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Simple relation between the bare and physical mass of fermions
Volume 34B, n u m b e r 1
PHYSICS
SIMPLE THE
BARE
AND
LETTERS
RELATION PHYSICAL
l S J a n u a r ~ 19)1
BETWEEN MASS
OF
FERMIONS
$
P. L. F, H A B E R L E R * Physzcs Department, Duke Un~verstty, Durham, North Carolina 27706, USA Received 11 December 19?0
Results obtained m various field-theoretlc models oi electrodynamies and pseudoscalar meson-nucleon theory with m a s s l e s s pions strongly suggest that the bare mass M0 of the fermlon involved is simply related to its physic,a[ mass A/by M0 Z 2M. As a possible consequence of this relation It is suggested that the baryons ~s well as the electron and muon can be thought of as composlte objects.
M o t i v a t e d b y t h e r e c e n t g r o w i n g i n t e r e s t [1] i n t h e a p p l i c a h o n of c o m b i n e d s c a l e a n d c h i r a l l n v a r i a n c e to s t r o n g i n t e r a c t i o n s w e w e r e l e d to s t u d y t h e n e c e s s a r y c o n d i h o n s [2] f o r t h e s e s y m m e t r i e s in t h e f r a m e w o r k of q u a n t u m f i e l d t h e o r y . On t h e b a s i s of t h e K a m e f u c h i - K / i l l e n L e h m a n n r e p r e s e n t a h o n it w a s c o n j e c t u r e d t h a t t h e s e c o n d i t i o n s - n a m e l y t h e v a n i s h i n g of t h e b a r e m a s s e s - a r e r e l a t e d to t h e v a n i s h i n g of t h e wave function renormallzation constants. In t h i s l e t t e r we p r e s e n t t h e r e s u l t s of s o m e explicit calculations which strongly suggest that in f i e l d t h e o r i e s w i t h Y u k a w a c o u p l i n g , in t h e l i m i t of v a n i s h i n g b o s o n m a s s - a n d o n l y if t h e limit exists - the following relahon holds: M0 = Z 2M
or
(1) 6M
-= M - M
0 =M(1-Z2),
where respectively M and M 0 are the phymcal and bare masses and Z 2 is the wave funchon ren o r m a l i z a t i o n c o n s t a n t of t h e f e r m i o n r e v o l v e d . F o r e l e c t r o d y n a m i c s (QED) eq. (1) h o l d s in t h e F r i e d - Y e n n i e [3] (FY) g a u g e a n d in t h e r a d i a t i o n g a u g e in a s p e c i f i c f r a m e , a s it t u r n s out, b e c a u s e t h e s e g a u g e s a r e f r e e of i n f r a r e d d i v e r g e n c e s . On t h e o t h e r h a n d i t i s w e l l k n o w n t h a t 5MQ ED i s g a u g e i n v a r i a n t . T h e r e f o r e Z QED h a s to a s s u m e a s p e c i a l g a u g e i n v a r i a n t v a l u e a s discussed below. T o s u p p o r t o u r m a i n r e s u l t (eq. (1)) w e h a v e d e r i v e d it e x a c t l y in l o w e s t o r d e r a n d f o r t h e }- Supported in p a r t by ARO(D) and the National Science Foundation.
l e a d i n g t e r m s in all o r d e r s of p e r t u r b a h o n t h e o r y f o r A) p s e u d o s c a l a r m e s o n n u c l e o n h e l d t h e o r y w i t h m a s s l e s s n e u t r a l p i o n s a n d B) F Y e l e c t r o d y n a m i c s . W e h a v e a l s o d e r i v e d it in t h e Zachariasen-Thirring m o d e l [4]. T h e d e t a i l s of the calculahon will be pubhshed elsewhere. A s ~s w e l l - k n o w n , t h e b a r e m a s s of a f e r m l o n c a n b e r e p r e s e n t e d in t h e f o l l o w i n g w a y [5]: +oo
M0 = fdaar(a) -
oo
(2)
or oo
M0 = MZ 2 + fdaa[s(a) M
- s(-a)],
w h e r e r(a) a n d s(a) a r e u n d e r s t o o d to b e u n r e n o r m a l i z e d . T h e e x i s t e n c e of t h e i n t e g r a l i s a s s u m e d w h i c h a m o u n t s to i n t r o d u c i n g a c u t o f f if n e c e s s a r y . E q . (1) c a n b e f u l h l l e d n o n t r i v i a l l y if a n d o n l y if s(a)
(3)
= s(-a).
