PHYSICAL REVIEW D VOLUME 41, NUMBER 12 15 JUNE 1990

Systematic derivation of current-algebra structures in anomalous o models

Subir Ghosh and Gautam Bhattacharya Saha Institute of Nuclear Physics, 92, Acharya Prafulla Chandra Road, Calcutta 700 009, India

(Received 18 August 1989)

The structures of all possible current algebras consistent with the divergence anomalies of currents in a models in 1 + 1 and 3 + 1 dimensions are discussed. The current-algebra models with Wess-Zumino anomalies are shown to be consistent even without a Schwinger term in 1 + 1 dimen- sions but in 3 + 1 dimensions it will stay in the minimal framework.

It is generally believed that the theories that are inflicted with anomalies have underlying anomalous com- mutation relations among the generators of symmetry. Well-known examples are the current algebra (CA) of chiral fermions interacting with gauge fields in ~ b e l i a n ' or n o n - ~ b e l i a n ~ theories. It will be worthwhile to study the status of such an anomalous algebra in the context of flavor currents as that would have direct implications on hadronic physics phenomenology. Recently a method has been suggested that relates the divergence anomaly of the currents directly with anomalous C A ~ in the chiral gauge theory. The well-known Adler-Bardeen anomaly could thus be shown to lead to the two-cocycle in the CA of the generators of gauge symmetry.2p4 We would like to see if this technique can be used in the C A model hav- ing a Wess-Zumino (WZ) anomaly in the divergence of the flavor hadronic current^.^ The effective Lagrangian of this model started with the principal chiral model asso- ciated with the SU,(3)XSUR(3)-flavor CA. The WZ anomaly was introduced6 to eliminate the additional sym- metries that the chiral model has but QCD does not.6 The canonical Hamiltonian formulation of the problem was pioneered by ~ a j e e v ' who modified the usual equal- time CA with some additional central extension and thus accounted for the WZ anomaly in the Hamiltonian framework of ~ u ~ a w a r a . ~ However, the modification seems to be somewhat ad hoc. Can there be a more sys- tematic way of arriving at such a modification? More im- portant is the question of uniqueness of such modification of the CA. There may very well occur other inequivalent CA's, consistent with divergence anomaly. Its underly- ing Hamiltonian may, however, be different from that of Sugawara.

In this paper our aim is to investigate all possible modljied C A ' s (both in 1+ 1 and 3+ 1 dimensions) ir- respective of a specific choice of Hamiltonian but con- sistent with the equations of motion of nonlinear (T mod- els, i.e., zero-curvature condition (implying that the fields

take values on the flavor group) and current divergence anomaly. We also suggest a new CA in 1 + 1 dimensions where the Schwinger term is absent. This follows from ei- ther demanding the CA to satisfy the Jacobi identity or the standard form of light-cone algebra. In 3 + 1 dimen- sions, however, we do not find any new form of CA other than the one suggested in Ref. 7.

We start with a set of non-Abelian currents J ; ( x ) (of mass dimension one) obeying the equations of motion

where

G a ( x ) = k f a b C J ~ ~ ( x ) in 1 + 1 dimensions, ( 2 )

b c d e G " ( x ) = ~ ( i/f,)2~abcde~pvpuJpJJpJ.(x) in 1 + 3 dimensions , (3)

and

Here k is a numerical constant that gets quantized due to geometrical reason^,^..^ f abc (the structure constant) and Wabcde are, respectively, three- and five-forms (completely antisymmetrized traces of three and five generators) of the Lie algebra and f , is the pion decay constant. The CA is formally written as

S $ ( X , ~ ) (note that it should not be treated as a Lorentz tensor of rank 2) contains, apart from the naive terms, lo- cal polynomials in the currents of mass dimension two. The consistency relations [to be used to fix the arbitrari- ness of S $ ( X , ~ )] are derived by differentiating Eq. (5) with respect to time and substituting a Jg(x ) and a,,Jy(x ) from Eqs. (1) and (4). Thus we obtain

1 1 a & $ ( x , y )=7[[a,J,Yx ) + G a ( x )],J;(Y I]+:[J;(x ) , [ a , ~ g ( y 1-f b d c ~ d ( y )J:(y 111 ,

I ( 6 )

1 1 a & g ( x , y ) = T [ [ a , ~ p ( ~ ) + c a ( x )],J;(Y ) I - ~~~~~~~~y] ,

1 1 (7)

1 1 a & ~ b ( x , y ) = : [ [ ~ ~ ( x 1-f a d C ~ $ ( x )J:(x )],J,?y )]-i[a-b,x-y] .

