- Email: [email protected]
Finiteness of radiative corrections in all orders to μ-decay
-~
Nuclear Physics B7 (1968) 160-166. North-Holland Publ. Comp., Amsterdam
FINITENESS
OF
IN A L L
RADIATIVE
ORDERS
TO
CORRECTIONS y-DECAY
P. VAN NIEUWENHUIZEN Instituut voor Theoretische Fysica, Maliesingel 23, Utrecht, The Netherlands
Received 6March 1968
Abstract: The well known result that second order radiative corrections to y-decay are finite is generalized to all orders.
1. INTRODUCTION Recently, Cabibbo et al. [1] d e r i v e d conditions on the v e c t o r and axialv e c t o r c u r r e n t by postulating that E - d e c a y for the pion in o r d e r e 2 be finite. Since the q u a r k model does not fulfil these conditions, they r e j e c t it in favour of another model (in which e.g. all e l e m e n t a r y constituents have integ r a l charge). It is natural, then, to a s k if the t h e o r y is finite in any o r d e r of radiative c o r r e c t i o n s . In t~-decay one can check whether all radiative c o r r e c t i o n s a r e finite or not, since here no c o m p l i c a t i o n s of s t r o n g i n t e r a c t i o n s a r e p r e s e n t . It is well known that the radiative c o r r e c t i o n s in s e c ond o r d e r to t~-decay a r e finite. They have been calculated and the r e s u l t is that the weak coupling constant is i n c r e a s e d by 0.2% (ref. [2]). It is also well known that the i n f r a r e d d i v e r g e n c e s of inner r a d i a t i v e c o r r e c t i o n s to y - d e c a y cancel in e a c h o r d e r against the i n f r a r e d d i v e r g e n c e s f r o m i n t e r nal B r e m s s t r a h l u n g [3]. It will be shown here that the u l t r a v i o l e t d i v e r g e n c e s which a r i s e in t~-decay cancel in each o r d e r . This m e a n s that it is p o s s i b l e to calculate the radiative c o r r e c t i o n s to ~ - d e c a y to a r b i t r a r y o r der. Or p h r a s e d differently: t h e r e is no need to r e n o r m a l i z e the weak coupling constant since adding e l e c t r o d y n a m i c s to f i r s t o r d e r weak i n t e r a c t i o n s does not introduce d i v e r g e n c e s . In sect. 2 we show that in o r d e r to p r o v e the f i n i t e n e s s of t~-decay in all o r d e r radiative c o r r e c t i o n s , it is sufficient to p r o v e that the s u m of all p r o p e r weak v e r t e x p a r t s equals a p r o d u c t of a c e r t a i n infinite constant (which cancels the other u l t r a v i o l e t d i v e r g e n c e s that a r e present) and a finite function. In sect. 3 we write down a B e t h e - S a l p e t e r type integral equation f o r the sum of all p r o p e r weak v e r t e x p a r t s and then show that it is sufficient for our p u r p o s e s to p r o v e : (i) that Zl(tt) - Zl(e) is finite (~ and e denote muon and electron) and (ii) a r e l a t i o n f o r the h o m o g e n e o u s p a r t of the integral equation. T h e s e r e l a t i o n s a r e p r o v e d in sect. 4. In sect. 5 we
RADIATIVE CORRECTIONS
161
d i s c u s s the r e n o r m a l i z a t i o n of q u a n t u m e l e c t r o d y n a m i c s a n d c o n c l u d e t h a t i t g o e s on, e v e n w h e n two ( o r m o r e ) t y p e s of f e r m i o n s a r e p r e s e n t t h a t i n t e r a c t o n l y t h r o u g h t h e e l e c t r o m a g n e t i c f i e l d , a r e s u l t t h a t i s u s e d in t h e . p r e c e d i n g s e c t i o n s . S i n c e w e o n l y n e e d t h e V - A c h a r a c t e r of t h e w e a k i n t e r a c t i o n s a f t e r t h e F i e r z t r a n s f o r m a t i o n in s e c t . 2, a n d a s s u m e t h e l e f t - h a n d e d n e s s of n e u t r i n o s , o u r r e s u l t s a p p l y to an i n t e r a c t i o n of t h e V - A f o r m f o r t h e o r i g i n a l i n t e r a c t i o n , to w h i c h c a n b e a d d e d a t e r m of t h e f o r m [C~e (1 +75)~PVe] [~u~ t (1 - 75)~/~] •
