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On Schwinger terms in nonabelian chiral gauge theories
Volume 188, n u m b e r 1
PHYSICS LETTERS B
2 April 1987
O N S C H W I N G E R T E R M S IN N O N A B E L I A N C H I R A L G A U G E T H E O R I E S P. M I T R A Department of Physics, North Bengal University, Darjeeling, India Received 5 January 1987 Chiral fermionic currents, coupled with nonabelian background gauge fields, are known to have Schwinger terms in their commutators. It is shown that if the gauge group is semisimple, the anomaly is completely determined by these Schwinger terms. Violation of Jacobi identities can also be demonstrated using them.
Faddeev's suggestion [1] that there is an anomalous term in the commutator between generators of time-independent gauge transformations in chiral gauge theories triggered off several attempts to calculate this commutator explicitly. While Fujikawa went for this commutator directly [2], others started off with current-current commutators. We mention Jo [3,4], Faddeev and Shatashvili [5] and Niemi and Semenoff [6] among others. It is the current-current commutators that we are concerned with in this letter. In ref. [3], Jo perturbatively calculated the Schwinger terms in these commutators, keeping the gauge fields as classical background fields. He has given expressions for the Schwinger terms for both the t i m e - t i m e and time-space components. The most striking feature of these long expressions is that they vanish if the anomaly-vanishing condition is satisfied, thus suggesting an intimate connection between the Schwinger terms and the (nonabelian) anomaly. Indeed, Zumino [7] has given formal relations between 2-cocycles in 2n - 1 dimensions and 1-cocycles in 2n dimensions. But the connection between the cocycles and field theoretic quantities like commutators has not been demonstrated explicitly. An explicit use of the expression for the divergence anomaly was made by Fujikawa [2] in his derivation of the commutator between generators of time-independent gauge transformations, but this was done in the framework of quantized gauge fields. While there have been suggestions [8,5,6] that anomalous gauge theories can be made sense of, we think there are subtleties involved - subtleties which are ignored in the naive approach of ref. [2]. It was in this context that we set out to find a connection between the Schwinger terms and the divergence anomaly, both defined with background gauge fields. The main result in this letter is the derivation of the divergence anomaly from the Schwinger terms given in ref. [3]. We emphasize that the derivation is purely algebraic and involves no regularization at any point. This, of course, is as it should be: while the calculation of any anomalous object must involve some kind of regularization at some stage, two related anomalies should allow a connection to be made by purely formal manipulations. An example which comes to mind is the connection between the consistent and covariant forms of the non-abelian anomaly [9,10]. Jo was unable to convert his current-current commutators into a nontrivial 2-cocycle in the algebra of generators of time-independent gauge transformations without quantizing the gauge fields. When this questionable step was taken [4], a 2-cocycle emerged all right, but there was something more curious certain Jacobi identities were found to be violated. The second result of this letter is to show that a failure of some Jacobi identities can be inferred from the structure of the Schwinger terms calculated with background gauge fields. In fact, if all Jacobi identities could be assumed to be valid, it would be possible 111
Volume 188,number 1
PHYSICS LETTERSB
2 April 1987
tO calculate the commutators of the gauge-transformation generators from the Schwinger terms by purely algebraic methods. One meets an inconsistency in trying to carry out this programme, thus demonstrating the violation of a Jacobi identity. We shall mostly follow Jo's notation [3]. Thus we use antihermitian generators T ~, satisfying (1)
I T ~, T b ] =fab~TC
and currents j a = __i~Ta½(1 __ ~/5)y~Ab"
(2)
The usual divergence anomaly (which we are going to calculate) is (3)
O~J~ + fabcaabJf =- n a
and the anomalous commutators, as derived in ref. [3], are [ J~(Y), Jor(ff')] ET = --ifabcj~(Y~)8(~ -- ~ ' ) + S~ob(~, ~ ' ) ,
(4)
- - i f a b c J f ( ~ ) 8 ( f f - - ~ ') + s~b(2, if'),
(5)
[ J~(Y),
J/b(xt)]ET
=
with £ijk
S~0a(2, x ' )
16~r 2£ijk tr(( Ta' T b }Fig(~)) ~kS( y ~ . -- 2 ' ) -- --;-70iA~Aak~(224~r - if')
× t r ( T a T b T C T a - TaTdTCT b + 2 T ~ T ~ T d T b - 2 T ~ T b T d T ¢ + 3 T ~ T d T b T ~ - 3 T a T ~ T b T a) ciJ k
_ 16qrzAiAjAk~¢ d e ( ~ _ _ ~ , ) t r ( T b T a T C T d T e - T ~ T b T ~ T d T e + Z T ~ T ~ T b T a T e -
2 T b T ~ T a T d T e)
(6)
and £ijk
ciJ k
t r ( ( T a, T b } F o j ( X ) ) O k ~ ( x - - ~ ') + 1 - ~ 2 tr((Za, Tb}0o _ 16-ciJ k
48or 2 ( _ OjAoAk c d + 3jAkAo c d _ 2OoAjA c dk _ 5A~o 3jA ka _
2A~ OkA ao -
AJOoA ¢ a~ )
× tr(T~TCTaT b - TaTbTdTC)8(y~ -- ~ ' ) £ijk
16~r z
+
c
d
c
d
a
d
b
(OoAj&-OjAo& )tr(T T T T
f i j k- A o cA ~ A d , .et r ( T b T a T c T d T e
16~2
g ,~ .
