Asymptotic behaviour of the vertex function in field theory

Volume i i , number 2

ASYMPTOTIC

PHYSICS LETTERS

BEHAVIOUR

OF

THE

VERTEX

15 July 1964

FUNCTION

IN FIELD

THEORY

M. CASSANDRO and M. CINI s

o

,

Istituto di Fisica dell Unzverszt~, Rorna Istituto Nazionale di Fisica Nucleate, Sezione di Rorna

Received 1 June 1964

R e c e n t i n v e s t i g a t i o n s have r e n e w e d the i n t e r e s t in the q u e s t i o n of the m a g n i t u d e of r e n o r m a i i s a tion c o n s t a n t s in quantum f i e l d t h e o r y 1-7). An i m p o r t a n t point r e l e v a n t to t h i s p r o b l e m i s the d e t e r m i n a t i o n of the a s y m p t o t i c b e h a v i o u r of the v e r t e x function. We w i s h to c o n t r i b u t e to this point by a t t e m p t i n g to d e r i v e an a s y m p t o t i c l i m i t of the v e r t e x when the f e r m i o n e x t e r n a l l i n e s a r e on s h e l l and the m o d u l u s of the b o s o n m o m e n t u m t e n d s to infinity. We c o n c e n t r a t e o u r attention on s p i n o r e l e c t r o d y n a m i c s with m a s s i v e photon, a t h e o r y which r e c e n t l y h a s r e c e i v e d c o n s i d e r a b l e attention, and shown to have r e m a r k a b l e p r o p e r t i e s 8). In t h i s w o r k the e x p r e s s i o n for the r e n o r m a l i z e d v e r t e x function F ~ R ( P l , P2) with TPl =YP2 = m and t = - ( P l " P2) 2 ~ ~o, i s o b t a i n e d by s u m m i n g the l e a d i n g t e r m s in the l i m i t of l a r g e t to a l l o r d e r s in p e r t u r b a t i o n t h e o r y . Use h a s b e e n m a d e of the m e t h o d s d e v e l o p e d by Tiktopoulos 9) to , e v a l u a t e the a s y m p t o t i c b e h a v i o u r of a given F e y n m a n g r a p h . We n o t i c e that s i n c e the r e n o r m a l i z e d v e r t e x i s o b t a i n e d by s u b t r a c t i n g a t - i n dependent c o n t r i b u t i o n

w h e r e a = e2/~ic and At is the photon m a s s . The d e t a i l s of the d e r i v a t i o n of (1) will be given e l s e w h e r e (10): s o m e c a r e m u s t be taken in p e r f o r m i n g the l i m i t b e c a u s e the v e r t e x i s a s i n g u l a r configuration of type B in Tiktopoulos' s e n s e 9). Our r e s u l t , however, d i f f e r s f r o m what one would obtain by applying blindly the p r e s c r i p tions given by h i m for this c a s e . We s t r e s s that (1) d o e s not give the whole inf r a r e d c o n t r i b u t i o n a s the photon m a s s ~t2 ~ 0 : t h i s i s b e c a u s e in taking the l i m i t f o r t --" ~o we have m

=

in m--~2-~- in ~-~2-"

It i s h o w e v e r t r u e that to k e e p ~t2 ¢ 0 i s e s s e n t i a l in o r d e r to obtain f i n i t e r e s u l t s . We r e m e m b e r that (1) i s v a l i d a l s o f o r d i a g r a m s allowing any n u m b e r of c r o s s i n g s of the photon l i n e s , b e c a u s e , a s was pointed out by Nambu 11) and n o t i c e d b y Tiktopoulos 9), for two and t h r e e l e g g e d d i a g r ~ r n s , p l a n a r and non p l a n a r g r a p h s can be t r e a t e d on the s a m e footing. T h e r e f o r e the total c o n t r i b u t i o n f r o m such a c l a s s of d i a g r a m s to F ~ R of o r d e r e 2n i s

r~ R = z I r ~ = r~ + A~(Pl, P2) - A~CPl, "Pl) n'.

we will automatically obtain the asymptotic lirnit for r. R if we drop all the non leading terms in t. Le~ us consider the contribution to r R of order e 2n given by a diagram containing~n photon lines encompassing the vertex: it turns out that the leading term for very large t (neglecting terms with at least one power less of In t) is proportional to ~ , the coefficient g2n(0)(t)being given by *

g~O2(t) = ( _ l ) n '

n l(~_~)

~~ ) l nt

Since no other diagram of order e2~ gives contributions which ~tsymptottcally go a s f a s t a s (in It I) 2n we d i s r e g a r d t h e m c o n s i s t e n t l y with o u r a p p r o x i m a t i o n . We o b t a i n t h e r e f o r e :

r R(t)~

~

~

x(t)2"

(~.)

with X(t) = ~/(o/2~) in It I, it being understood that t is now dimensionless. It can be shown 10) that

2~,

(1) • When t is timeltke gg.n(O)(t) has also an imaginary part, which, however, ts of the order of the negleoted terms.

g(~)(t).

the s e r i e s (2) i s the confluent h y p e r g e o m e t r i c

function q(1, I, -~ x2) 12). Our result should be compared with a similar calculation performed by Sudakov 13) with the conditions that t >> pl 2, p2 2 >> m 2. He finds that 159