2. A C O N D I T I O N ON T H E W E A K V E R T E X T h e s u m M of a l l F e y n m a n d i a g r a m s r e p r e s e n t i n g the r a d i a t i v e c o r r e c t i o n s to # - d e c a y c a n b e f a c t o r i z e d into a p r o d u c t of f o u r f a c t o r s : (1)
M : T e F v Tg L v
w h e r e T e i s the s u m of a l l s e l f - e n e r g y t e r m s of the e x t e r n a l e l e c t r o n l i n e a n d g i v e s r i s e to the w a v e f u n c t i o n r e n o r m a l i z a t i o n of the e l e c t r o n , T ~ d o e s the s a m e f o r the m u o n , F u i s t h e s u m of a l l p r o p e r w e a k v e r t e x p a r t s and Lv is the neutrino current (times a factor g / R ) . We postpone a disc u s s i o n of B r e m s s t r a h l u n g in g - d e c a y u n t i l the e n d of s e c t . 4. F i g . l a i l l u s t r a t e s t h i s , in f i g . l b we g i v e a t e r m o c c u r r i n g in F u. Note t h a t t h e d i a g r a m s c o r r e s p o n d i n g to the t e r m s in F u h a v e no f r e e e l e c t r o n o r m u o n l i n e s . We c a n f o r g e t a b o u t the n e u t r i n o l i n e s , s i n c e we h a v e w r i t t e n the V - A i n t e r a c t i o n H a m i l t o n i a n b y a F i e r z t r a n s f o r m a t i o n in t h e f o r m :
fd3x~g(x) : ~ f
d~ [~vu(X) 7v (1 +75)~Pve(X)][~e(X) 7v (1 + 7 5 ) ~ ( x ) ] . (2)
t I t .g
a
b Fig. 1.
162
P. VAN NIEUWENHUIZEN
A s in q u a n t u m e l e c t r o d y n a m i c s T/~ = ¢ Z - 2 - ~ ( s e e s e c t . 5). If we c a n s h o w t h a t
[
r
~//~(0) a n d T e = ~ - ) ~ e ( 0 )
7
(3/
= LZ2(e)Z2(~)I
w i t h f i n i t e F v, t h e n i t f o l l o w s upon i n s e r t i n g eq. (3) in eq. (1) t h a t the r a d i a t i v e c o r r e c t i o n s to ~ - d e c a y a r e f i n i t e .
3. AN I N T E G R A L E Q U A T I O N F O R T H E W E A K V E R T E X T h e s u m of a l l p r o p e r , u n r e n o r m a l i z e d , w e a k v e r t e x p a r t s F ~ c a n b e w r i t t e n g r a p h i c a l l y in the i n t e g r a l e q u a t i o n s h o w n in fig. 2, w h e r e S'e a n d S/~ a r e the u n r e n o r m a l i z e d p r o p a g a t o r s .
~f
~e
J
i
I
L
0
L
I i
~I
zip
Fig. 2. T h e s q u a r e b l a c k b o x d e n o t e s the s u m of a l l t h o s e d i a g r a m s w i t h one e l e c t r o n g o i n g out a t fi a n d one m u o n g o i n g in at a (with f o u r v e c t o r s p e a n d p/~) a n d one e l e c t r o n g o i n g in a t 5 a n d one m u o n g o i n g out a t ~ (four v e c t o r s now p e + s a n d p P + s) t h a t c a n n o t b e c o m e d i s c o n n e c t e d b y c u t t i n g one e l e c t r o n a n d one m u o n l i n e . In fig. 3 we g i v e s o m e of the p e r t i n e n t a n d n o n pertinent diagrams.
NaT
INCLV~E~
~
+ • - •
Fig. 3.
RADIATIVE CORRECTIONS
163
In formula:
Ha (pe,pp) = [~v (I +y5)]~a +
F lj
fd4s [S'e (pe+ s)]i 0 FV0e(pe +s, pP +s) T
× [S# (p~ + s)]Ey K~sy a (s, pe, p#)
(4a)
or schematically:
(5)
Fv =Yv(l+Y5) +~f S'# KS' -e F v "
A simple example is given in fig. 4. This diagram does not contain divergences for fixed s, and its contribution I to the S - m a t r i x can be written as I = [{~e (0)]~r;a (spec)[~# (0)] a [{~v# (0)~v(1 + Y5)~Ve(0)]~2 54(~ u _pe _ Q) with Q is the sum of the neutrino f o u r - v e c t o r s and where FV(spec) is a t e r m in F v and can be written as in (4a):
F v ( s p e c ) : f d 4 s e 4.