A) P s e u d o s c a l a r m e s o n n u c l e o n t h e o r y w i t h vamshing meson mass. (a) In l o w e s t o r d e r o n e o b t a i n s s (2)
= (g2/32n2) a2--M2>
[ a [a2
O,
(4)
f r o m w h i c h eq. (1) f o l l o w s . (b) G i v e n s ( 2 ) ( a ) , t h e L a g r a n g i a n of t h e Zachariasen-Thirrmg [4] m o d e l c a n b e e a s i l y written down * P r e s e n t a d d r e s s M a x - P l a n e k - I n s t i t u t f f i r Physlk und Astrophymk, F ~ h r i n g e r Ring 6, D-8 MUnchen 23, West Germany. 75
Volume 34B. number 1
PHYSICS L E T T E R S
J2 = i~(x)y4~/~ ~9(x) - M O ~ ( X ) ~ ( x ) +oo
+ f d a [ l g / ( a ) ~ 0~ ~(a) - a~(a)C,(a)] O(a 2 - M 2) -oo +oo
t u r n s out -- one can l i t e r a l l y take over eqs. (5) and (6) provided one makes the s u b s h t u t i o n lg ~ e and 1/327 2 ~ 3/16~ 2, thereby again e s t a b l i s h i n g eq. (1). In a d i r e c t calculation we have obtained the following r e s u l t for Z 2 to o r d e r e4:
- lg~(x) f d a t~(a)f(a), -
18January 1971
Z!4) = (5e4/2(16~2) 2) l o g 2 ( h 2 / M 2)
v,o
(9)
where
f2(a)
= g-a(a - 214)2 s(2)(a) O(a2 - M2).
(5)
F r o m eq. (5) and the s p e c t r a l r e p r e s e n t a t i o n of the i n v e r s e f e r m i o n p r o p a g a t o r one fmds +~
MO
: M + g 2 fdaf2(a)/(a
- M) = M Z 2 ,
-co
which a g r e e s wRh an independent calculation of -(5214(4)/214) [8]. Using the r e n o r m a l i z a t i o n group one can again extend eq. (9) to all o r d e r s of p e r t u r b a h o n theory. Now the coefficients P l l , P21, P31 a r e given r e s p e c t i v e l y by -3/16~ 2, -3/16~2 and -1/12~ 2 and we finally obtain (n = 1 , 2 . . . ) z(2n) = _SM(2n)/M
2
where
= (e2n/n")P21(-Pa1+P21)(-2P31 +P21).. • Z2 :
1 _g2_
a
_
+ (a+M)2
. (6) ×(-(n - 1)P31 +P21) l°gn(A2/M2) •
(C) Now we t u r n to the calculation of 5M and Z 2 in higher o r d e r s of p e r t u r b a t i o n theory. Eq. (1) was checked d i r e c t l y to o r d e r g4 for the leading l o g a r i t h m i c t e r m . After a long and t e dious calculation we found Z~4) = -( g4 / 2 ( 3 27r2)2 ) log2 ( A 2 / M2) , 5M(4) = -Z~4) M ,
(7)
A being a r e l a h v l s t i c cutoff. Independently we have obtained eq. (7) by using the powerful m e t h ods of the r e n o r m a l i z a t i o n group, thereby h e a v i ly r e l y i n g on the r e m a r k a b l e work of Astaud and Jouvet [6, 7]. In fact, using t h e i r r e s u l t s we a r e in the position to extend eq. (7) to all o r d e r s of p e r t u r b a t m n theory. Namely, Jouvet and Astaud [6, 7] have shown that the leading t e r m s of the v e r t e x , the f e r m i o n s e l f - e n e r g y and the vacuum p o l a r i z a t i o n can be calculated to all o r d e r s , provided one knows the coefficients of the l e a d ing l o g a r i t h m i c t e r m s of these functions in lowest o r d e r . In our case the coefficients a r e r e spectively given by P l l = 1/16~2, P21 = -1/32~2 and P31 = -1/8~2" Since the coefficients of the leading t e r m of 6M(2) and Z(2) a r e equal, we obtain ( n = l , 2 . . . ) :
z~2n)--
= - 5M(2n)/M = ( g 2 n / n ' ) P ~ l ( - 2 + 1)(-2.2 + 1 ) . . . × ( - 2 ( n - 1) + 1) logn(h2/M2).
(8)
B) F Y - e l e c t r o d y n a m i e s . In this ease -- as it 76
(10)
Concerning radiation gauge e l e c t r o d y n a m i c s s e v e r a l authors [9] pointed out that an i n f r a r e d d i v e r g e n c e - f r e e positive e x p r e s s i o n for the f e r mion propagator exists. Using t h e i r r e s u l t s , one easily e s t a b l i s h e s eq. (1) to lowest o r d e r . Higher o r d e r c o r r e c t i o n s a r e unknown but we c o n j e c t u r e that eq. (1) is true in radiation gauge, provided one chooses the right f r a m e . On the c o n t r a r y , m a Yukawa theory with s c a l a r coupling, eq. (1) is not likely to hold b e cause of the well-known i n f r a r e d d i v e r g e n c e s for vanishing boson m a s s . Indeed c o m p a r i n g the e x p r e s s i o n of Z2, and 5M in lowest o r d e r **
A2 z(2) = 1 - (f2/16~2) f d L 2~ 0 × l d z z 2 [ - M 2 ( 1 - z ) 2 + 4M2(1 - z ) + U 2 z + L z ] . J0 [M2(1 _~)2 + ~2~ + L~]2 '
A2
~M (2)
=
-
(~/16~ 2) ~
cr
fdL× 0
×f
1
dzz(1 + z)o
, (11)
0 M2( 1 - z ) 2 + ~ z + Lz ,,(2) one r e a d i l y sees that, while a2~ is i n f r a r e d d i v e r g e n t , 5M~) is not. Again using the r e n o r m a h z a t i o n group, one can show that eq. (1) does not hold for the leading t e r m s of Z2(7 and 5Mcr in all o r d e r s of p e r t u r b a t i o n theory. • * The subscmpt (~ denotes the scalar meson contribution.
