LIGHT-CONE SUPERSPACE AND THE FINITENESS OF THE N=4 MODEL

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LIGHT-CONE SUPERSPACE AND THE FINITENESS OF THE N=4 MODEL

S. Mandelstam

To cite this version:

S. Mandelstam. LIGHT-CONE SUPERSPACE AND THE FINITENESS OF THE N=4 MODEL.

Journal de Physique Colloques, 1982, 43 (C3), pp.C3-331-C3-333. �10.1051/jphyscol:1982367�. �jpa-

00221922�

JOURNAL DE PHYSIQUE

CoZZoque

C3,

suppl6ment au n o

12,

Tome

43,

ddcembre

1982

page

c3-331

LIGHT-CONE SUPERSPACE AND THE F I N I T E N E S S OF THE N=4 MODEL

S . Mandelstam

Department o f Physics, University of CaZifornia, BerkeZey, California,

U.S.A.

I s h o u l d . l i k e t o d i s c u s s s u p e r s p a c e i n l i g h t - c o n e c o - o r d i n a t e s , s i n c e e x t e n d e d s u p e r s y m m e t r i e s may t h e n be t r e a t e d on t h e same f o o t i n g a s o r d i n a r y s u p e r s y m m e t r i e s . We s h a l l be a b l e t o show, i n a n y o r d e r o f p e r t u r b a t i o n t h e o r y , t h a t t h e

N

= 4 model h a s no u l t r a - v i o l e t d i v e r g e n c e s i n a c e r t a i n form o f t h e l i g h t - c o n e g a u g e , a n d t h e r e - f o r e t h a t t h e 6 - f u n c t i o n v a n i s h e s i n a n y g a u g e . The Wess-Zumino a l g e b r a f o r N = 1 supersymmetry h a s been q u o t e d by S i e g e 1 and ~ a t e s l ) , and a t r e a t m e n t o f N = 4 sym- m e t r y h a s been g i v e n i n d e p e n d e n t l y o f u s by B r i n k , L i n d g r e n and ~ i l s s o n ~ ) , b u t n e i t h e r o f t h e s e g r o u p s c o n s i d e r e d t h e u l t r a - v i o 1 . e t problem.

The d i f f i c u l t y i n o b t a i n i n g c o v a r i a n t e x t e n d e d s u p e r s p a c e s i s c o n n e c t e d w i t h t h e l a c k o f c o r r e s p o n d e n c e between t h e number o f f i e l d s and t h e number o f p a r t i c l e s . I n a n y s u p e r s y m m e t r i c model, t h e p a r t i c l e m u l t i p l e t s w i l l form a r e p r e s e n t a t i o n o f t h e supersymmetry a l g e b r a . I n p a r t i c u l a r , t h e number o f bosons and f e r m i o n s w i l l be e q u a l . When g o i n g from f i e l d s t o p a r t i c l e s t h i s c o r r e s p o n d e n c e is l o s t , and o n e r e s t o r e s t h e b a l a n c e by a d d i n g a u x i l i a r y f i e l d s . No s u c h s e t o f a u x i l i a r y f i e l d s h a s been found f o r t h e N

=

4 model. With l i g h t - c o n e c o - o r d i n a t e s t h e number o f f i e l d s and p a r t i c l e s i s t h e same, and no problem a r i s e s .

I n l i g h t - c o n e c o - o r d i n a t e s t h e number o f s p i n o r f i e l d components, and t h e r e f o r e t h e number o f supersymmetry o p e r a t o r s , i s h a l f t h e number i n t h e c o - v a r i a n t forma- l i s m . I n v a r i a n c e u n d e r t h e s e o p e r a t o r s , t o g e t h e r w i t h L o r e n t z i n v a r i a n c e , g u a r a n t e e s f u l l supersymmetry i n v a r i a n c e . The Wess-Zumino a l g e b r a i s s i m p l e :

F o r N = 4 , t h e a l g e b r a i s a s f o l l o w s :