1 Ifd4 t -1 .yo.l~ 5 (2~) 4 Y P ~ - l ~ - ime
Ha
× L_~_+~_ime]~eF -1 [rv (1 +~5)]0 e I
x Y# ~
-1
-
-i
imp
(t- s) 2
_---~mejEr
-~] ya
(4b)
The f i r s t and last t e r m in (4b) in square b r a c k e t s together form the (here finite) contribution to K, see sect. 5 and eq. (4b).
Fig. 4.
164
l?. VAN NIEUWENHUIZEN
In sect. 5 it is shown that
(6)
~: : [Z2(e)Z2(tt)] -1
with finite K. By finite we always m e a n independent of the cut-off introduced by m e a n s of the method of r e g u l a t o r s [5]. We then find the following i n t e g r a l equation for ~-~_ [Z2(e)Z2(/~)]½ r v , m
ru
1
-
-
= [Z2(e)Z2(/~)]~ 7u(1 +Y5) + f St.t K S e F v ,
(7)
where S~ = Z2(~)S-,, and S'e = Z2(e ) S-e" (Even in o r d e r e2,~Z2(e) ¢ Z2(/~)). We w i l l show in sect. 4 that in each o r der: ± (i) [Z2(tt)Z2(e)]2 - Z2(e ) is finite ,
(8a)
(ii) f S e K S e ~-V f ~ - K S ~-D is finite e/1 e
(8b)
(K"e equals__ K but the ingoing lepton is an electron instead of a muon and ~eu equals F u in the s a m e way). Then, all that r e m a i n s to be shown is the f i n i t e n e s s of QW: QW = Z2(E)Tu(1 +75) + f S e K e S e F ev
(7a)
If 7u(1 +75) w e r e r e p l a c e d by 7u in (Ta) this would be p u r e e l e c t r o d y n a m i c s and the f i n i t e n e s s of this e x p r e s s i o n has b e e n p r o v e d [4]. The d i v e r g e n t p a r t in the i n t e g r a l with yu(1 +75) r e p l a c e d by 7u75 is however the s a m e a s that belonging to the p u r e quantum e l e c t r o d y n a m i c a l c a s e . F o r , the effect of bringing 75 to the right of the integrand in (7a) is a r e p l a c e m e n t of m e by - m e in one f r e e e l e c t r o n - l i n e only. And the d i v e r g e n t p a r t of this e x p r e s sion is the s a m e as that of the s a m e e x p r e s s i o n with +m e (and still 75 at the end of the integrand). This c o m e s about b e c a u s e the highest p o s s i b l e d i v e r gence of the graph is l o g a r i t h m i c , and differentiation to m e r e n d e r s it finite (for d e t a i l s see sect. 4). So the d i v e r g e n t p a r t s in (7a) cancel, and all we have to do, it to p r o v e the r e l a t i o n s (8a) and (8b).
4. A RELATION BETWEEN Z - F A C T O R S In this section we p r o v e the r e l a t i o n s (8a) and (8b). Equivalent to (8a) is: Z2(/~) - Z2(e) is finite o r , since Z 1 = Z 2 it is to be shown that ZI(/I ) - Z l ( e ) is finite. We will show this by p r o v i n g that AZ(rn) = Z l ( m ) - Z l ( e ) is d i f f e r entiable. Then, since AZ(rne) = O, we will have p r o v e d (8a). By Z l ( m ) we m e a n Z I ( ~ ) with only in the incoming and outgoing muon line the m a s s m/~
RADIATIVE CORRECTIONS
165
Fig. 5. by m. In fig. 5 we i l l u s t r a t e this:: in AC and BC we have a m a s s m , in DE h o w e v e r a m a s s ~n~ and in FG a m a s s me. (Minus the (logarithmically) d i v e r g e n t p a r t of this graph c o n t r i b u t e s to Z1.) In the graphs that contribute to Z l ( m ) , as f o r e x a m p l e does fig. 5, we have only an o v e r a l l (logarithmic) d i v e r g e n c e , all o t h e r (inner) d i v e r g e n c e s being r e m o v e d by m a s s and c h a r g e r e n o r m a l i z a t i o n . The m a s s m o c c u r s only in the f e r m i o n p r o p a g a t o r s : (t+ i m ) / t 2 + m 2 - i ¢ ) , and differentiating the d e n o m i n a t o r to m i n c r e a s e s the c o n v e r g e n c e of the integrand (making it finite) w h e r e a s the f a c t o r im in the n o m i n a t o r n e v e r c o n t r i b u t e s to the d i v e r g e n c e , as is s e e n by p o w e r counting. So, ZI(/~)- Zl(e) is finite. Relation ( 8 b ) i s p r o v e d in the s a m e way. Now we only r e p l a c e the m a s s my of the incoming muon by m (see fig. 6). The e x p r e s s i o n s in (8b) a r e only r e l a t e d to the p a r t of the graph CDEFG. Replacing in CD m.~_by m~= m e a n s r e p l a c i n g K b y Ke, and replacing_ m/4 by m m e a n s r e p l a c i n g K by Kin" In the s a m e way in DM we r e p l a c e S~ b___ySin, and changing the m a s s in ME m e a n s going to a new v e r t e x function F m. But at K f o r example we do not change anything. We now p r o c e e d by induction. Suppose (8b) is valid to o r d e r (2n-2). Then, by (Sa) and (Sb), ~ is finite to o r d e r (2n-2). This m e a n s that in (Sb) we only have a l o g a r i t h m i c o v e r - a l l d i v e r g e n c e . Consider as an example fig. 6: MEN is finite, its d i v e r g e n c e having d i s a p p e a r e d by going o v e r f r o m E
! !