Volume 34B, number 1
PHYSICS
L e t us now d i s c u s s the i m p h c a t i o n of o u r r e sults : 1) E q s . (6), (8) and (10) s t r o n g l y s u g g e s t that eq. (1) is t r u e in g e n e r a l , i n d e p e n d e n t of any perturbation theory considerations. 2) In o r d e r to be t r u e in g e n e r a l f o r Q E D eq. (1) can only hold m the f o r m z e r o e q u a l s z e r o , i . e . , MQ~ED= Z QED = 0 (we e x c l u d e the t r i v i a l c a s ~ z Q E D = 1). T h e v a l u e zQED= 0 is s u g g e s t e d by the f a c t that 6 M QED is gauge i n v a r i a n t . C l e a r l y m p e r t u r b a t i o n t h e o r y we h a v e no c o n v i n c i n g i n d i c a t i o n that z2QED e q u a l s z e r o . F u r t h e r m o r e in g a u g e s o t h e r than F Y o r r a d i a tion g a u g e eq. (1) d o e s not e v e n hold, b a s i c a l l y due to i n f r a r e d d i v e r g e n c e s . T h e r e f o r e we e x p e c t Z QED = 0 to be a p r o p e r t y of an e x a c t s o l u tion r a t h e s e n s e of p r o v i d i n g an e l g e n v a l u e e q u a t m n f o r the r e n o r m a l i z e d c o u p l i n g c o n s t a n t [10]. (See point 5) below.) 3) T a k i n g f o r g r a n t e d that Z QED e q u a l s z e r o we a r e n a t u r a l l y led to the c o m p o s i t e p a r t i c l e c o n d i t i o n [11] and t h i s t h e r e f o r e s u g g e s t s that not only s t r o n g l y i n t e r a c t i n g p a r t i c l e s but a l s o l e p t o n s (with the p o s s i b l e e x c e p t i o n of the n e u t r i n o s , b e c a u s e they h a v e no d i r e c t e l e c t r o m a g n e t i c m t e r a c t m n ) can be thought of as c o m p o s i t e o b l e c t s and should s h o w s o m e s t r u c t u r e at very h~gh e n e r g i e s . 4) Due to the W a r d i d e n t i t y the v e r t e x r e n o r m a l i z a t i o n c o n s t a n t z 1 Q E D e q u a l s Z QED. If Z QED v a n i s h e s , Z QED m u s t a l s o v a n i s h . T h i s c o n c l u s i o n a g r e e s with that of o t h e r n o n e l e c t r o m a g n e t t c m o d e l s [12] of the v e r t e x f u n c t i o n r e normalization constant. Furthermore, K~llen's [13] r e s u l t f o r the v e r t e x f u n c t i o n at l a r g e m o m e n t u m t r a n s f e r m i g h t a f t e r a l l be t r u e , if Z QED = 0, s u c h that m t h i s c a s e t h e r e no l o n g e r a r i s e s any d i f f i c u l t y [14] due to gauge d e p e n d e n c e of the r e s u l t . 5) If Z QED = 0, M o Q E D v a m s h e s a l s o [15]. In t h i s c a s e A s t a u d and J o u v e t [6, 7, 16] h a v e shown (without a s s u m i n g M = 0 ) that J o h n s o n ' s [17] r e s u l t on the a s y m p t o t i c b e h a v i o u r of the v a c u u m p o l a r i z a t i o n t e n s o r , n a m e l y that it only d i v e r g e s l i k e a s i n g l e l o g a r i t h m , h o l d s to a l l o r d e r s m p e r t u r b a t i o n t h e o r y . A l s o t h e y s h o w e d that the u n r e n o r m a l i z e d c h a n g e eo2/47r -= c~o h a s a f i x e d v a l u e ~of in t h i s c a s e , d e t e r m i n e d * by the r o o t of go(M~0/k2
= 0, ffof) -= ~o(°~of) = 0.
H0( ~, 8) -- b e i n g i d e n t i c a l to the f a m o u s G e l l Mann - Low f u n c t i o n ~I,(fl) [18] -- is the e x a c t k e r n e l of the i n t e g r a l e q u a t i o n f o r the photon • Subscript zero denotes the cas MoQED=0.
