ASYMPTOTIC FREEDOM IN FIELD THEORY

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ASYMPTOTIC FREEDOM IN FIELD THEORY

R. J.-Crewther

To cite this version:

R. J.-Crewther. ASYMPTOTIC FREEDOM IN FIELD THEORY. Journal de Physique Colloques,

1973, 34 (C1), pp.C1-111-C1-115. �10.1051/jphyscol:1973110�. �jpa-00215190�

ASYMPTOTIC FRE- IN FIELD THEORY

ASMPTOTIC FREMOM I N FIELD THEORY R

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T h i s b r i e f r e p o r t d e s c r i b e s some r e c e n t work on t h e r e n o r m a l i z a t i o n g r o u p , w i t h p a r t i c u l a r emphasis on t h e d i s c o v e r y o f P o l i t z e r ( 2 1 and Gross end W i l c z e k ( 3

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R.J. CRGYPIHER

i n t h e a s y n n t o t i c e x v a n s i m of

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,

t h e a s y m p t o t i c b c h a v i o r o f t h ? z e r o - n s s s t h e o r y ( w i t h g #

r:

1 i s d e t e r n i n e d by i t s n r o n e r t i e s a t t h e e i n m - v a l u e :

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t h e r a - s u l t ( 1 3 )

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i s

s i m n l y t h e e x n o n e n t o b t a i q d u s i n q i r e e - f i e l d d i m e n s i o n s . Does t h i s mean t h a t a f i e l d - t h e o - r e t i c e x p l m a t i m of

exact

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1

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ASYMPTOTIC FREEDCM IN F I E L D THEORY C1-113

m d s o t h e f a c t o r

i n E q . 181 o r o d u c e s m a s y m p t o t i c deo m d e n c e 1 4 )

i q

r

r e l a t i v e t o t h e n a i v e r e s u l t

T f r e e

of f r e e - f i e l d t h e o r y . From Eqs. ( 1 2 ) and ( 1 3 1 , we s e e t h a t t h e ex- p o n m t

-

^{i: /2b} c m be obtaiqer: from a low-order n e r -

0 0

t u r h a t i v e c a l c u l a t i o n . Llslce, I31 o r k m s c a l i n ~ : i s v i o - l a t e d by c a l c u l ~ b l e powers o f l o g a r i t h m s o* t h e mo- mentum t r a n s f e r .

T h u s , i t i s i m p o r t a n t t h a t t h e t e r m s " a s y m p t o t i c freedom" and " a s y m o t o t i c f r e e - f i e l d b e h a v i o r " ( a s in p a r t o n m o d e l s ) s h o u l d n o t b e c o n f u s e d . F r e e - f i e l d b e h a v i o r c a n n o t be produced in a c o n v e n t i o n a l r e n o r - m a l i z a t i o n - y r o u n a o n r o a c h i n ¹ d i n a n s i o n s . The m l v k q o w e x a ~ o l e [I F 1 t o which

both

t c r m s may he a n o l i ~ d i s twn-rlimensional nuan turn r ? l e c t r o r J ~ n a m i c s ISchwinqer model 11711. T h i s t h e o r v i s c o n p l e t e l v s o l u b l e , qua- s e s s e s a s v v n t o t i c f r e e - f i e l d b e h s v i o r l h e c a ~ s e

i t

i s s u n e r r m o r ~ a l i z ~ b l e l , anr! can be a n a l y z e r ' u s i ~ e r m o r - m l i z s t i o n - e r o u n methods [ I F ) :

O b v i o u s l y , X = 0 i s a UV-stable f i x n 4 r t o t n t . I t i s a l s o i m o o r t a n t t h a t t h e r e s u l t ( ? 7 1 s h c u l d n o t be c m f u s e d w i t h t h e a s v m o t o t i c b e h a v i o r of

rfinite,

where

rfinite

d e n o t e s t h e a m n l i t u d e

r

comnu- t e d i n a f i n i t e o r d e r of n e r t u r b s t i o n t h o o r y . The exnwrent -c /2b il 117) i s a r a t l c n a l n u ~ b r , b u t ,

0 3

ii e ; e n e r a l ,

i t

i s

not

^?R i n t e p e r . w h e r ~ a s t b e exoo- n e n t P d e f i n a d b y

i s a l w a y s a7 i n t e ~ e r . To o b t a i n 1171 , t b e l o v a r i t h n s r i v e n by (IQImust be summed t o a l l o r d e r s i n p e r t u r - b a t i o n t h e o r v .

7 . I

.-

TIiE POLITZER, GROSS-WILCZEK -CALCULATION

.-

Con=irlcr a yaup c Cheorv w i t 3 L a q r a n q i m

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^{J, ~nr! X a a r e aauca} mcsc,r., fermion and U i

Faddeev-Pooov q h o s t f i e l d s ,

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a,

and t h e c o v a r i a n t d e r i v a t i v e

c m t a i q s t h e m a t r i c e 3 T~ o f t h e f e r m i o n r e p r e s m t a t i o n

R

o f G . T h e r e i s o n l y one c o u p l i n e c c n s t a n t , q .