Fig. 6.
166
P. VAN NIEUWENHUIZEN
Fig. 7. F m to F r n . T h e o v e r a l l d i v e r g e n c e i s the f i n a l i n t e g r a t i o n , a s i n d i c a t e d b y J d 4 s in f o r m u l a (4a). But now d i f f e r e n t i a t i o n to rn m a k e s (8b) f i n i t e to o r d e r (2n), t h u s c o m p l e t i n g t h e p r o o f b y i n d u c t i o n . B r e m s s t r a h l u n g n e v e r m a k e s K o r S' m o r e d i v e r g e n t a n d the p r o o f g o e s on in t h i s c a s e u n c h a n g e d .
5. P H O T O N , E L E C T R O N AND MUON GIVES AGAIN Q U A N T U M ELECTRODYNAMICS In t h e p r e c e d i n g s e c t i o n s w e h a v e e x t e n s i v e l y u s e d e l e c t r o d y n a m i c a l r e s u l t s , a l t h o u g h we d e a l with two k i n d s of f e r m i o n s . But t h e s e r e s u l t s go on. For example, both for muon and electron self-energy terms a Ward identity h o l d s , a n d c h a r g e - r e n o r m a l i z a t i o n a t t h e e l e c t r o n a s w e l l a s the m u o n v e r t e x h o l d s , a s i s i l l u s t r a t e d in fig. 7: we s p l i t the d i v e r g e n t p a r t s of p r o p a g a t o r s in two p a r t s . T h e n , a t e a c h v e r t e x the b a r e c h a r g e e o i s r e p l a c e d b y e. T h i s m e a n s t h a t the k e r n e l K i s f i n i t e ( a f t e r m a s s a n d c h a r g e r e n o r m a l i z a t i o n s of b o t h e l e c t r o n a n d muon). A c o m p l e t e p r o o f w o u l d s t a r t w i t h t h e c o u p l e d i n t e g r a l e q u a t i o n s f o r q u a n t u m e l e c t r o d y n a m i c s , b u t t h e s e a r e only t r i v i a l l y m o d i f i c a t e d , s o t h e p r o o f of r e n o r m a l i z a t i o n i s in o u r c a s e the s a m e . O u r k e r n e l K d e s c r i b e s ~ - ~ i n s t e a d of ;~-e s c a t t e r i n g , s o a c o m b i n a t o r i a l c o r r e c t i o n l i k e in C o m p t o n s c a t t e r i n g n e e d not b e i n c l u d e d w h e n t a k i n g the l i m i t rn~ -~ m e . T h e a u t h o r i s g r e a t l y i n d e b t e d to P r o f e s s o r s J. B j o r k e n , J. P r e n t k i a n d M. V e l t m a n f o r v a l u a b l e a d v i c e a n d h e l p f u l s u g g e s t i o n s . T h i s r e s e a r c h w a s s u p p o r t e d b y t h e F o u n d a t i o n f o r F u n d a m e n t a l R e s e a r c h of M a t t e r ( F . O . M . ) .
REFERENCES [1] [2] [3] [4]
N. Cabibbo, L. Maiani and G. P r e p a r a t a , Phys. L e t t e r s 25B (1967) 31, 132. T.Kinoshita and A. Sirlin, Phys. Rev. 113 11959} 1652. F. Bloch and A. Nordsieck, Phys. Rev. 52 (1937) 54. J. D. Bjorken and S. D. Drell, Relativistic quantum fields (Mac-Graw-Hill 1965) pp. 331-334. [5] S. N. Gupta, P r o c . Phys. Soc. {London) A66 (1953) 129.