c
- T~TbTdTeT ~
TbTaTaT~T ~
+ T a T d T b T C T e + TbTdTaTeTC -- T a T d T b T e T C ) 8 ( ~ -- ~ ' ) .
(7)
Here, in matrix notation, F.~ = 0~A~- 0.A. + [ A., A~].
(8)
We shall also need S~°(~, if') defined analogously to (5) and given by STob(y, ~ ' ) =
112
--Soba(y ', y ) .
(9)
Volume 188, number 1
PHYSICS LETTERSB
2 April 1987
To derive our formal relation between H a and the Schwinger terms S ab, we differentiate (4) with respect to time:
[~°J0a(.~), J0b(.~')] +
[J~(ff), 0°J0b(~')] = - i f abC O ° J ~ 6 ( f f - Y ' ) +
O°S~b(Y, ~').
(10)
Making use of (3), we obtain
[Ha(Z)
__ Otjia(~) - J . . . . . .Alxjlxc; . "-x.~X),
Job(.x")]-[" [ J(~(~'), H b ( x t ) - - O t J i b ( x ' )
= --ifabc(H c - O'Jic - f C e d A ; J " a ) 8 ( ~ -
-'bJ . . A,,,I . . . P". .tX ..
)]
~') + ~°Sg~(.~, ~').
(11)
Since the anomaly H a with background gauge fields is a c-number, its commutator with currents vanishes. Hence we only have current-current commutators in (11). Simplifying these, we find
i fabc~i( JCS( y - ~ ' ) ) - O;S~ob(y, ~ ' ) - faecAeo( ~)(--i fcbaJoaS( ~ -- y~') + s~b( ~, ~') ) -faecAie( y ) ( - - i fcbaJiar( ff -- ~') + Slob(X, X') ) + i fabco'i( JiCS( Y -- ~') ) -- Otts~ib( x, .~') --fbe~AOe(y')(-ifaCaJoaS(ff - if') + S~(Y~, if')) - fbe~A;e(Y')(-ifa~aJ~aS(Y - ~') + Sff[(Y, ~')) = - i f a b c ( H c - a'J[ -fceaAe~J~d)8(~ - ~') + O°S~ob(Y , ~').
(12)
It is not difficult to see that the canonical terms, viz. those not involving H or S, cancel out, leaving
oiS~ob(~, ~') + faecAie(.~)SCob(~., ~') + O"S~ib(.~, ~') + fb%4;e(~')SgC(~., ~.') -}- oOSgob(~, ~') + faecA°e(.~. )s~b( x, x ' ) -'FfbecAOe( x')S~oC(.~, .~') = i fae~H~8(.2 - .~').
(13)
This is our relation connecting the Schwinger terms with the anomaly. The left hand side can be calculated using (6), (7) and (9). After long but straightforward work, one finds
214~r2fab~e""°°tr(T ~ O~( A~OoA o + ½ A , A o A o ) ) 8 ( ~ - ~')=ifab~HcS(ff -- ~').