(12)
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18 Januam 1971
p r o p a g a t o r . T h e r e is then no l m p h e d c o n t r a d i c t i o n [19] if we h a v e s o f i x e d , s i n c e the r e n o r m a h z a t i o n c h a r g e a is f i x e d a l s o : n a m e l y by
zQED(~) = MQED(~) : 0 [ l S ] 6) Due to the e q u i v a l e n c e t h e o r e m [20] we c o n j e c t u r e that eq. (1) a l s o holds f o r the p s e u d o v e c t o r c o u p h n g t h e o r y , e s p e c i a l l y b e c a u s e the h m ~ t of v a m s h i n g b o s o n m a s s e x i s t s . F u r t h e r m o r e t h e r e is a d e e p e r p h y s i c a l r e a s o n f o r the e x i s t e n c e of t h i s l i m i t m p s e u d o s c a l a r t h e o r i e s -- in c o n t r a s t to s c a l a r t h e o r i e s as e x h i b i t e d by eq. i l l ) -- n a m e l y the c o n s e r v a t m n of c h i r a h t y [21]. A l t h o u g h the s i m p l e p s e u d o s c a l a r c o u p l i n g t h e o r y has no c h l r a l s y m m e t r y m the L a g r a n g i a n , it can a c q u i r e t h i s s y m m e t r y if and only i f Z 2 = 0 a n d M ¢ 012,22]. Therefore also in 75 t h e o r i e s we c o m e to the c o n c l u s i o n that Z 2 = 0, w h i c h is highly p l a u s i b l e in w e w of the s u c c e s s of the b o o t s t r a p h y p o t h e s i s . 7) F r o m the d i s c u s s i o n a b o v e we l e a r n that the b a r e m a s s of f e r m i o n s (leptons and b a r y o n s ) should a c t u a l l y v a m s h . T h i s l e a d s to the r e m a r k a b l e s i t u a t m n w h e r e the b a r e l e p t o n s and b a r y o n s cannot be d i s t i n g u i s h e d $ and to the p o s s i b i l i t y of an u n d e r l y i n g f u n d a m e n t a l m a t t e r f i e l d [24], which would r e p r e s e n t t h e s e m a s s l e s s degenerate bare fermlons. T h e a u t h o r ,s v e r y m u c h i n d e b t e d to P r o f e s s o r L. E v a n s f o r s o m e v e r y i l l u m i n a t i n g d , s c u s s i o n s and f o r c m t l c a l l y r e a d i n g the m a n u s c r i p t . He a l s o thanks P r o f e s s o r B. J o u v e t f o r m a k i n g his i n t e r e s t i n g p a p e r a v a i l a b l e p r i o r to p u b h c a t i o n and P r o f e s s o r R. J a c k i w f o r u s e f u l d i s c u s s , o n s in B o u l d e r . See m this connection a similar observation by Nambu and Jona-Lasmlo [23]
References [1] p. Carruthers, Nucl. Phys., to be pubhshed. [2] P. L. F. Haberler, Symposium on Conformal and de Sitter groups, Boulder (1970). [3] H. Freed and D. R. Yennm, Phys. Rev. 112 (1958) 1391. [4] F. Zacharlasen, Phys. Rev. 121 (1961) 1851' W. Thlrmng, Phys. Rev. 126 (1962) 1209. [5] W. Thlrrlng, Principles of quantum electrodynamlcs (Academic P r e s s , Inc., New York, 1958). [6] M. Astaud and B. Jouvet, Nuovo Cimento 63A (1969) 5. [7] M.Astaud, Nuovo Cimento 66A (1970) 111; d i s s e r tation (June 1969), unpubhshed. [8] Z. Bialynicka-Blrula, Bull. Acad. Pol. Sc. Math. 13 (1965) 139. [9] I. Bialynlckl-Birula, Phys. Rev. 166 (1968) 1505, and references thereto. [10] P. L. F. Haberler, to be pubhshed. [11] B. Jouvet, Nuovo Clmento 5 (1957) 1. 77
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[12] N.G. D e s h p a n d e and S . A . B l u d m a n , P h } s . Rev. 143 (1966) 1239. [13] G. Khllen, Kgl D a n s k e V l d e n s k . Selsk. M a t - F s . s . Medd 27 No. 12 (1954). [14] B Z u m m o , J o u r n . Math P h y s . I (1960) 1. [15] W. T h l r r m g , P h y s . L e t t e r s 4 (1963) 167: K. J o h n s o n , to be p u b h s h e d . [16] B. J o u v e t , College de F r a n c e , p r e p r m t (July, 1970). [17] M. B a k e r and K. J o h n s o n , P h 3 s . R e V . 183 0 9 6 9 ) 1292. [18] M. G e l l - M a n n and F. Low, P h y s Rev. 95 (1954) 1300.
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[19] J . D . B ] o r k e n and S. D. Drell, R e l a t l v l s t m q u a n t u m f i e l d s ( M c G r a w - H i l l . 1966). [20] F . J . D y s o n , P h y s . Rev. 73 (1948) 929. [21] Y . N a m b u a n d D . L u r l e , P h y s . Rev 125 (1962) 1429 [22] P. L. F. H a b e r l e r , to be p u b h s h e d , [23] Y. N a m b u and G. J o n a - L a s l m o , P h y s . Rev. 122 (1961) 345" 124 (1961) 246. [24] W. H e l s e n b e r g , I n t r o d u c t i o n to the unified field t h e o r y of e l e m e n t a r y p a r t i c l e s (Wiley Ltd , London, 1966).