Volume 11, n u m b e r 2

PHYSICS LETTERS

ruR(t)

v a n i s h e s as exp (-X2); the a p p a r e n t disc r e p a n c y is due to the fact that in his case ~2 = 0, and this l i m i t cannot be obtained by analytic continuation f r o m the a s y m p t o t i c e x p r e s s i o n evaluated for ~ ¢ 0 10). A c c o r d i n g to the a s y m p t o t i c behavfour of the confluent h y p e r g e o m e t r i c functions, we have F~(t) X ~

-~

2/x2 "

r (t) t 4 £ 1

r u l e 15, 16) should a r i s e f r o m i n t e r m e d i a t e s t a t e s with only e l e c t r o n - p o s i t r o n pair. If, however, one is willing to accept that F~R(t) with ~ ¢ 0 might roughly r e p r e s e n t the c o n t r i b u t i o n of an e l e c t r o n - p o s i t r o n p a i r a c c o m p a n i e d by soft photons of e n e r g y s m a l l e r than ~ in the s u m r u l e for Z3, one obtains approximately, u s i n g our a s y m p totic e x p r e s s i o n (3)

(3)

The v a n i s h i n g of r u R(t) for l a r g e t thus found is a consequence of the a l t e r n a t i n g s i g n s of the t e r m s in the s e r i e s (2). Because of this c a n c e l l a tion of the leading t e r m s , however, the s u m of the c o n t r i b u t i o n n e g l e c t e d o r d e r by o r d e r might not be negligible. This is not the case when the leading t e r m s all have the s a m e sign 14). Whate v e r the non leading t e r m s also c a n c e l in the l i m i t for l a r g e t is a q u e s t i o n which n e e d s f u r t h e r investigation. We now wish to make two f u r t h e r r e m a r k s . F i r s t of all we s t r e s s that our r e s u l t applies only to s p i n o r e l e c t r o d y n a m i c s : for other field t h e o r i e s the b e h a v i o u r of the vertex is c o m p l e t e l y different. When all the p a r t i c l e s a r e s c a l a r one e a s i l y finds (4)

thus leading to a divergent Z 3. For other theories such as spinor-pseudoscalar coupling w e cannot m a k e as yet definite statements but one can see that for a given d i a g r a m of o r d e r g2n t h e r e a r e no t e r m s of o r d e r (in t)2n. It is r e m a r k a b l e , in our opinion, that also in this case s p i n o r e l e c t r o d y n a m i c s with m a s s i v e photon t u r n s out to be p r i v i l e g e d with r e s p e c t to other r e n o r m a l i z a b l e theories. F i n a l l y we would like to spend a few words about the s i t u a t i o n in q u a n t u m e l e c t r o d y n a m i c s with zero m a s s photons. S t r i c t l y speaking r_~R(t) v a n i s h e s for any value of t when the l i m i t ~,2--, 0 is p r o p e r l y taken in the exponential which f a c t o r i z e s in the i n f r a r e d l i m i t 3). T h e r e f o r e no cont r i b u t i o n to Z 3 in the K e l l ~ n - L e h m a n n s u m

oO

2~ Z 3 ~- 1 - ~-~/--~-f dX ~2(X) . (5) o W e certainly do not want to attach m u c h meaning • to this result but merely point out that the condition Z 3 = 0 does not give the physical value of a : one would rather conclude from (6) that Z 3 is finite and not m u c h smaller than 1.

We a r e grateful to D r s . G. Domokos and F. Calogero for useful d i s c u s s i o n s .

RefeTence$ 1) K. Johnson, Phys.Rev. 112 (1958) 1367. 2) B. Zumino, Nuovo Cimento 17 (1960) 517. 3) D.R.Yennie, S.C. Frautschi, H.Suura, Ann.Phys. 13 (1961) 379. 4) J. Houard and B. Jouvet, Nuovo Cimento 18 (1960) 496. 5) A.Salam, Nuovo Cimento 25 (1962) 224. 6) R.Delbourgo, Nuovo Cimento 27 (1963) 1431. 7) M. Cini, Report to the Sienna Conference 1963. 8) M. Gell-Mann, M.L. Goldberger, F.E. Low, E. Marx and F. Zachariasen, Phys.Rev. 133 (1964) B145. 9) G. Tiktopoulos, Phys.Rev. 131 (1963) 480; G. Tiktopoulos, Phys.Rev. 131 (1963) 2373. 10) M. Cassandro, M. Cini, to be published on Nuovo Cimento. 11) Y.Nambu, Nuovo Cimento 6 (1957) 1064. 12) Bateman Manuscript Project, vol. 1, pag. 248. 13) V.V.Sudakov, Soviet Phys. JETP 3 (1956) 65. 14) L. Bertocchi, E. Predazzi, A. Stanghellini, M. Tonin, Nuovo Cimento 27 (1963) 913. J. C. Polktnghorne, J. Math. Phys. 4 (1963) 503. 15) H. Lehman, K. Symanzik and W. Zimmermann, Nuovo Cimento 2 (1955) 425. 16) L. E. Evans, Nucl. Phys. 17 (1960) 163.

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15 July 1964