Nuclear Physics B7 (1968) 160-166. North-Holland Publ. Comp., Amsterdam
FINITENESS
OF
IN A L L
RADIATIVE
ORDERS
TO
CORRECTIONS y-DECAY
P. VAN NIEUWENHUIZEN Instituut voor Theoretische Fysica, Maliesingel 23, Utrecht, The Netherlands
Received 6March 1968
Abstract: The well known result that second order radiative corrections to y-decay are finite is generalized to all orders.
1. INTRODUCTION Recently, Cabibbo et al. [1] d e r i v e d conditions on the v e c t o r and axialv e c t o r c u r r e n t by postulating that E - d e c a y for the pion in o r d e r e 2 be finite. Since the q u a r k model does not fulfil these conditions, they r e j e c t it in favour of another model (in which e.g. all e l e m e n t a r y constituents have integ r a l charge). It is natural, then, to a s k if the t h e o r y is finite in any o r d e r of radiative c o r r e c t i o n s . In t~-decay one can check whether all radiative c o r r e c t i o n s a r e finite or not, since here no c o m p l i c a t i o n s of s t r o n g i n t e r a c t i o n s a r e p r e s e n t . It is well known that the radiative c o r r e c t i o n s in s e c ond o r d e r to t~-decay a r e finite. They have been calculated and the r e s u l t is that the weak coupling constant is i n c r e a s e d by 0.2% (ref. [2]). It is also well known that the i n f r a r e d d i v e r g e n c e s of inner r a d i a t i v e c o r r e c t i o n s to y - d e c a y cancel in e a c h o r d e r against the i n f r a r e d d i v e r g e n c e s f r o m i n t e r nal B r e m s s t r a h l u n g [3]. It will be shown here that the u l t r a v i o l e t d i v e r g e n c e s which a r i s e in t~-decay cancel in each o r d e r . This m e a n s that it is p o s s i b l e to calculate the radiative c o r r e c t i o n s to ~ - d e c a y to a r b i t r a r y o r der. Or p h r a s e d differently: t h e r e is no need to r e n o r m a l i z e the weak coupling constant since adding e l e c t r o d y n a m i c s to f i r s t o r d e r weak i n t e r a c t i o n s does not introduce d i v e r g e n c e s . In sect. 2 we show that in o r d e r to p r o v e the f i n i t e n e s s of t~-decay in all o r d e r radiative c o r r e c t i o n s , it is sufficient to p r o v e that the s u m of all p r o p e r weak v e r t e x p a r t s equals a p r o d u c t of a c e r t a i n infinite constant (which cancels the other u l t r a v i o l e t d i v e r g e n c e s that a r e present) and a finite function. In sect. 3 we write down a B e t h e - S a l p e t e r type integral equation f o r the sum of all p r o p e r weak v e r t e x p a r t s and then show that it is sufficient for our p u r p o s e s to p r o v e : (i) that Zl(tt) - Zl(e) is finite (~ and e denote muon and electron) and (ii) a r e l a t i o n f o r the h o m o g e n e o u s p a r t of the integral equation. T h e s e r e l a t i o n s a r e p r o v e d in sect. 4. In sect. 5 we
RADIATIVE CORRECTIONS
161
d i s c u s s the r e n o r m a l i z a t i o n of q u a n t u m e l e c t r o d y n a m i c s a n d c o n c l u d e t h a t i t g o e s on, e v e n w h e n two ( o r m o r e ) t y p e s of f e r m i o n s a r e p r e s e n t t h a t i n t e r a c t o n l y t h r o u g h t h e e l e c t r o m a g n e t i c f i e l d , a r e s u l t t h a t i s u s e d in t h e . p r e c e d i n g s e c t i o n s . S i n c e w e o n l y n e e d t h e V - A c h a r a c t e r of t h e w e a k i n t e r a c t i o n s a f t e r t h e F i e r z t r a n s f o r m a t i o n in s e c t . 2, a n d a s s u m e t h e l e f t - h a n d e d n e s s of n e u t r i n o s , o u r r e s u l t s a p p l y to an i n t e r a c t i o n of t h e V - A f o r m f o r t h e o r i g i n a l i n t e r a c t i o n , to w h i c h c a n b e a d d e d a t e r m of t h e f o r m [C~e (1 +75)~PVe] [~u~ t (1 - 75)~/~] •
