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THE VANISHING β-FUNCTION OF N=4 SUPERSYMMETRIC YANG-MILLS THEORY
K. Stelle
To cite this version:
K. Stelle. THE VANISHING β-FUNCTION OF N=4 SUPERSYMMETRIC YANG-MILLS THE- ORY. Journal de Physique Colloques, 1982, 43 (C3), pp.C3-326-C3-327. �10.1051/jphyscol:1982365�.
�jpa-00221920�
JOURNAL DE
PHYSIQUEColloque C3, suppl6ment au n o 12, Tome
43,de'cembre
1982page
c3-326THE VANISHING B-FUNCTION OF N=4 SUPERSYMMETRIC YANG-MILLS T H E O R Y
K.S.
StelleImperial College of Science
&Technology, Prince Consort Road, London
SW72BZ, U.K
The maximally extended supersymmetric Yang-Mills theory saturates the well-known bound on asymptotic freedom with its four Weyl spinor fields in the adjoint representation of the gauge group, thus having a vanishing one-loop 8-function.
Remarkably, this cancellation persists as well through the three-loop level as shown by explicit calculations 1.
Here we explain the reasons for this cancellation from two points of view. The first relies upon non-renormalization theorems together with a manifestly supersymmetric calculational technique that preserves
N=2
supersymmetry. The second analyses the constraints imposed on the stress tensor supermultiplet byN=2
supersymmetry and the conservation of the~ ( 2 )
symmetry currents. This second point of view is analogous to arguments based upon N=l supersymmetry,
but theN=2
analysis is complete in that no assumptions need to be made about preservation of internal symmetries beyond those that are manifest in the formalism. Moreover, in this particular theory, higher derivative regularization can be used to preserveN=2
supersymmetry and the full rigid~ ( 2 )
symmetry as well as gauge invariance.It is possible to quantize the
N=4
theory usingN=2
superfield Feynman rules 3 , although unfortunately it is not known how to manifestly preserve the fullN=4
super- symmetry in a superfield calculation. However, the manifest preservation ofN=2
supersymmetry coupled with the non-renormalization theorems using the background field method is sufficient to demonstrate finiteness. When written inN=2
super- fields, theN=4
super-Yang-Mills action consists of two terms: the action for theN=2
super-Yang-Mills theory5 .
where Aa i is the
N=2
gauge connection superfield, and the superspace Lagrangian for theN=2
scalar hypymultiplet4 8
aih2
= j d x d B t r { i p i +h.c.+
L ijka 'ijka'(2) . .
where
hai
= D a i ~ l JArticle published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982365
K.S. Stelle C3-327 Both Im2 and
b2
f a i l t h e requirements of t h e non-renormalization theorems f o r counterterms above one l o o p , namely 1) a l l counterterms must b e w r i t t e n a s f u l l superspace i n t e g r a l s 2 ) a l l . c o u n t e r t e r m s must b w r i t t e i n terms of t h e "pre-gauge7- - 7
i n v a r i a n t " q u a n t i t i e s A,i, L ' ~ , ~~j~~ nd V = D l j D kR X i j k +
.
S i n c e i n t h e background f i e l d method of r e f e r e n c e'
, t h e one-loop c s l c a a t l o n i s t r e a t e d s e p a r a t e l y , a t t h i s o r d e r an e x p l i c i t check i s needed, w i t h t h e r e s u l t t h a t n e i t h e r Im2 norb2
i s renormalized.Q u a n t i z i n g t h e t h e o r y w i t h N=2 s u p e r f i e l d s a l s o a l l o w s a supersymmetric r e g u l a r i z a t i o n scheme t o be devised. The scheme i s simply h i g h e r d e r i v a t i v e
r e g u l a r i z a t i o n , adding a term D,,DP i n s e r t e d between each of t h e f a c t o r s i n ( 1 ) and ( 2 ) . The u s u a l disadvantage of h i g h e r d e r i v a t i v e r e g u l a r i z a t i o n i s t h a t it g i v e s modified v e r t i c e s w i t h d e r i v a t i v e c o u p l i n g s a s w e l l a s modified propagators.