PHYSICS
SIMPLE THE
BARE
AND
LETTERS
RELATION PHYSICAL
l S J a n u a r ~ 19)1
BETWEEN MASS
OF
FERMIONS
$
P. L. F, H A B E R L E R * Physzcs Department, Duke Un~verstty, Durham, North Carolina 27706, USA Received 11 December 19?0
Results obtained m various field-theoretlc models oi electrodynamies and pseudoscalar meson-nucleon theory with m a s s l e s s pions strongly suggest that the bare mass M0 of the fermlon involved is simply related to its physic,a[ mass A/by M0 Z 2M. As a possible consequence of this relation It is suggested that the baryons ~s well as the electron and muon can be thought of as composlte objects.
M o t i v a t e d b y t h e r e c e n t g r o w i n g i n t e r e s t [1] i n t h e a p p l i c a h o n of c o m b i n e d s c a l e a n d c h i r a l l n v a r i a n c e to s t r o n g i n t e r a c t i o n s w e w e r e l e d to s t u d y t h e n e c e s s a r y c o n d i h o n s [2] f o r t h e s e s y m m e t r i e s in t h e f r a m e w o r k of q u a n t u m f i e l d t h e o r y . On t h e b a s i s of t h e K a m e f u c h i - K / i l l e n L e h m a n n r e p r e s e n t a h o n it w a s c o n j e c t u r e d t h a t t h e s e c o n d i t i o n s - n a m e l y t h e v a n i s h i n g of t h e b a r e m a s s e s - a r e r e l a t e d to t h e v a n i s h i n g of t h e wave function renormallzation constants. In t h i s l e t t e r we p r e s e n t t h e r e s u l t s of s o m e explicit calculations which strongly suggest that in f i e l d t h e o r i e s w i t h Y u k a w a c o u p l i n g , in t h e l i m i t of v a n i s h i n g b o s o n m a s s - a n d o n l y if t h e limit exists - the following relahon holds: M0 = Z 2M
or
(1) 6M
-= M - M
0 =M(1-Z2),
where respectively M and M 0 are the phymcal and bare masses and Z 2 is the wave funchon ren o r m a l i z a t i o n c o n s t a n t of t h e f e r m i o n r e v o l v e d . F o r e l e c t r o d y n a m i c s (QED) eq. (1) h o l d s in t h e F r i e d - Y e n n i e [3] (FY) g a u g e a n d in t h e r a d i a t i o n g a u g e in a s p e c i f i c f r a m e , a s it t u r n s out, b e c a u s e t h e s e g a u g e s a r e f r e e of i n f r a r e d d i v e r g e n c e s . On t h e o t h e r h a n d i t i s w e l l k n o w n t h a t 5MQ ED i s g a u g e i n v a r i a n t . T h e r e f o r e Z QED h a s to a s s u m e a s p e c i a l g a u g e i n v a r i a n t v a l u e a s discussed below. T o s u p p o r t o u r m a i n r e s u l t (eq. (1)) w e h a v e d e r i v e d it e x a c t l y in l o w e s t o r d e r a n d f o r t h e }- Supported in p a r t by ARO(D) and the National Science Foundation.
l e a d i n g t e r m s in all o r d e r s of p e r t u r b a h o n t h e o r y f o r A) p s e u d o s c a l a r m e s o n n u c l e o n h e l d t h e o r y w i t h m a s s l e s s n e u t r a l p i o n s a n d B) F Y e l e c t r o d y n a m i c s . W e h a v e a l s o d e r i v e d it in t h e Zachariasen-Thirring m o d e l [4]. T h e d e t a i l s of the calculahon will be pubhshed elsewhere. A s ~s w e l l - k n o w n , t h e b a r e m a s s of a f e r m l o n c a n b e r e p r e s e n t e d in t h e f o l l o w i n g w a y [5]: +oo
M0 = fdaar(a) -
oo
(2)
or oo
M0 = MZ 2 + fdaa[s(a) M
- s(-a)],
w h e r e r(a) a n d s(a) a r e u n d e r s t o o d to b e u n r e n o r m a l i z e d . T h e e x i s t e n c e of t h e i n t e g r a l i s a s s u m e d w h i c h a m o u n t s to i n t r o d u c i n g a c u t o f f if n e c e s s a r y . E q . (1) c a n b e f u l h l l e d n o n t r i v i a l l y if a n d o n l y if s(a)
(3)
= s(-a).