(14)
For a semi-simple gauge group, the factors fa~¢ can be cancelled, yielding Ha_
__~ooi 24~-2
t r ( T ~ O,(A~OoAo+½A~A~Ao) ),
(15)
the well-known expression. This may be regarded as a perturbative derivation of the anomaly, since the Schwinger terms used here were calculated using perturbation theory. Of course, if getting the anomaly is the only aim, Schwinger terms provide an unnecessarily circuitous route. We wanted to connect the two things. The commutator of the generators G~(.2) of time-independent gauge transformations is considered next.
Ga( ~) = ta(.~) + J~) ( ~),
(16)
where Lo(Y) generates gauge transformations on functionals of the background gauge fields:
La(~) = i(O;3/rA~(~) +fabcA~(x)a/rA;(~)).
(17)
These operators have of course the standard commutator [La(-2), Lb(-~')] = --if"6~U(.2)8(.2 -- Y,')
(18) 113
Volume 188, number l
PHYSICS LETTERS B
2 April 1987
and any Schwinger terms in G-G c o m m u t a t o r s must arise from S~ob and from a n o m a l o u s terms in L-J o c o m m u t a t o r s . T o determine the latter, we consider the c o m m u t a t o r o f Ld(.,~") with b o t h sides of (4):
[Ld(~"),[Jg(~),jb(y~')]]=--ifabC[Ld(y~"),J~(2)]8(~--~')+[La(~"),s~b(Y~,.2')].
(19)
If we are allowed to m a k e use of the Jacobi identity on the left h a n d side, we get -[J~(~),
[g(~'),
Ld(~")]]-
[g(~'),
[Ld(~"),
J~(~)]]
= --ifabc[Ld(y~"), J ~ ( f f ) ] 8 ( f f - ~ ' ) + [ L d ( 2 " ) , S~0b(~, N ' ) ] .
(20)
T h e L-J c o m m u t a t o r s on the left h a n d side are unknown. Naively, one would expect these c o m m u t a t o r s to vanish, but a n o m a l o u s effects show up in the f o r m of local polynomials in the b a c k g r o u n d gauge fields A. Even though we do not k n o w these terms, we can, however, safely say that they will c o m m u t e with the current operators. Thus, in spite of a n o m a l o u s effects, the left h a n d side must vanish. We therefore conclude that
ifabc[ Ld(x"), J~(2)]8(Y~- 2') = [ Ld(x"), s~b(x, -~')].
(21)
This, however, is an impossibility. If we look at the structure of S~ob in (6), we see that the right h a n d side contains not only 8 ( ~ - Y') but its derivatives as well, whereas the left hand side m a k e s no allowance for such derivatives. We interpret this inconsistency as an indication that the Jacobi identity fails. T h e a m o u n t b y which the Jacobi identity is violated cannot unfortunately be calculated from (19) because the a n o m a l o u s term in the L - J c o m m u t a t o r is unknown. T o determine these two u n k n o w n s a second input is necessary. It m a y be provided by imposing the Malcev identity [11]. We p o s t p o n e this and other issues to a m o r e detailed p a p e r [12] and conclude here by recalling that the L-J c o m m u t a t o r was derived (and the Malcev identity demonstrated) by Jo using F e y n m a n diagrams after naively taking the gauge fields to be quantized [41.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
114
L.D. Faddeev, Phys. Lett. B 145 (1984) 81. K. Fujikawa, Phys. Lett. B 171 (1986) 424. S.-G. Jo, Nucl. Phys. B 259 (1985) 616. S.-G. Jo, Phys. Lett. B 163 (1985) 353. L.D. Faddeev and S.L. Shatashvili, Phys. Lett. B 167 (1986) 225. A.J. Niemi and W. Semenoff, Phys. Rev. Lett. 56 (1986) 1019. B. Zumino, Nucl. Phys. B 253 (1985) 477. R. Jackiw and R. Rajaraman, Phys. Rev. Lett. 54 (1985) 1219. W. Bardeen and B. Zumino, Nucl. Phys. B 244 (1984) 421. H. Baneljee, R. Banerjee and P. Mitra, Z. Phys. C 32 (1986) 445. A. Malcev, Mat. Sb. 78 (1955) 569. P. Mitra, to be published.