2. A C O N D I T I O N ON T H E W E A K V E R T E X T h e s u m M of a l l F e y n m a n d i a g r a m s r e p r e s e n t i n g the r a d i a t i v e c o r r e c t i o n s to # - d e c a y c a n b e f a c t o r i z e d into a p r o d u c t of f o u r f a c t o r s : (1)
M : T e F v Tg L v
w h e r e T e i s the s u m of a l l s e l f - e n e r g y t e r m s of the e x t e r n a l e l e c t r o n l i n e a n d g i v e s r i s e to the w a v e f u n c t i o n r e n o r m a l i z a t i o n of the e l e c t r o n , T ~ d o e s the s a m e f o r the m u o n , F u i s t h e s u m of a l l p r o p e r w e a k v e r t e x p a r t s and Lv is the neutrino current (times a factor g / R ) . We postpone a disc u s s i o n of B r e m s s t r a h l u n g in g - d e c a y u n t i l the e n d of s e c t . 4. F i g . l a i l l u s t r a t e s t h i s , in f i g . l b we g i v e a t e r m o c c u r r i n g in F u. Note t h a t t h e d i a g r a m s c o r r e s p o n d i n g to the t e r m s in F u h a v e no f r e e e l e c t r o n o r m u o n l i n e s . We c a n f o r g e t a b o u t the n e u t r i n o l i n e s , s i n c e we h a v e w r i t t e n the V - A i n t e r a c t i o n H a m i l t o n i a n b y a F i e r z t r a n s f o r m a t i o n in t h e f o r m :
fd3x~g(x) : ~ f
d~ [~vu(X) 7v (1 +75)~Pve(X)][~e(X) 7v (1 + 7 5 ) ~ ( x ) ] . (2)
t I t .g
a
b Fig. 1.
162
P. VAN NIEUWENHUIZEN
A s in q u a n t u m e l e c t r o d y n a m i c s T/~ = ¢ Z - 2 - ~ ( s e e s e c t . 5). If we c a n s h o w t h a t
[
r
~//~(0) a n d T e = ~ - ) ~ e ( 0 )
7
(3/
= LZ2(e)Z2(~)I
w i t h f i n i t e F v, t h e n i t f o l l o w s upon i n s e r t i n g eq. (3) in eq. (1) t h a t the r a d i a t i v e c o r r e c t i o n s to ~ - d e c a y a r e f i n i t e .
3. AN I N T E G R A L E Q U A T I O N F O R T H E W E A K V E R T E X T h e s u m of a l l p r o p e r , u n r e n o r m a l i z e d , w e a k v e r t e x p a r t s F ~ c a n b e w r i t t e n g r a p h i c a l l y in the i n t e g r a l e q u a t i o n s h o w n in fig. 2, w h e r e S'e a n d S/~ a r e the u n r e n o r m a l i z e d p r o p a g a t o r s .
~f
~e
J
i
I
L
0
L
I i
~I
zip
Fig. 2. T h e s q u a r e b l a c k b o x d e n o t e s the s u m of a l l t h o s e d i a g r a m s w i t h one e l e c t r o n g o i n g out a t fi a n d one m u o n g o i n g in at a (with f o u r v e c t o r s p e a n d p/~) a n d one e l e c t r o n g o i n g in a t 5 a n d one m u o n g o i n g out a t ~ (four v e c t o r s now p e + s a n d p P + s) t h a t c a n n o t b e c o m e d i s c o n n e c t e d b y c u t t i n g one e l e c t r o n a n d one m u o n l i n e . In fig. 3 we g i v e s o m e of the p e r t i n e n t a n d n o n pertinent diagrams.
NaT
INCLV~E~
~
+ • - •
Fig. 3.
RADIATIVE CORRECTIONS
163
In formula:
Ha (pe,pp) = [~v (I +y5)]~a +
F lj
fd4s [S'e (pe+ s)]i 0 FV0e(pe +s, pP +s) T
× [S# (p~ + s)]Ey K~sy a (s, pe, p#)
(4a)
or schematically:
(5)
Fv =Yv(l+Y5) +~f S'# KS' -e F v "
A simple example is given in fig. 4. This diagram does not contain divergences for fixed s, and its contribution I to the S - m a t r i x can be written as I = [{~e (0)]~r;a (spec)[~# (0)] a [{~v# (0)~v(1 + Y5)~Ve(0)]~2 54(~ u _pe _ Q) with Q is the sum of the neutrino f o u r - v e c t o r s and where FV(spec) is a t e r m in F v and can be written as in (4a):
F v ( s p e c ) : f d 4 s e 4.