A) P s e u d o s c a l a r m e s o n n u c l e o n t h e o r y w i t h vamshing meson mass. (a) In l o w e s t o r d e r o n e o b t a i n s s (2)
= (g2/32n2) a2--M2>
[ a [a2
O,
(4)
f r o m w h i c h eq. (1) f o l l o w s . (b) G i v e n s ( 2 ) ( a ) , t h e L a g r a n g i a n of t h e Zachariasen-Thirrmg [4] m o d e l c a n b e e a s i l y written down * P r e s e n t a d d r e s s M a x - P l a n e k - I n s t i t u t f f i r Physlk und Astrophymk, F ~ h r i n g e r Ring 6, D-8 MUnchen 23, West Germany. 75
Volume 34B. number 1
PHYSICS L E T T E R S
J2 = i~(x)y4~/~ ~9(x) - M O ~ ( X ) ~ ( x ) +oo
+ f d a [ l g / ( a ) ~ 0~ ~(a) - a~(a)C,(a)] O(a 2 - M 2) -oo +oo
t u r n s out -- one can l i t e r a l l y take over eqs. (5) and (6) provided one makes the s u b s h t u t i o n lg ~ e and 1/327 2 ~ 3/16~ 2, thereby again e s t a b l i s h i n g eq. (1). In a d i r e c t calculation we have obtained the following r e s u l t for Z 2 to o r d e r e4:
- lg~(x) f d a t~(a)f(a), -
18January 1971
Z!4) = (5e4/2(16~2) 2) l o g 2 ( h 2 / M 2)
v,o
(9)
where
f2(a)
= g-a(a - 214)2 s(2)(a) O(a2 - M2).
(5)
F r o m eq. (5) and the s p e c t r a l r e p r e s e n t a t i o n of the i n v e r s e f e r m i o n p r o p a g a t o r one fmds +~
MO
: M + g 2 fdaf2(a)/(a
- M) = M Z 2 ,
-co
which a g r e e s wRh an independent calculation of -(5214(4)/214) [8]. Using the r e n o r m a l i z a t i o n group one can again extend eq. (9) to all o r d e r s of p e r t u r b a h o n theory. Now the coefficients P l l , P21, P31 a r e given r e s p e c t i v e l y by -3/16~ 2, -3/16~2 and -1/12~ 2 and we finally obtain (n = 1 , 2 . . . ) z(2n) = _SM(2n)/M
2
where
= (e2n/n")P21(-Pa1+P21)(-2P31 +P21).. • Z2 :
1 _g2_
a
_
+ (a+M)2
. (6) ×(-(n - 1)P31 +P21) l°gn(A2/M2) •
(C) Now we t u r n to the calculation of 5M and Z 2 in higher o r d e r s of p e r t u r b a t i o n theory. Eq. (1) was checked d i r e c t l y to o r d e r g4 for the leading l o g a r i t h m i c t e r m . After a long and t e dious calculation we found Z~4) = -( g4 / 2 ( 3 27r2)2 ) log2 ( A 2 / M2) , 5M(4) = -Z~4) M ,
(7)
A being a r e l a h v l s t i c cutoff. Independently we have obtained eq. (7) by using the powerful m e t h ods of the r e n o r m a l i z a t i o n group, thereby h e a v i ly r e l y i n g on the r e m a r k a b l e work of Astaud and Jouvet [6, 7]. In fact, using t h e i r r e s u l t s we a r e in the position to extend eq. (7) to all o r d e r s of p e r t u r b a t m n theory. Namely, Jouvet and Astaud [6, 7] have shown that the leading t e r m s of the v e r t e x , the f e r m i o n s e l f - e n e r g y and the vacuum p o l a r i z a t i o n can be calculated to all o r d e r s , provided one knows the coefficients of the l e a d ing l o g a r i t h m i c t e r m s of these functions in lowest o r d e r . In our case the coefficients a r e r e spectively given by P l l = 1/16~2, P21 = -1/32~2 and P31 = -1/8~2" Since the coefficients of the leading t e r m of 6M(2) and Z(2) a r e equal, we obtain ( n = l , 2 . . . ) :
z~2n)--
= - 5M(2n)/M = ( g 2 n / n ' ) P ~ l ( - 2 + 1)(-2.2 + 1 ) . . . × ( - 2 ( n - 1) + 1) logn(h2/M2).
(8)
B) F Y - e l e c t r o d y n a m i e s . In this ease -- as it 76
(10)
Concerning radiation gauge e l e c t r o d y n a m i c s s e v e r a l authors [9] pointed out that an i n f r a r e d d i v e r g e n c e - f r e e positive e x p r e s s i o n for the f e r mion propagator exists. Using t h e i r r e s u l t s , one easily e s t a b l i s h e s eq. (1) to lowest o r d e r . Higher o r d e r c o r r e c t i o n s a r e unknown but we c o n j e c t u r e that eq. (1) is true in radiation gauge, provided one chooses the right f r a m e . On the c o n t r a r y , m a Yukawa theory with s c a l a r coupling, eq. (1) is not likely to hold b e cause of the well-known i n f r a r e d d i v e r g e n c e s for vanishing boson m a s s . Indeed c o m p a r i n g the e x p r e s s i o n of Z2, and 5M in lowest o r d e r **
A2 z(2) = 1 - (f2/16~2) f d L 2~ 0 × l d z z 2 [ - M 2 ( 1 - z ) 2 + 4M2(1 - z ) + U 2 z + L z ] . J0 [M2(1 _~)2 + ~2~ + L~]2 '
A2
~M (2)
=
-
(~/16~ 2) ~
cr
fdL× 0
×f
1
dzz(1 + z)o
, (11)
0 M2( 1 - z ) 2 + ~ z + Lz ,,(2) one r e a d i l y sees that, while a2~ is i n f r a r e d d i v e r g e n t , 5M~) is not. Again using the r e n o r m a h z a t i o n group, one can show that eq. (1) does not hold for the leading t e r m s of Z2(7 and 5Mcr in all o r d e r s of p e r t u r b a t i o n theory. • * The subscmpt (~ denotes the scalar meson contribution.