To c o n s t r u c t a s u p e r s p a c e f o r t h e Mess-Zumino model, we n o t e t h a t t h e first q u a n t i z e d model h a s o p e r a t o r s x and p , t o g e t h e r w i t h t h e two f e r m i o n i c o p e r a t o r s Q 1 and Q,. A f t e r second q u a n t i z a t i o n , t h e wave f u n c t i o n s become f i e l d s which are f u n c - t i o n s o f c o - o r d i n a t e s c o r r e s p o n d i n g t o

x

and t h e Q ' s . We r e q u i r e one c o - o r d i n a t e c o r r e s p o n d i n g t o e a c h p a i r o f c o n j u g a t e v a r i a b l e s , s o t h a t we have f o u r xP's and o n e a n t i c o m m u t i n g 0 . Our a p p r o a c h t h u s d i f f e r s from t h e u s u a l a p p r o a c h , which would h a v e i n t r o d u c e d a 0 and a

8.

F o r t h e N = 4 model w e r e q u i r e f o u r BAS. A t r e a t m e n t c l o s e r t o t h e c o n v e n t i o n a l o n e i s g i v e n i n R e f s . 1) and 2 ) .

C o r r e s p o n d i n g t o Q , a and Q Z a , we d e f i n e t h e two o p e r a t o r s

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982367

C3-33 2 JOURNAL DE PHYSIQUE

I n w r i t i n g down t h e s u p e r f i e l d f o r t h e N = 4 model, we s t a r t from t h e h i g h e s t h e l i c i t y f i e l d and g o downwards. Thus

Each f i e l ' d c a r r i e s a g a u g e i n d e x which h a s been s u p p r e s s e d . The p o s i t i v e and nega- t i v e h e l l c i t y v e c t o r and s p i n o r f i e l d s , and t h e s i x s p i n l e s s f i e l d s , h a v e been r e - p r e s e n t e d by t h e s y m b o l s V ,

vt,

$,, +?, a n d A r e s p e c t i v e l y . The m a t r i x p i s t h a t o f G l i o z z i , O l i v e and s c h e r k 3 ) .

The s u p e r m u l t i p l e t f o r t h e N = 4 t h e o r y i s s e l f - c o n j u g a t e . The f i e l d s @ and

@'f' a r e t h e r e f o r e n o t i n d e p e n d e n t , b u t a r e r e l a t e d by t h e e q u a t i o n

We s h a l l o m i t t h e d e r i v a t i o n o f t h e f o r m u l a f o r t h e L a g r a n g i a n ; t h e r e s u l t i s :

Dot and c r o s s p r o d u c t s r e f e r t o t h e gauge d e g r e e s o f f r e e d o m . The d e r i v a t i v e s 3

a r e

a l + i a , ,

w h i l e r

,

We o b s e r v e t h a t !f i n v o l v e s o n l y 3/30' and n o t 0', s o t h a t i t commutes w i t h t h e oper- a t o r 8'. F u r t h e r m o r e , i t o n l y i n v o l v e s 3/36, i n t h e c o m b i n a t i o n ( 7 ) , s o t h a t it commutes w i t h P+Ba. The L a g r a n g i a n t h u s c o m u t e s w i t h t h e ' s u p e r s y m m e t r y o p e r a t o r s

( 3 ) . It i s u n d e r s t o o d t h a t t h e 0 d e r i v a t i v e s i n t h e t h i r d a n d f o u r t h t e r m s o f ( 7 ) a r e t o b e w r i t t e n i n c y c l i c o r d e r . The d e r i v a t i v e s t h u s a l l o c c u r combined w i t h t h e symbol a n d , s i n c e t h i s i s a n S U ( 4 ) i n v a r i a n t c o m b i n a t i o n , t h e L a g r a n g i a n i s m a n i f e s t l y S U ( 4 ) i n v a r i a n t .