A t t h e one l o o p l e v e l , t h e s e two m o d i f i c a t i o n s c a n c e l t h e e f f e c t i v e n e s s o f t h e r e g u l a r i z a t i o n , although h i g h e r loops a r e r e g u l a t e d . I n t h i s p a r t i c u l a r t h e o r y , t h e known c a n c e l l a t i o n s a t one l o o p a l l o w u s t o proceed t o t h e h i g h e r o r d e r s where t h e r e g u l a r i z a t i o n i s e f f e c t i v e . The N=2 s u p e r f i e l d s a r e c r u c i a l i n t h i s r e g a r d , however, f o r t h e y m a i n t a i n t h e one-loop c a n c e l l a t i o n s of a l l i n f i n i t i e s even i n t h e presence of t h e h i g h e r d e r i v a t i v e t e r m s , which would not o t h e r w i s e o c c u r , a s f o r example i s t h e c a s e w i t h N = l s u p e r f i e l d s .
The c a n c e l l a t i o n s i n t h e - p r e s e n c e of t h e h i g h e r d e r i v a t i v e terms may be v e r i f i e d by s e p a r a t i n g a l l t h e modified p r o p a g a t o r s i n t o p a r t i a l f r a c t i o n s , c o l l e c t i n g t h e propagating s t a t e s i n t o "physical" and "ghost" s u p e r m u l t i p l e t s
,
t a k i n g i n t o account t h e c o n t r i b u t i o n s from a u x i l i a r y f i e l d s . The c a n c e l l a t i o n s due t o t h e m a s s l e s s"physical" m u l t i p l e t s a r e a s i n t h e t h e o r y without h i g h e r d e r i v a t i v e terms. The r e g u l a t o r g h o s t s t a t e s f i l l o u t f o u r massive m u l t i p l e t s , each c o n t a i n i n g ( 1 s p i n one,
4
s p i ni , 5
s p i n 0 ) . The c o n t r i b u t i o n s o f t h e g h o s t s t o t h e 8-function a r e g i v e n by t h e u n i v e r s a l formula f o r h e l i c i t y A ,2 2X
@ ( A ) Q ~(1-12A )(-I) ( 5 )
where C i s t h e q u a d r a t i c Casimir eigenvalue f o r t h e 9fuge group r e p r e s e n t a t i o n . For a l l s u p e r m u l t i p l e t s , t h e sum over h e l i c i t i e s of (-1) c a n c e l s ; f o r t h e p a r t i c u l a r N=2 ghost m u l t i p l e t s h e r e t h e sum of ~ ~ ( - 1 ) ~ ~ c a n c e l s a s w e l l .
The h i g h e r d e r i v a t i v e r e g u l a r i z a t i o n p r e s e r v e s not o n l y N=2 supersymmetry, but t h e
~ ( 2 ) r i g i d symmetry of ( 1 ) and ( 2 ) a s w e l l . Thus a t no l o o p o r d e r can t h e r e b e an anomaly i n t h e a x i a l ~ ( 1 ) p a r t of ~ ( 2 ) = SU(2) x ~ ( 1 ) . This i n t u r n r u l e s o u t any anomaly i n t h e t r a c e of t h e energy momentum t e n s o r , s i n c e t h e t r a c e s u b m u l t i p l e t of t h e N=2 s u p e r c u r r e n t m u l t i p l e t c o n t a i n s b o t h Tp,, and 3 ~ ~
.
2With a v a n i s h i n g t r a c e anomaly, we a g a i n o b t a i n t h a t @= 0.A f u l l e r p r e s e n t a t i o n of t h e above d i s c u s s i o n w i l l b e p r e s e n t e d elsewhere. 8 REFERENCES
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