1 (8)

41 3851 - 3 1990 The American Physical Society

3852 BRIEF REPORTS - 4 1

Additionally we will use the standard form of the algebra of conserved charge and currents dictated by unitary symme- try, and the constraints demanded by the Jacobi identity (if imposed).

(1 + I)-dimensional case. The most general form of s$,(x ,y ) is

s $ ( x , y )=cf a b C ~ ; 6 ( x -y )+al f a b C ~ ; 6 ( x -y )+a26aba16(x -y ) , (9a)

S{:(X,~ )=cf a b c ~ f ~ ( ~ -y )+/3YbCJ66(x -y ) + E b a 1 6 ( x -y ) , (9b)

s~:(x -JJ )=c , f a b c ~ 6 6 ( x -y f a b c ~ f 6 ( x -y ) + y 2 6 a b a , 6 ( ~ -y ) . ( 9 ~ )

All the coefficients except c are treated as unknowns. Note that the antisymmetric property of S% and S Y ~ (under simultaneous interchange of a , b and x,y) and the transformation property of invariant tensors in Lie algebra under glo- bal redefinition of generators3 are exploited to arrive at a simpler structure of Eqs. (9b) and ( 9 ~ ) . Substituting Eqs. (9) and (2) in Eq. (6 ) we derive the identity

bd dce +J f J ;6 (x -y ){kf adc/3fbe+kYlfaedf dbc-kfl f ]]=O . (10)

All arguments of JE and all derivatives are evaluated at x. Each expression in the curly brackets of Eq. (10) is indepen- dent as the zero-curvature condition (4) has already been incorporated. Now we study the different cases that may arise.

(1) k = 0, no WZ anomaly. The CA is obtained as

The remaining consistency relations, i.e., Eqs. (7) and (81, produce no new constraints. The algebra [Eq. (1 l ) ] produces the correct transformation property in the internal space satisfying

Finally an explicit computation of the Jacobi identity of J:(x ) and J ! (~ gives

~ ( J ; ( x ),J;(Y ),J;(z ) ) = ~ a z f ~ ~ ~ [ a ; s c x -Z )sex -y ) + ~ s ( . x -y -Z ) + a ; 6 ( ~ -2 )6(z -X ) I , (13)

$ ( J ~ ( x ),J:(Y),J; ( z ) ) = ( c 1 ( c 1 - ~ ) - a Z ] f ~ ~ ~ [ a ; t i ( x - Z ) S ( X -y)+&sc~ -y ) s ( y - z ) + a ; 6 ( y - z ) 6 ( ~ - ~ ) I . (14)

If the Jacobi identity is imposed (which seems justified because the commutators should be canonical for no anomaly) we find a 2 = 0 and c l = c or c =0. Now c1 = c also removes the Schwinger term in S{:(X,~ ). But we will demonstrate later that the CA with both the anomaly and Schwinger term vanishing leads to contradictory re- sults. So the only possibility is c 1 =a2 =0, and we recov- er the standard naive CA.

(2) k#O, WZ anomaly is present. In this case the con- sistency relation (6) gives

I

1 -[J;(x ) , ~ f ( ~ ) ] = ~ f a b C ~ f 6 ( x -y ) + k c l f a b C ~ h 6 ( ~ - y )

1

+ ( c ~ -C ) 6 a b a l s ( ~ - Y ) , (1 5b)

1 - [J?(X ) , J : (~ ) ] = c I f a b C ~ 6 6 ( x - y ) + k c , f " b c ~ ~ 6 ( ~ -y ) .

1

( 1 5 ~ )

Once again Eqs. (7) and (8) do not give any new relations between c,k, and c , . This CA is also consistent with the transformation law