1 Ifd4 t -1 .yo.l~ 5 (2~) 4 Y P ~ - l ~ - ime
Ha
× L_~_+~_ime]~eF -1 [rv (1 +~5)]0 e I
x Y# ~
-1
-
-i
imp
(t- s) 2
_---~mejEr
-~] ya
(4b)
The f i r s t and last t e r m in (4b) in square b r a c k e t s together form the (here finite) contribution to K, see sect. 5 and eq. (4b).
Fig. 4.
164
l?. VAN NIEUWENHUIZEN
In sect. 5 it is shown that
(6)
~: : [Z2(e)Z2(tt)] -1
with finite K. By finite we always m e a n independent of the cut-off introduced by m e a n s of the method of r e g u l a t o r s [5]. We then find the following i n t e g r a l equation for ~-~_ [Z2(e)Z2(/~)]½ r v , m
ru
1
-
-
= [Z2(e)Z2(/~)]~ 7u(1 +Y5) + f St.t K S e F v ,
(7)
where S~ = Z2(~)S-,, and S'e = Z2(e ) S-e" (Even in o r d e r e2,~Z2(e) ¢ Z2(/~)). We w i l l show in sect. 4 that in each o r der: ± (i) [Z2(tt)Z2(e)]2 - Z2(e ) is finite ,
(8a)
(ii) f S e K S e ~-V f ~ - K S ~-D is finite e/1 e
(8b)
(K"e equals__ K but the ingoing lepton is an electron instead of a muon and ~eu equals F u in the s a m e way). Then, all that r e m a i n s to be shown is the f i n i t e n e s s of QW: QW = Z2(E)Tu(1 +75) + f S e K e S e F ev
(7a)
If 7u(1 +75) w e r e r e p l a c e d by 7u in (Ta) this would be p u r e e l e c t r o d y n a m i c s and the f i n i t e n e s s of this e x p r e s s i o n has b e e n p r o v e d [4]. The d i v e r g e n t p a r t in the i n t e g r a l with yu(1 +75) r e p l a c e d by 7u75 is however the s a m e a s that belonging to the p u r e quantum e l e c t r o d y n a m i c a l c a s e . F o r , the effect of bringing 75 to the right of the integrand in (7a) is a r e p l a c e m e n t of m e by - m e in one f r e e e l e c t r o n - l i n e only. And the d i v e r g e n t p a r t of this e x p r e s sion is the s a m e as that of the s a m e e x p r e s s i o n with +m e (and still 75 at the end of the integrand). This c o m e s about b e c a u s e the highest p o s s i b l e d i v e r gence of the graph is l o g a r i t h m i c , and differentiation to m e r e n d e r s it finite (for d e t a i l s see sect. 4). So the d i v e r g e n t p a r t s in (7a) cancel, and all we have to do, it to p r o v e the r e l a t i o n s (8a) and (8b).
4. A RELATION BETWEEN Z - F A C T O R S In this section we p r o v e the r e l a t i o n s (8a) and (8b). Equivalent to (8a) is: Z2(/~) - Z2(e) is finite o r , since Z 1 = Z 2 it is to be shown that ZI(/I ) - Z l ( e ) is finite. We will show this by p r o v i n g that AZ(rn) = Z l ( m ) - Z l ( e ) is d i f f e r entiable. Then, since AZ(rne) = O, we will have p r o v e d (8a). By Z l ( m ) we m e a n Z I ( ~ ) with only in the incoming and outgoing muon line the m a s s m/~
RADIATIVE CORRECTIONS
165
Fig. 5. by m. In fig. 5 we i l l u s t r a t e this:: in AC and BC we have a m a s s m , in DE h o w e v e r a m a s s ~n~ and in FG a m a s s me. (Minus the (logarithmically) d i v e r g e n t p a r t of this graph c o n t r i b u t e s to Z1.) In the graphs that contribute to Z l ( m ) , as f o r e x a m p l e does fig. 5, we have only an o v e r a l l (logarithmic) d i v e r g e n c e , all o t h e r (inner) d i v e r g e n c e s being r e m o v e d by m a s s and c h a r g e r e n o r m a l i z a t i o n . The m a s s m o c c u r s only in the f e r m i o n p r o p a g a t o r s : (t+ i m ) / t 2 + m 2 - i ¢ ) , and differentiating the d e n o m i n a t o r to m i n c r e a s e s the c o n v e r g e n c e of the integrand (making it finite) w h e r e a s the f a c t o r im in the n o m i n a t o r n e v e r c o n t r i b u t e s to the d i v e r g e n c e , as is s e e n by p o w e r counting. So, ZI(/~)- Zl(e) is finite. Relation ( 8 b ) i s p r o v e d in the s a m e way. Now we only r e p l a c e the m a s s my of the incoming muon by m (see fig. 6). The e x p r e s s i o n s in (8b) a r e only r e l a t e d to the p a r t of the graph CDEFG. Replacing in CD m.~_by m~= m e a n s r e p l a c i n g K b y Ke, and replacing_ m/4 by m m e a n s r e p l a c i n g K by Kin" In the s a m e way in DM we r e p l a c e S~ b___ySin, and changing the m a s s in ME m e a n s going to a new v e r t e x function F m. But at K f o r example we do not change anything. We now p r o c e e d by induction. Suppose (8b) is valid to o r d e r (2n-2). Then, by (Sa) and (Sb), ~ is finite to o r d e r (2n-2). This m e a n s that in (Sb) we only have a l o g a r i t h m i c o v e r - a l l d i v e r g e n c e . Consider as an example fig. 6: MEN is finite, its d i v e r g e n c e having d i s a p p e a r e d by going o v e r f r o m E
! !