Volume 34B, number 1
PHYSICS
L e t us now d i s c u s s the i m p h c a t i o n of o u r r e sults : 1) E q s . (6), (8) and (10) s t r o n g l y s u g g e s t that eq. (1) is t r u e in g e n e r a l , i n d e p e n d e n t of any perturbation theory considerations. 2) In o r d e r to be t r u e in g e n e r a l f o r Q E D eq. (1) can only hold m the f o r m z e r o e q u a l s z e r o , i . e . , MQ~ED= Z QED = 0 (we e x c l u d e the t r i v i a l c a s ~ z Q E D = 1). T h e v a l u e zQED= 0 is s u g g e s t e d by the f a c t that 6 M QED is gauge i n v a r i a n t . C l e a r l y m p e r t u r b a t i o n t h e o r y we h a v e no c o n v i n c i n g i n d i c a t i o n that z2QED e q u a l s z e r o . F u r t h e r m o r e in g a u g e s o t h e r than F Y o r r a d i a tion g a u g e eq. (1) d o e s not e v e n hold, b a s i c a l l y due to i n f r a r e d d i v e r g e n c e s . T h e r e f o r e we e x p e c t Z QED = 0 to be a p r o p e r t y of an e x a c t s o l u tion r a t h e s e n s e of p r o v i d i n g an e l g e n v a l u e e q u a t m n f o r the r e n o r m a l i z e d c o u p l i n g c o n s t a n t [10]. (See point 5) below.) 3) T a k i n g f o r g r a n t e d that Z QED e q u a l s z e r o we a r e n a t u r a l l y led to the c o m p o s i t e p a r t i c l e c o n d i t i o n [11] and t h i s t h e r e f o r e s u g g e s t s that not only s t r o n g l y i n t e r a c t i n g p a r t i c l e s but a l s o l e p t o n s (with the p o s s i b l e e x c e p t i o n of the n e u t r i n o s , b e c a u s e they h a v e no d i r e c t e l e c t r o m a g n e t i c m t e r a c t m n ) can be thought of as c o m p o s i t e o b l e c t s and should s h o w s o m e s t r u c t u r e at very h~gh e n e r g i e s . 4) Due to the W a r d i d e n t i t y the v e r t e x r e n o r m a l i z a t i o n c o n s t a n t z 1 Q E D e q u a l s Z QED. If Z QED v a n i s h e s , Z QED m u s t a l s o v a n i s h . T h i s c o n c l u s i o n a g r e e s with that of o t h e r n o n e l e c t r o m a g n e t t c m o d e l s [12] of the v e r t e x f u n c t i o n r e normalization constant. Furthermore, K~llen's [13] r e s u l t f o r the v e r t e x f u n c t i o n at l a r g e m o m e n t u m t r a n s f e r m i g h t a f t e r a l l be t r u e , if Z QED = 0, s u c h that m t h i s c a s e t h e r e no l o n g e r a r i s e s any d i f f i c u l t y [14] due to gauge d e p e n d e n c e of the r e s u l t . 5) If Z QED = 0, M o Q E D v a m s h e s a l s o [15]. In t h i s c a s e A s t a u d and J o u v e t [6, 7, 16] h a v e shown (without a s s u m i n g M = 0 ) that J o h n s o n ' s [17] r e s u l t on the a s y m p t o t i c b e h a v i o u r of the v a c u u m p o l a r i z a t i o n t e n s o r , n a m e l y that it only d i v e r g e s l i k e a s i n g l e l o g a r i t h m , h o l d s to a l l o r d e r s m p e r t u r b a t i o n t h e o r y . A l s o t h e y s h o w e d that the u n r e n o r m a l i z e d c h a n g e eo2/47r -= c~o h a s a f i x e d v a l u e ~of in t h i s c a s e , d e t e r m i n e d * by the r o o t of go(M~0/k2
= 0, ffof) -= ~o(°~of) = 0.
H0( ~, 8) -- b e i n g i d e n t i c a l to the f a m o u s G e l l Mann - Low f u n c t i o n ~I,(fl) [18] -- is the e x a c t k e r n e l of the i n t e g r a l e q u a t i o n f o r the photon • Subscript zero denotes the cas MoQED=0.