[ d j X J = c f n b c ~ ; i y ) , i I (16)

where 7; = JE - k eP,J",s the conserved current associat-

!!! BRIEF REPORTS 3853

ed with the symmetry. The imposition of the Jacobi identity gives

This restriction of c , = c satisfies the Jacobi identity and removes the Schwinger term, both in the same stroke. This is one of our main results. (But note that in current algebra, the Jacobi identity may very well be violated2 and consequently the argument of the vanishing of the Schwinger term may be questioned. However, the dynamical consequence of Jacobi-identity violation is c ~ m p l i c a t e d ' ~ and it is certainly desirable to have a simpler theory.) In passing we comment that the CA without the Schwinger term [ c z c 1 in Eq. (15)] can be put in a simple form in the light-cone system:

where I"+J;kJy. This type of CA is isomorphic to Heisenberg spin algebra1' and plays an important role in proving the intregrability of the CA models.I2 I t is also possible to use an algebra with k = O in Eqs. ( 18) and (19) by taking suitable nontrivial boundary conditions. The problem of destroying the positivity of the Hilbert space can be bypassed.I3 The final task is to demonstrate that the new anomalous CA without the Schwinger term [Eq. (151, c = c l ] does not clash with the positivity of the Hil- bert space of the state vectors. At the same time this shows that the CA with both the anomaly and Schwinger term absent [ a2=0 , c = c l in Eq. (1 I)] is not allowed, as claimed before. The age-old existence proof of the Schwinger term (for details see Ref. 1) is reproduced here using Eqs. (2) and (15), c = c l and we obtain

where 4 " ( x ) are real test functions and H is some appropriate Hamiltonian. We have assumed (o(J:(x ) lo) =(OIJy(x 110) = O (Ref. 1). Now the left-hand side (LHS) of Eq. (20) is nonzero (because of the positivity of Hilbert space) and the RHS of Eq. (20) is nonzero only if k f O . Hence the WZ anomaly is necessary for a CA without the Schwinger term.

(3 + 1)-dimensional case. Here s$(x, y ) comes as a power series in 1 / f, and we consider terms up to 0 ( 1 / f :I, since divergence anomaly is of O ( 1 /f 5,). The structure of s $ ( x , y ) is

Generally in physically relevant theories JP(x )'s are tak- en to be commuting among themselves. But that would, as we will see later, necessarily make the Schwinger term in Eq. (21b) nonvanishing. On the other hand if we take terms of O ( l /f 4, ) in (21c), the subsequent calculation would become messy. The minimal scheme is to intro- duce terms up to O ( 1 /f:) with the hope of removing the Schwinger term in (21b) which is also of the same order. We substitute Eqs. (3) and (21) in Eq. (6 ) . The long ex- pression of the identity that follows [analogous to Eq. ( lo) ] is not represented here since it is not very illuminat- ing. We just mention the results and conclusions drawn from it. After solving for the coefficients, the CA that emerges is

I

1 :[J;(X ) , ~ , b ( y ) ] = ( i / f 2 , ) f ~ ~ ~ ~ g 6 ( x - y )

1

+ (4k / f 4, )eijk w ~ ~ ~ ~ ~ J ~ J ~ J { ~ ( x -y ) ,

(22a)

1 :[J;(X j ,~,b(y ) ] = ( i / f 2 , ) f a b C ~ ~ 6 ( x -y j

1

+ ( 11 f : )sabai6(x -y ) , (22b)

[JP(x ),Jjb(y )]=O , ( 2 2 ~ )

which is the same as that of ~ a j e e v . ' It seems that fol- lowing this approach it is impossible to remove the

3854 BRIEF REPORTS !!!

Schwinger term since it comes with an unambiguous nu- merical coefficient, independent of the WZ anomaly. The consistency of the CA itself demands the Schwinger term (remember that the Hamilton's equations of motion have not entered the picture). There are no arbitrary parame- ters left and so the modification of ~ a j e e v ~ is unique up to the order of 1 / f, that we have taken as input.

We summarize by noting the salient features of our work. From the analysis of the CA in 1 + 1 dimensions an arbitrary parameter has emerged, which can lead to a consistent CA without the Schwinger term in the pres- ence of the WZ anomaly. So CA's distinct from the es- tablished one7 are possible in 1 + 1 dimensions. This agrees with the observation in Ref. 14 which is a special case of Eq. (1 1). On the contrary, the (3+ 1)-dimensional

algebra is uniquely identified with the one proposed by ~ a j e e v ~ as long as [ J , , J , ] is kept to O ( l / f t ) and other algebras to O ( 1 /f 4, ). The proof of existence of the Schwinger term breaks down in anomalous CA. It may very well happen that one can avoid the Schwinger term in 3 + 1 dimensions by taking terms of higher orders in l/f,. We wish to pursue this problem and also the Hamiltonian formulation of the current algebras suggest- ed here and in a future work.I5

We wish to thank Dr. P. Mitra and Sasanka Ghosh for very fruitful discussions. S.G. wishes to acknowledge the support of Council of Scientific and Industrial Research, India.

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