One way o f o b t a i n i n e t h e c m s t m t bo d e f i n e d by ( 1 2 ) i s t o o h s e r v e t h a t . when a c u t o f f A i s i n t r o d u - c e d , e, i s r d a t e d t o i t s m r m o r m a l i z e d c o u n t e r p e r t

c0

by t h e f o r m u l a

where Z3 and 7, a r e t h e m u l t i p l i c a t i v e r e n o r m a l i z a t i o n c ~ s t m t s o f t h e o r o p a c a t o r m d t h r e e - p o i n t v e r t e x of t h e q a u o e n e s o n * i e l d :

3 / 2

.vvv\r Gouge meson

-

^Fermion

--

Faddeev-Popov ghost

C1-114 R.J. CReWTHER

X l t e m a t i v e l y , t h e r m o r m a l i z e d a n p l i t u d e s o b t a i n e d from t h e s e g r a p h s c a - ~ b e s u b s t i t u t e d i n t o t h e i r r e s - p e c t i v e r m o r m a l i z a t i m group q u a t i m s : t h e r e s u l - t i n g f o r m u l a e c m t h m b e s o l v e d f o r B(g1.

The r e s u l t

i s

(2-4. 181

where C2(Gl

is

t h e q u a d r a t i c C a s i m i r o p e r a t o r o f t h e a d j o i n t r e p r e s m t a t i m o f G ( 1 . e . C2[G16ab = CacdCbcdl

,

C2(Rl

i s

t h e c o r r e s p o n d i n g o p e r a t o r f o r t h e f e r m i m r e p r e s e n t a t l a , R o f G , d(R1

i s

t h e d i - mension o f

R ,

m d r l G l

is

t h e number of g e n e r a t o r s o f G; When n o f e n i m s a r e p r e s e n t , d[Rl v m i s h e s

.

a l l t h a t r e m a i n s i d t h e c m t r i b u t i m

4

C2(Gl d u e t o gauge

mesms,

m d we have UV s t a b i l i t y a t t h e o r i - g i n

.

F o r - t y p i c a l [ s m a l l r e p r e s e n t a t i m s R of i n t e -

rest,

t h e p o s i t i v e f e n i m c m t r i b u t i m s a r e s m a l l e r then t h e n e g a t i v e gaugemeson c m t r i b u t i m : t h u s . t h e c m d i t i m f o r UV s t a b i l i t y

1 1 rlG1 C2[Gl > 4 C2(Rl d[R1

i s

n o t m i m p o r t w t c m s t r a i n t f o r model-building.

S i m i l a r c a l c u l a t i o n s (2.41 y c e l d t h e c m s t m t s Co d e f i n e d by (131.

2.2.- REASONS FOR EXCLUDING SCALAR FIELDS.- When s e t o f s c c l a r f i e l d s

4 ^is

i q t r o d u c e d i n t o t h e La- g r a g i m ~ ( g l 09 Eq. (201.

it i s

n e c e s s a r y t o i n c l u - d e a s e c m d r m o r m a l i z e d c o u p l i n g c m s t m t

X

which d e s c r i b e s q u a r t i c s e l f - c o u p l i n g o f t h e s c a l a r mesons:

I f o t h e r q u a r t i c i n v a r i m t o f G e x i s t , t h e c o r r e s p m - d i n g c o u p l i n g c m s t m t s must a l s o b e i n t r o d u c e d . The

-+

-

t e r m V(cpl r e p r e s e n t s t h e s e a d d i t i m a l c o u p l i n g s , Q o g e t h e r w i t h a s y m p t o t i c a l l y n e g l i g i b l e c p

,

c p 2 o r

(p3

i n t e r a c t i m s .

S i n c e t h e r e a r e two c o u p l i n g c m s t m t s ( g , X I , t h e r e must b e two f 3 - f m c t i m s ( 8 Ig.XI m d RX[g,X1, m d

!?

h m c e

two

c m d i t i o n s

f o r a s y m p t o t i c fresdomr ( 1 . e . , we w m t [ g , k ] = ( 0 . 0 ) t o b e a UV-stable f i x e d p o i n t ) , What happens (41

is

t h a t , i n o r d e r t o s a t i s f y t h e c m d i t i m f o r

E X ,

t h e p o s i t i v e c m s t m t bo ( d e f i n e d f o r

B

a s i n (121 w i t h

g

h = 01 must b e given a v e r y s m a l l v a l u e , m d t h e group T; must be s u f f i c i m t l y large:?Iodels which s a - t i s f y t h e s e c m d i t i m s t e n d t c be r a t h e r c o m p l i c a t e d , m d h m c e v r a t t r a c t i v e .