(12)
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18 Januam 1971
p r o p a g a t o r . T h e r e is then no l m p h e d c o n t r a d i c t i o n [19] if we h a v e s o f i x e d , s i n c e the r e n o r m a h z a t i o n c h a r g e a is f i x e d a l s o : n a m e l y by
zQED(~) = MQED(~) : 0 [ l S ] 6) Due to the e q u i v a l e n c e t h e o r e m [20] we c o n j e c t u r e that eq. (1) a l s o holds f o r the p s e u d o v e c t o r c o u p h n g t h e o r y , e s p e c i a l l y b e c a u s e the h m ~ t of v a m s h i n g b o s o n m a s s e x i s t s . F u r t h e r m o r e t h e r e is a d e e p e r p h y s i c a l r e a s o n f o r the e x i s t e n c e of t h i s l i m i t m p s e u d o s c a l a r t h e o r i e s -- in c o n t r a s t to s c a l a r t h e o r i e s as e x h i b i t e d by eq. i l l ) -- n a m e l y the c o n s e r v a t m n of c h i r a h t y [21]. A l t h o u g h the s i m p l e p s e u d o s c a l a r c o u p l i n g t h e o r y has no c h l r a l s y m m e t r y m the L a g r a n g i a n , it can a c q u i r e t h i s s y m m e t r y if and only i f Z 2 = 0 a n d M ¢ 012,22]. Therefore also in 75 t h e o r i e s we c o m e to the c o n c l u s i o n that Z 2 = 0, w h i c h is highly p l a u s i b l e in w e w of the s u c c e s s of the b o o t s t r a p h y p o t h e s i s . 7) F r o m the d i s c u s s i o n a b o v e we l e a r n that the b a r e m a s s of f e r m i o n s (leptons and b a r y o n s ) should a c t u a l l y v a m s h . T h i s l e a d s to the r e m a r k a b l e s i t u a t m n w h e r e the b a r e l e p t o n s and b a r y o n s cannot be d i s t i n g u i s h e d $ and to the p o s s i b i l i t y of an u n d e r l y i n g f u n d a m e n t a l m a t t e r f i e l d [24], which would r e p r e s e n t t h e s e m a s s l e s s degenerate bare fermlons. T h e a u t h o r ,s v e r y m u c h i n d e b t e d to P r o f e s s o r L. E v a n s f o r s o m e v e r y i l l u m i n a t i n g d , s c u s s i o n s and f o r c m t l c a l l y r e a d i n g the m a n u s c r i p t . He a l s o thanks P r o f e s s o r B. J o u v e t f o r m a k i n g his i n t e r e s t i n g p a p e r a v a i l a b l e p r i o r to p u b h c a t i o n and P r o f e s s o r R. J a c k i w f o r u s e f u l d i s c u s s , o n s in B o u l d e r . See m this connection a similar observation by Nambu and Jona-Lasmlo [23]
References [1] p. Carruthers, Nucl. Phys., to be pubhshed. [2] P. L. F. Haberler, Symposium on Conformal and de Sitter groups, Boulder (1970). [3] H. Freed and D. R. Yennm, Phys. Rev. 112 (1958) 1391. [4] F. Zacharlasen, Phys. Rev. 121 (1961) 1851' W. Thlrmng, Phys. Rev. 126 (1962) 1209. [5] W. Thlrrlng, Principles of quantum electrodynamlcs (Academic P r e s s , Inc., New York, 1958). [6] M. Astaud and B. Jouvet, Nuovo Cimento 63A (1969) 5. [7] M.Astaud, Nuovo Cimento 66A (1970) 111; d i s s e r tation (June 1969), unpubhshed. [8] Z. Bialynicka-Blrula, Bull. Acad. Pol. Sc. Math. 13 (1965) 139. [9] I. Bialynlckl-Birula, Phys. Rev. 166 (1968) 1505, and references thereto. [10] P. L. F. Haberler, to be pubhshed. [11] B. Jouvet, Nuovo Clmento 5 (1957) 1. 77
V o l u m e 34B, n u m b e r 1
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[12] N.G. D e s h p a n d e and S . A . B l u d m a n , P h } s . Rev. 143 (1966) 1239. [13] G. Khllen, Kgl D a n s k e V l d e n s k . Selsk. M a t - F s . s . Medd 27 No. 12 (1954). [14] B Z u m m o , J o u r n . Math P h y s . I (1960) 1. [15] W. T h l r r m g , P h y s . L e t t e r s 4 (1963) 167: K. J o h n s o n , to be p u b h s h e d . [16] B. J o u v e t , College de F r a n c e , p r e p r m t (July, 1970). [17] M. B a k e r and K. J o h n s o n , P h 3 s . R e V . 183 0 9 6 9 ) 1292. [18] M. G e l l - M a n n and F. Low, P h y s Rev. 95 (1954) 1300.
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[19] J . D . B ] o r k e n and S. D. Drell, R e l a t l v l s t m q u a n t u m f i e l d s ( M c G r a w - H i l l . 1966). [20] F . J . D y s o n , P h y s . Rev. 73 (1948) 929. [21] Y . N a m b u a n d D . L u r l e , P h y s . Rev 125 (1962) 1429 [22] P. L. F. H a b e r l e r , to be p u b h s h e d , [23] Y. N a m b u and G. J o n a - L a s l m o , P h y s . Rev. 122 (1961) 345" 124 (1961) 246. [24] W. H e l s e n b e r g , I n t r o d u c t i o n to the unified field t h e o r y of e l e m e n t a r y p a r t i c l e s (Wiley Ltd , London, 1966).