PHYSICS LETTERS B
2 April 1987
O N S C H W I N G E R T E R M S IN N O N A B E L I A N C H I R A L G A U G E T H E O R I E S P. M I T R A Department of Physics, North Bengal University, Darjeeling, India Received 5 January 1987 Chiral fermionic currents, coupled with nonabelian background gauge fields, are known to have Schwinger terms in their commutators. It is shown that if the gauge group is semisimple, the anomaly is completely determined by these Schwinger terms. Violation of Jacobi identities can also be demonstrated using them.
Faddeev's suggestion [1] that there is an anomalous term in the commutator between generators of time-independent gauge transformations in chiral gauge theories triggered off several attempts to calculate this commutator explicitly. While Fujikawa went for this commutator directly [2], others started off with current-current commutators. We mention Jo [3,4], Faddeev and Shatashvili [5] and Niemi and Semenoff [6] among others. It is the current-current commutators that we are concerned with in this letter. In ref. [3], Jo perturbatively calculated the Schwinger terms in these commutators, keeping the gauge fields as classical background fields. He has given expressions for the Schwinger terms for both the t i m e - t i m e and time-space components. The most striking feature of these long expressions is that they vanish if the anomaly-vanishing condition is satisfied, thus suggesting an intimate connection between the Schwinger terms and the (nonabelian) anomaly. Indeed, Zumino [7] has given formal relations between 2-cocycles in 2n - 1 dimensions and 1-cocycles in 2n dimensions. But the connection between the cocycles and field theoretic quantities like commutators has not been demonstrated explicitly. An explicit use of the expression for the divergence anomaly was made by Fujikawa [2] in his derivation of the commutator between generators of time-independent gauge transformations, but this was done in the framework of quantized gauge fields. While there have been suggestions [8,5,6] that anomalous gauge theories can be made sense of, we think there are subtleties involved - subtleties which are ignored in the naive approach of ref. [2]. It was in this context that we set out to find a connection between the Schwinger terms and the divergence anomaly, both defined with background gauge fields. The main result in this letter is the derivation of the divergence anomaly from the Schwinger terms given in ref. [3]. We emphasize that the derivation is purely algebraic and involves no regularization at any point. This, of course, is as it should be: while the calculation of any anomalous object must involve some kind of regularization at some stage, two related anomalies should allow a connection to be made by purely formal manipulations. An example which comes to mind is the connection between the consistent and covariant forms of the non-abelian anomaly [9,10]. Jo was unable to convert his current-current commutators into a nontrivial 2-cocycle in the algebra of generators of time-independent gauge transformations without quantizing the gauge fields. When this questionable step was taken [4], a 2-cocycle emerged all right, but there was something more curious certain Jacobi identities were found to be violated. The second result of this letter is to show that a failure of some Jacobi identities can be inferred from the structure of the Schwinger terms calculated with background gauge fields. In fact, if all Jacobi identities could be assumed to be valid, it would be possible 111
Volume 188,number 1
PHYSICS LETTERSB
2 April 1987
tO calculate the commutators of the gauge-transformation generators from the Schwinger terms by purely algebraic methods. One meets an inconsistency in trying to carry out this programme, thus demonstrating the violation of a Jacobi identity. We shall mostly follow Jo's notation [3]. Thus we use antihermitian generators T ~, satisfying (1)
I T ~, T b ] =fab~TC
and currents j a = __i~Ta½(1 __ ~/5)y~Ab"
(2)
The usual divergence anomaly (which we are going to calculate) is (3)
O~J~ + fabcaabJf =- n a
and the anomalous commutators, as derived in ref. [3], are [ J~(Y), Jor(ff')] ET = --ifabcj~(Y~)8(~ -- ~ ' ) + S~ob(~, ~ ' ) ,
(4)
- - i f a b c J f ( ~ ) 8 ( f f - - ~ ') + s~b(2, if'),
(5)
[ J~(Y),
J/b(xt)]ET
=
with £ijk
S~0a(2, x ' )
16~r 2£ijk tr(( Ta' T b }Fig(~)) ~kS( y ~ . -- 2 ' ) -- --;-70iA~Aak~(224~r - if')
× t r ( T a T b T C T a - TaTdTCT b + 2 T ~ T ~ T d T b - 2 T ~ T b T d T ¢ + 3 T ~ T d T b T ~ - 3 T a T ~ T b T a) ciJ k
_ 16qrzAiAjAk~¢ d e ( ~ _ _ ~ , ) t r ( T b T a T C T d T e - T ~ T b T ~ T d T e + Z T ~ T ~ T b T a T e -
2 T b T ~ T a T d T e)
(6)
and £ijk
ciJ k
t r ( ( T a, T b } F o j ( X ) ) O k ~ ( x - - ~ ') + 1 - ~ 2 tr((Za, Tb}0o _ 16-ciJ k
48or 2 ( _ OjAoAk c d + 3jAkAo c d _ 2OoAjA c dk _ 5A~o 3jA ka _
2A~ OkA ao -
AJOoA ¢ a~ )
× tr(T~TCTaT b - TaTbTdTC)8(y~ -- ~ ' ) £ijk
16~r z
+
c
d
c
d
a
d
b
(OoAj&-OjAo& )tr(T T T T
f i j k- A o cA ~ A d , .et r ( T b T a T c T d T e
16~2
g ,~ .