The t h i r d term o f L can be o b t a i n e d from t h e s e c o n d by r e p l a c i n g

4

by

@t,

which i s r e l a t e d t o @ by E q . ( 5 ) . The t e r m s a r e t h u s t h e H e r m i t i a n c o n j u g a t e s o f one a n o t h e r .

To p r o v e t h e f i n i t e n e s s o f t h e model,

\re

examine a g e n e r a l v e r t e x d i a g r a m . L e t u s c o n s i d e r a n e x t e r n a l t h r e e - p o i n t v e r t e x o f t h e form o f t h e s e c o n d t e r m o f ( 6 ) . By u s i n g t h e i d e n t i t y

we n o t e t h a t we c a n c h o o s e one p a i r of l i n e s m e e t i n g a t t h e v e r t e x , i n c l u d i n g t h e e x t e r n a l l i n e , and a p p l y t h e f a c t o r s

a~

and 2p' t o them. Thus t h e e x t e r n a l l i n e h a s a t l e a s t one f a c t o r

a R

o r 2p+. Ye c a n t r e a t t h e o t h e r t e r m s o f ( 6 ) s i m i l a r l y ; e a c h e x t e r n a l l i n e h a s a t l e a s t one t ' a c t o r o f

aE,-ar,

2p+ o r 3/20.

S. Mandelstam

C3-333

We thus find that there are more powers of p on the external lines of the vertex corrections than on the external lines of the bare vertex. (A factor 3/30 is di- mensionally equivalent to a factor p1/2.) It might therefore be expected that the number of powers of p on internal lines is insufficient to give a divergence. The power counting is easily performed and confirms this result; the vertex correct*

are finite, provided it is legitimate to employ naive power counting with all com- ponents of p treated equally.

In the usual light-cone gauge, such power counting is not in fact permissible.

The reason is that ultra-violet divergences appear from t h e x g i o n where p+ and p+p- -

^1?2

are finite, while p- and

pZ

are large. The poles in the factors (p+)-' prevent us from continuing to imaginary pa and thgs avoiding these dangerous regions.

The condition A+

= 0

does not define the light-cone gauge uniquely, since it remains true under a gauge transformation which depends on x1 and x+ but not on x-.

The ambiguity is reflected in the ic prescription in the factors (P+)-'. Usua+lly, one takes a principal-value prescription. If we could use the prescription (p

)-' + +

(p+

+

icp-I-', there would be no difficulty in continuing the integration to

imaginary pO. After such continuation it is easy to see that naive power counting is valid.

It is not difficult to show that one can define a light-cone gauge wit.h the above ic prescription. Such a "modified light-cone gaugen is inconvenient for most purposes, since it is only invariant under Lorentz transformations which leave both p+ and.p- unchanged. For our purposes this is the best gauge to use, since the vertex functions are finite. It is also easy to prove that all n-point functions, with n

2 3,

are finite.

For the two-point functions, the above reasoning would still allow a divergent term of the form ~ p

+

Bpipj. To show that such a term does not in fact occur, ~ 6 ~ we use the Ward identi2y in the form A1(p,p,Ol = -a/api i'I(p,p). This version of the Ward identity is valid only if proper Green's functions involving gluons of all four polarizations are free of singularities when any of the p+'s become zero; a

condition

which is true in the modified light-cone gauge (though not in the usual light-cone gauge). From the differential form of the Ward identity and the finite- ness of A, we can conclude that divergent terms proportional to p26.. or to pipj cannot occur in It. The two-point function, and the complete model,'&-e thus finite in any order of perturbation theory.

In other gauges the wave function renormalization will generally not be finite.

The divergence is a pure gauge artifact. The

8

function will always vanish, however, since its vanishing is a gauge-invariant condition.

REFERENCES

1) Siege1 W. and Gates S.J., Nucl. Phys. (1981) 295.

2)

Brink L., Lindgren

0.

and Nilsson B.E.W., Goteborg preprint

3) Gliozzi F., Olive D. and Scherk J., Nucl. Phys. (1977) 253.