Fig. 6.
166
P. VAN NIEUWENHUIZEN
Fig. 7. F m to F r n . T h e o v e r a l l d i v e r g e n c e i s the f i n a l i n t e g r a t i o n , a s i n d i c a t e d b y J d 4 s in f o r m u l a (4a). But now d i f f e r e n t i a t i o n to rn m a k e s (8b) f i n i t e to o r d e r (2n), t h u s c o m p l e t i n g t h e p r o o f b y i n d u c t i o n . B r e m s s t r a h l u n g n e v e r m a k e s K o r S' m o r e d i v e r g e n t a n d the p r o o f g o e s on in t h i s c a s e u n c h a n g e d .
5. P H O T O N , E L E C T R O N AND MUON GIVES AGAIN Q U A N T U M ELECTRODYNAMICS In t h e p r e c e d i n g s e c t i o n s w e h a v e e x t e n s i v e l y u s e d e l e c t r o d y n a m i c a l r e s u l t s , a l t h o u g h we d e a l with two k i n d s of f e r m i o n s . But t h e s e r e s u l t s go on. For example, both for muon and electron self-energy terms a Ward identity h o l d s , a n d c h a r g e - r e n o r m a l i z a t i o n a t t h e e l e c t r o n a s w e l l a s the m u o n v e r t e x h o l d s , a s i s i l l u s t r a t e d in fig. 7: we s p l i t the d i v e r g e n t p a r t s of p r o p a g a t o r s in two p a r t s . T h e n , a t e a c h v e r t e x the b a r e c h a r g e e o i s r e p l a c e d b y e. T h i s m e a n s t h a t the k e r n e l K i s f i n i t e ( a f t e r m a s s a n d c h a r g e r e n o r m a l i z a t i o n s of b o t h e l e c t r o n a n d muon). A c o m p l e t e p r o o f w o u l d s t a r t w i t h t h e c o u p l e d i n t e g r a l e q u a t i o n s f o r q u a n t u m e l e c t r o d y n a m i c s , b u t t h e s e a r e only t r i v i a l l y m o d i f i c a t e d , s o t h e p r o o f of r e n o r m a l i z a t i o n i s in o u r c a s e the s a m e . O u r k e r n e l K d e s c r i b e s ~ - ~ i n s t e a d of ;~-e s c a t t e r i n g , s o a c o m b i n a t o r i a l c o r r e c t i o n l i k e in C o m p t o n s c a t t e r i n g n e e d not b e i n c l u d e d w h e n t a k i n g the l i m i t rn~ -~ m e . T h e a u t h o r i s g r e a t l y i n d e b t e d to P r o f e s s o r s J. B j o r k e n , J. P r e n t k i a n d M. V e l t m a n f o r v a l u a b l e a d v i c e a n d h e l p f u l s u g g e s t i o n s . T h i s r e s e a r c h w a s s u p p o r t e d b y t h e F o u n d a t i o n f o r F u n d a m e n t a l R e s e a r c h of M a t t e r ( F . O . M . ) .
REFERENCES [1] [2] [3] [4]
N. Cabibbo, L. Maiani and G. P r e p a r a t a , Phys. L e t t e r s 25B (1967) 31, 132. T.Kinoshita and A. Sirlin, Phys. Rev. 113 11959} 1652. F. Bloch and A. Nordsieck, Phys. Rev. 52 (1937) 54. J. D. Bjorken and S. D. Drell, Relativistic quantum fields (Mac-Graw-Hill 1965) pp. 331-334. [5] S. N. Gupta, P r o c . Phys. Soc. {London) A66 (1953) 129.