lm
From Ž25. and Ž38., we obtain D5A Ž p . s y2'N M s ew 5k Ž p . .
1 5 2
Ž gm y 1. dn l dnq mˆ , m Un m
† q 12 Ž gm q 1 . d l , nq mˆ dn m Unq mˆ , m 4 ,
Ž 36 .
a
b
DmA s DmA Ž p . 4 q DmA Ž p . 4 ,
DAk Ž n . s aM Ý c Ž n . t kg 5 D Ž n,m . c Ž m .
DmA Ž p . 4
m
a
a
By integrating out the fermions, the above WT identity is expressed in terms of the pions and the gauge field. Matrix element D5A s ² w 5k < DAk <0: has the contribution from the following two terms,
Ž p. s
a
Ž p. 4 q
a
D5A Ž p . 4 s'N aH Tr q
D5A
Hq
Ž p. 4 ,
² c Ž q . c Ž q . : D0 Ž q .
2 Ž 2 y Ma . Ž v Ž q . q b Ž q . . JŽ q.
X
5 D0
DmA Ž p . 4
b
s yNM'L
HqTr ² c Ž q . c Ž q . :
=g 5 d k 0 V Ž p q q,q . .
Ž 40 .
It is not so difficult to show that the above two terms cancel with each other and DmA Ž p . s 0. On the other hand, the last term of Ž31. is evaluated by a similar method for the four-dimensional case by Kikukawa and Yamada w8x, and we obtain
b
=w 5k Ž p . s'N a w 5k Ž p . =
X
Hqq Tr ² c Ž q . c Ž q . :g
=V Ž p q qX ,q . d k 0 ,
Ž 37 .
m
D5A
s NM'L
= Ž p q qX . ² c Ž p q qX . c Ž p q qX . :
i i 'N c Ž n . Ž w Ž n . g 5 y w5 Ž n . .
=t it k Ý D Ž n,m . c Ž m . .
D5A
Ž 39 .
In a similar way, matrix element DmA s ² lm < DA0 <0: is evaluated as,
and the operator DAk Ž n. is given by
q
Ž 38 .
aTr g 5 D Ž n,n . s ,
i
Ý emn En lm . p'N mn
Ž 41 .
Eq. Ž41. is nothing but the chiral anomaly in two dimensions.
I. Ichinose, K. Nagaor Physics Letters B 460 (1999) 164–169
Then the final form of the WT identity is given by
Em j5,k m s i d k 0
sid
'NL p
'NL k0 p
(
2N
q
p
169
fermion is quite useful for numerical studies of lattice QCD4 and other realistic theories.
Ý emn En lm q 2 Me'N w5k mn
Acknowledgements K.N., one of the authors, would like to thank Prof. T. Yoneya for useful discussions.
Ý emn En lm mn
mp2 =
w 5k
'2p M 2
.
Ž 42 .
Then it is obvious that the PCAC relation is satisfied in the overlap fermion formalism. In this paper, we studied the overlap fermion formalism by using the two-dimensional gauged Gross–Neveu model in the large-N limit, and showed that the pion mass is automatically proportional to the bare quark mass Ži.e., the current quark mass. without any fine tuning and that the PCAC relation is satisfied. This result means that the chiral limit of the overlap fermion formalism is reached by MB ™ 0 w10x. This is in sharp contrast to the Wilson fermion formalism in which fine tuning of the Wilson parameter is required, and it is expected that the overlap
References w1x K.G. Wilson, in: A. Zichichi ŽEd.., New Phenomena in Subnuclear Physics, Plenum New York 1977. w2x R. Narayanan, H. Neuberger, Nucl. Phys. B 412 Ž1994. 574; Nucl. Phys. B 443 Ž1995. 305. w3x P.H. Ginsparg, K.G. Wilson, Phys. Rev. D 25 Ž1982. 2649. w4x T.W. Chiu, hep-latr9810052; S. Chandrasekharan, heplatr9805015; Phys. Rev. D 59 Ž1999. 094502; W. Bietenholz, I. Hip, hep-latr9902019. w5x I. Ichinose, Phys. Lett. B 110 Ž1982. 284; Ann. Phys. 152 Ž1984. 451. w6x H. Neuberger, Phys. Lett. B 417 Ž1998. 141; heplatr9801031; Phys. Rev. D 57 Ž1998. 5417. w7x M. Luscher, Phys. Lett. B 428 Ž1998. 342. ¨ w8x Y. Kikukawa, A. Yamada, Phys. Lett. B 448 Ž1999. 265. w9x Y. Kikukawa, A. Yamada, hep-latr9808026. w10x P. Hasenfratz, Nucl. Phys. B 525 Ž1998. 401.