c
- T~TbTdTeT ~
TbTaTaT~T ~
+ T a T d T b T C T e + TbTdTaTeTC -- T a T d T b T e T C ) 8 ( ~ -- ~ ' ) .
(7)
Here, in matrix notation, F.~ = 0~A~- 0.A. + [ A., A~].
(8)
We shall also need S~°(~, if') defined analogously to (5) and given by STob(y, ~ ' ) =
112
--Soba(y ', y ) .
(9)
Volume 188, number 1
PHYSICS LETTERSB
2 April 1987
To derive our formal relation between H a and the Schwinger terms S ab, we differentiate (4) with respect to time:
[~°J0a(.~), J0b(.~')] +
[J~(ff), 0°J0b(~')] = - i f abC O ° J ~ 6 ( f f - Y ' ) +
O°S~b(Y, ~').
(10)
Making use of (3), we obtain
[Ha(Z)
__ Otjia(~) - J . . . . . .Alxjlxc; . "-x.~X),
Job(.x")]-[" [ J(~(~'), H b ( x t ) - - O t J i b ( x ' )
= --ifabc(H c - O'Jic - f C e d A ; J " a ) 8 ( ~ -
-'bJ . . A,,,I . . . P". .tX ..
)]
~') + ~°Sg~(.~, ~').
(11)
Since the anomaly H a with background gauge fields is a c-number, its commutator with currents vanishes. Hence we only have current-current commutators in (11). Simplifying these, we find
i fabc~i( JCS( y - ~ ' ) ) - O;S~ob(y, ~ ' ) - faecAeo( ~)(--i fcbaJoaS( ~ -- y~') + s~b( ~, ~') ) -faecAie( y ) ( - - i fcbaJiar( ff -- ~') + Slob(X, X') ) + i fabco'i( JiCS( Y -- ~') ) -- Otts~ib( x, .~') --fbe~AOe(y')(-ifaCaJoaS(ff - if') + S~(Y~, if')) - fbe~A;e(Y')(-ifa~aJ~aS(Y - ~') + Sff[(Y, ~')) = - i f a b c ( H c - a'J[ -fceaAe~J~d)8(~ - ~') + O°S~ob(Y , ~').
(12)
It is not difficult to see that the canonical terms, viz. those not involving H or S, cancel out, leaving
oiS~ob(~, ~') + faecAie(.~)SCob(~., ~') + O"S~ib(.~, ~') + fb%4;e(~')SgC(~., ~.') -}- oOSgob(~, ~') + faecA°e(.~. )s~b( x, x ' ) -'FfbecAOe( x')S~oC(.~, .~') = i fae~H~8(.2 - .~').
(13)
This is our relation connecting the Schwinger terms with the anomaly. The left hand side can be calculated using (6), (7) and (9). After long but straightforward work, one finds
214~r2fab~e""°°tr(T ~ O~( A~OoA o + ½ A , A o A o ) ) 8 ( ~ - ~')=ifab~HcS(ff -- ~').