I n

p a r t i c u l a r , l a r g e numbers o f f e r m i c n s must be i n t r o d u c e d .

The u s u a l r a t i o n a l e f o r i n t r o d u c i n g s c a l a r mesons

i s

t h a t m e would l i k e t o g i v e t h e v e c t o r mesons mass v i a t h e n i g g s mechmism, [ u s i n g t h e t r e e a p p r o - x i m a t i m l .

I t

a p p e a r s (3.41 t h a t r e q u i r i n g a l l v e c t o r mesons t o b e given mass l e d t o s p e c u l g t i m [ 2 , 3 1 t h a t s p o n t m e o u s b r e a k i n g s h o u l d a r i s e d p a m i c a l l y ( 1 9 ) . C u r r e n t wisdom (201 m t h i s s u b j e c t

i s

t h a t t h e gauge group s h o u l d be i n t e r p r e t e d a s m e x a c t c o l o u r sym- m e t r y . We do n o t w m t c o l o u r e d p a r t i c l e s s u c h a s gauge mesons t o a p p e a r a s a s y m p t o t i c s t a t e s , s o t h e r e

i s

n o p o i n t i n t r y i n g t o g i v e them mass.

I t

h a s y e t t o be d e t e r m i n e d w h e t h e r t h e i n f r a r e d b e h a v i o r o f such a t h e o r y

i s

r e a l i s t i c : i n c o l o u r e d c h m n e l s ,

i t

s h o u l d be h i g h l y s i n g u l a r , b u t c o l o u r - s i n g l e t c h a n n e l s s h o u l d have r e g u l a r i n f r a r e d p r o p e r t i e s compatLble w i t h t h e e x i s t e n c e o f a s y m p t o t i c s t a t e s .

Of c o u r s e , t h e o b s e r v e d s u p p r e s s i o n o f t h e l o n g i t u - d i n a l c r o s s - s e c t i o n in d e e p i n e l a s t i c e l e c t m p r o d u c t i o n t i o n can b e c i t e d a s y e t m o t h e r r e a s o n f o r e x c l u d i n g

+

'P

.

3.- OTHER FIELD THEORIES.- Coleman m d Gross (211 have r e c e n t l y s h o w t h a t m y f o u r - d i m s i o n a l f i e l d t h e o r y which d o e s n o t i n v o l v e non-Abelim gauge me- s o n s c a n n o t be a s y m p t o t i c a l l y f r e e . The argument r m s a s i o l l o w s :

1 1 Abelian gauge f i e l d s c a n n o t be p r e s e n t i f asymp- t o t i c freedom

i s

d e s i r e d , b e c a u s e t h e c o r r e s p o n d i n g 8 - f m c t i o n

i s

t h e same a s t h a t f o r quantum e l c t r o - dynamics.

i l l S i m i l a r l y , a d i r e c t c a l c u l a t i m by Zse (221 h a s shown t h a t a p u r e

Xcp

4 t h e o r y [ o r

i t s

g m e r a l i z e t i o n t o a p o s i t i v e q u a r t i c form Xijklcpicpj~cp1)

i s

n o t a s y m p t o t i c a l l y f r e e .

i i i l A l l t h a t r e m a i n s ' i s t h e Yukaua c o u p l i n g and g e n e r a l i z a t i o n s t h e m o f : t h i s

i s

t h e problem s o l v e d by C o l m m d G r o s s .

The example

ASYMPTOTIC F R E E m l IN FIELD THEORY C 1 - 1 1 5

i l l u s t r a t e s t h e i r method. I f t h e r e i s o n l y one cou- p l i n g c o n s t a n t g, ~ ( a ) r e c e i v e s c o n t r i b u t i o n s

f r o m t h e g r a o h s :

I n s p e c t i o n o f t h e s e granhs shows t h a t t h e c o r r e c t g m e r a l i z a t i o n o f (281 i s

where qi, R~ o r e = t r i c e s w i t h elements 4- i ab' 'ab' A s y m p t o t i c freedom does n o t o c c u r u n l e s s a l l o f t h e R : ~ a r e n e q a t i v e as each

cab

i t e n d s t o Era t h r o u ~ h

p o s i t i v e v a l u e s . TO p r o v e t h e i r a o i n t . Coleman m d Gross examine t h e sum.

q6n2 ~r q i

&

⁼ t T r ei $' T r qi e1

i i 1 1 . 1

+ ZTr q t1 qi g 1 + T r g c! e 7 m d o b s e r v e t h a t n e i t h e r

2 T r ci

e1

T r yi g1 + 2 T r

ei

g1 ci q1

n o r T r ei qi can b e n e y a t i v e :

T h e r e f o r e , a s y m p t o t i c freedom i s i m p o s s i b l e . The r e s u l t r e m a i n s v a l i r l w h m (271 i s r e o l a c e d b y t h e most g e n e r a l Yl~kawa c o u n ? i n a :

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