We show that the anomaly of nonlocal symmetry can be canceled by the
well-known Wess-Zumino acton which cancels the gauge anomaly in the
two-dimensional chiral electrodynamics.Comment: 4 pages, latex, no figure

Kim, Won T.

Kim, Yongduk

Park, Young-Jai

Shin, Hyeonjoon

English

arXiv

We show that the anomaly of nonlocal symmetry can be canceled by the well-known Wess-Zumino acton which cancels the gauge anomaly in the two-dimensional chiral electrodynamics

Shin, H

Park, Y J

Kim, Y

Kim, W T

CERN Document Server

Po
st
Sc
rip
t〉  p
ro
ce
ss
ed
 b
y 
th
e 
SL
A
C/
D
ES
Y
 L
ib
ra
rie
s o
n 
26
 Ju
n 
19
95
.
H
EP
-T
H
-9
50
61
66
SOGANG-HEP-200/95
hep-th/9506166
May 1995
On the anomaly of nonlocal symmetry in the chiral QED
Hyeonjoon Shin, Young-Jai Park, and Yongduk Kim
Department of Physics

and Basic Science Research Institute, Sogang University,
C.P.O. Box 1142, Seoul 100-611, Korea
Won T. Kim
Department of Science Education and Basic Science Research Institute,
Ehwa Women's University, Seoul 120-750, Korea
ABSTRACT
We show that the anomaly of nonlocal symmetry can be canceled by the well-known
Wess-Zumino acton which cancels the gauge anomay in the two-dimensional chiral elec-
trodynamics.
||||||||||||||-

e-mail address: hshin, yjpark, wtkim@physics.sogang.ac.kr
Quantum gauge symmetries are very important to build modern quatum eld the-
ories. Recently, Lavelle and McMullan [1] have shown that there exists a distinct
nonlocal symmetry which is dierent from the usual gauge symmetry in the quantum
electrodynamics (QED). The original gauge symmetry is implemented by the Becci-
Rouet-Stora-Tyutin (BRST) symmetry [2] in the gauge xed action. The nonlocal
symmetry, which the gauge xing term is invariant, is dual to the BRST symme-
try. The generalized Lorentz covariant symmetry has discussed in Ref.[3]. On the
other hand, the classical gauge symmetry or BRST symmetry may be spoiled by the
anomaly after quantization in the chiral QED, which has a chiral gauge coupling [4].
Therefore, it is natural to study whether the nonlocal symmetry may be anomalous in
chiral QED or not.
In this Brief Report, we show that the anomaly of the nonlocal symmetry exists,
and this can be canceled by the well-known Wess-Zumino action [5] in the exactly
solvable two-dimensional chiral QED.
The chiral QED is dened by the gauge invariant Lagrangian as follows
L
0
=  
1
4
F

F

+ i

 

(@

  ieA

1  
5
2
) ; (1)
and we choose the Lorentz invariant gauge xing term and the corresponding ghost
action,
L
GF
+ L
Ghost
=  
1
2
(@

A

)
2
  i@

c@

c: (2)
Then, the total classical action L
C
= L
0
+ L
GF
+ L
Ghost
is BRST invariant under the
following transformations as

B
 = iec ; 
B

 = ie

 c

B
A

=  
1
e
@

c; 
B
c = 0; 
B
c =
i
e
@

A

: (3)
It is also invariant under the nonlocal transformation [1] as follows

?
 = e
 
@
0
r
2
c
!
 ; 
?

 = e

 
@
0
r
2
c;
2
?
A
0
= ic; 
?
A
1
= i
@
1
@
0
r
2
c; (4)

?
c = 0; 
?
c =  A
0
+
@
1
@
0
r
2
A
1
+
e
r
2
j
0
;
where j
0
= e

 
0
1 
5
2
 .
Let us now consider the quantum eective action to investigate the anomaly of the
nonlocal symmetry at the quantum level. The eective action, which incorporates one
fermionic loop [4], is exactly calculated as
L
eff
=
1
2
ae
2
A

A

 
e
2
2
A

(g

+ 

)@

@

(g

  

)
A

; (5)
where the constant a is a regularization ambiguity. Then the BRST anomaly of the
total quantum eective action, L
Q
= L
eff
+ L
GF
+ L
Ghost
is given by

B
L
Q
= e [

@

A

+ (a  1)@

A

] c: (6)
On the other hand, the nonlocal transformations are dened by exactly the same
forms in Eq. (4) except for the ghost eld c by the following rule

?
c =  A
0
+
@
1
@
0
r
2
A
1
+
e
r
2
J
0
; (7)
where J
0
= aeA
0
  e(@
0
+ @
1
)
1
(g

  

)@

A

. Then, the anomaly of the nonlocal
transformation is given by

?
L
Q
= ie
2
c
@
0
r
2
[

@

A

+ (a  1)@

A

] : (8)
It is interesting to note that the nonlocal symmetry can be broken after quantization
similar to the BRST case in the two-dimensional chiral QED. To recover this symmetry,
it is sucient to add the following Wess-Zumino action L
WZ
to the eective action L
Q
[5],
L
WZ
=
1
2
(a  1)@

@

 + e [(a  1)g

+ 

] @

A

: (9)
Then, after the integration of the  eld, the resulting Lagrangian is given by
L =  
1
4
F

 1
(+m
2
)F

 
1
2
(@

A

)
2
  i@

c@

c ; (10)
3
where m
2
=
ae
2
a 1
for a > 1. This Lagrangian is manifestly invariant under the nonlocal
transformation as well as the BRST transformation.
In conclusion, we have shown that the nonlocal symmetry may be anomalous after
quantization in the chiral QED. However, it is sucient to add the well-known Wess-
Zumino action to recover the nonlocal symmetry. It would be interesting to study the
nonlocal anomaly related to the chiral anomaly in higher dimensions and its geometrical
meaning.
ACKNOWLEDGMENTS
The present study was supported by Ministry of Education, Project No. BSRI-95-2414
(1995).
References
[1] M. Lavelle and David McMullan, Phys. Rev. Lett. 71, 3758 (1993).
[2] T. Kugo and I. Ojima, Prog. Theor. Phys. Suppl. 66, 1(1979).
[3] Z. Tang and D. Finkelstein, Phys. Rev. Lett. 73, 3055 (1994).
[4] R. Jackiw and Rajaraman, Phys. Rev. Lett. 54, 1219 (1985).
[5] K. Harada and I. Tsutsui, Phys. Lett. B 183, 311 (1987).
4


On the anomaly of nonlocal symmetry in the chiral QED

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.266.4005

On the anomaly of nonlocal symmetry in the chiral QED

Abstract

Similar works

Full text

Available Versions

CERN Document Server