(14)
For a semi-simple gauge group, the factors fa~¢ can be cancelled, yielding Ha_
__~ooi 24~-2
t r ( T ~ O,(A~OoAo+½A~A~Ao) ),
(15)
the well-known expression. This may be regarded as a perturbative derivation of the anomaly, since the Schwinger terms used here were calculated using perturbation theory. Of course, if getting the anomaly is the only aim, Schwinger terms provide an unnecessarily circuitous route. We wanted to connect the two things. The commutator of the generators G~(.2) of time-independent gauge transformations is considered next.
Ga( ~) = ta(.~) + J~) ( ~),
(16)
where Lo(Y) generates gauge transformations on functionals of the background gauge fields:
La(~) = i(O;3/rA~(~) +fabcA~(x)a/rA;(~)).
(17)
These operators have of course the standard commutator [La(-2), Lb(-~')] = --if"6~U(.2)8(.2 -- Y,')
(18) 113
Volume 188, number l
PHYSICS LETTERS B
2 April 1987
and any Schwinger terms in G-G c o m m u t a t o r s must arise from S~ob and from a n o m a l o u s terms in L-J o c o m m u t a t o r s . T o determine the latter, we consider the c o m m u t a t o r o f Ld(.,~") with b o t h sides of (4):
[Ld(~"),[Jg(~),jb(y~')]]=--ifabC[Ld(y~"),J~(2)]8(~--~')+[La(~"),s~b(Y~,.2')].
(19)
If we are allowed to m a k e use of the Jacobi identity on the left h a n d side, we get -[J~(~),
[g(~'),
Ld(~")]]-
[g(~'),
[Ld(~"),
J~(~)]]
= --ifabc[Ld(y~"), J ~ ( f f ) ] 8 ( f f - ~ ' ) + [ L d ( 2 " ) , S~0b(~, N ' ) ] .
(20)
T h e L-J c o m m u t a t o r s on the left h a n d side are unknown. Naively, one would expect these c o m m u t a t o r s to vanish, but a n o m a l o u s effects show up in the f o r m of local polynomials in the b a c k g r o u n d gauge fields A. Even though we do not k n o w these terms, we can, however, safely say that they will c o m m u t e with the current operators. Thus, in spite of a n o m a l o u s effects, the left h a n d side must vanish. We therefore conclude that
ifabc[ Ld(x"), J~(2)]8(Y~- 2') = [ Ld(x"), s~b(x, -~')].
(21)
This, however, is an impossibility. If we look at the structure of S~ob in (6), we see that the right h a n d side contains not only 8 ( ~ - Y') but its derivatives as well, whereas the left hand side m a k e s no allowance for such derivatives. We interpret this inconsistency as an indication that the Jacobi identity fails. T h e a m o u n t b y which the Jacobi identity is violated cannot unfortunately be calculated from (19) because the a n o m a l o u s term in the L - J c o m m u t a t o r is unknown. T o determine these two u n k n o w n s a second input is necessary. It m a y be provided by imposing the Malcev identity [11]. We p o s t p o n e this and other issues to a m o r e detailed p a p e r [12] and conclude here by recalling that the L-J c o m m u t a t o r was derived (and the Malcev identity demonstrated) by Jo using F e y n m a n diagrams after naively taking the gauge fields to be quantized [41.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
114
L.D. Faddeev, Phys. Lett. B 145 (1984) 81. K. Fujikawa, Phys. Lett. B 171 (1986) 424. S.-G. Jo, Nucl. Phys. B 259 (1985) 616. S.-G. Jo, Phys. Lett. B 163 (1985) 353. L.D. Faddeev and S.L. Shatashvili, Phys. Lett. B 167 (1986) 225. A.J. Niemi and W. Semenoff, Phys. Rev. Lett. 56 (1986) 1019. B. Zumino, Nucl. Phys. B 253 (1985) 477. R. Jackiw and R. Rajaraman, Phys. Rev. Lett. 54 (1985) 1219. W. Bardeen and B. Zumino, Nucl. Phys. B 244 (1984) 421. H. Baneljee, R. Banerjee and P. Mitra, Z. Phys. C 32 (1986) 445. A. Malcev, Mat. Sb. 78 (1955) 569. P. Mitra, to be published.