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Remarks on the radiative corrections of order α2 to muon decay and the determination of Gμ
~
N u c l e a r P h y s m s B29 (1971) 296-304.
N o r t h - H o l l a n d Publishing Company
REMARKS ON THE RADIATIVE CORRECTIONS OF ORDER a 2 TO MUON DECAY AND THE DETERMINATION OF G~ Matts ROOS * CERN, Geneva, Switzerland A l b e r t o S I R L I N ** Department of Physics, New York Umvers,ty, New York, USA 1Received 17 M a r c h 1971 A b s t r a c t We d i s c u s s s e v e r a l t h e o r e t m a l a s p e c t s of the study of the r a d i a t i v e c o r r e c t m n s to the total decay p r o b a b i l i t y P in p - d e c a y . On the basis of the t h e o r e m on c a n c e l l a t i o n of m a s s s m g u l a r i t m s , we obtain by a s i m p l e a r g u m e n t , that p a r t of the r a d i a t i v e c o r r e c t i o n s of o r d e r a 2 to P which contains m a s s s i n g u l a r i t i e s (i.e. t e r m s whmh d i v e r g e in the m a t h e m a t m a l l i m i t m e --~ 0) in an expansion m p o w e r s of the r e n o r m a h z e d charge. We also state a t h e o r e m on the effect of the finite e l e c t r o n m a s s on the r a d i a t i v e c o r r e c t i o n s of o r d e r a. We then d i s c u s s an exponentiation method r e c e n t l y a p p h e d to study the c o r r e c t i o n s to P. We show by d i r e c t calculation that the method, as p r e s e n t l y i m p l e m e n t e d , is ambiguous and that it l e a d s , to o r d e r a 2 , to a v i o l a t m n of the conclusions obtained on the basis of the t h e o r e m on m a s s s i n g u l a r i t i e s . Finally we give what we c o n s i d e r to be the best value p r e s e n t l y available for Gtt m the f r a m e w o r k of the l o c a l V-A t h e o r y , and b r m f l y d i s c u s s the e x p e c t e d o r d e r of magnitude of the t h e o r e t m a l e r r o r .
1. ~ T R O D U C T I O N A s i s w e l l k n o w n , in t h e f r a m e w o r k of t h e l o c a l V - A t h e o r y , t h e w e a k i n t e r a c t i o n c o u p l i n g c o n s t a n t G p i s d e t e r m i n e d f r o m t h e k n o w l e d g e of t h e m u o n l i f e t i m e rt~ b y m e a n s of t h e f o r m u l a 1 rtz
192~ 3
P
rn/~
where m p and m e stand for the muon and electron masses and 6 is the rad i a t i v e c o r r e c t i o n t o t h e t o t a l d e c a y p r o b a b i l i t y . In eq. (1) w e h a v e n o t i n c l u d e d n e g l i g i b l e c o n t r i b u t i o n s of o r d e r [ ( m e / m p)4 In (rn t z / r n e ) ] a n d h i g h e r . * On leave of absence f r o m the D e p a r t m e n t of N u c l e a r P h y s i c s , U m v e r s i t y of H e l sink1, Helsinki, Finland. ** R e s e a r c h s u p p o r t e d in p a r t by the National Science Foundation.
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M.Roos, A. S~rlm, Muon decay
In u n i t s of ~ = c = 1, t h e m o s t r e c e n t v a l u e s of Tp [1], mp [2] a n d t h e f u n d a m e n t a l c o n s t a n t s [2] d e t e r m i n e 1
G ~ = (1.4324 4- 0 . 0 0 0 3 ) ( 1 + 5 ) - ~ x 10 . 4 9 e r g . c m 3 1
= (1.1635 4- 0.0002)(1 + 5 ) - ~ x 10 - 5 G e V - 2 .
(2)
T o f i r s t o r d e r in a, a n d n e g l e c t i n g s m a l l c o n t r i b u t i o n s w h i c h v a n i s h in t h e m a t h e m a t i c a l l i m i t m e / m ~ ~ 0, 5 i s g i v e n b y [3, 4] 5 = 51-
~
(~r2
~-4.2x10
.3
(3)
w h e r e t h e s u b s c r i p t 1 i n d i c a t e s t h a t we h a v e o n l y i n c l u d e d t h e c o r r e c t i o n s of f i r s t o r d e r in a. U s i n g t h i s v a l u e t h e c o r r e c t e d G/z b e c o m e s G # = 1.4354 x 10 . 4 9 e r g . c m 3 = 1.1660 X 10 "5 G e V - 2 .
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A q u e s t i o n of c o n s i d e r a b l e t h e o r e t i c a l a n d p r a c t i c a l i n t e r e s t i s t h e f o l l o w i n g : w h a t i s t h e o r d e r of m a g n i t u d e of t h e c o r r e c t i o n s of o r d e r a 2 to t h e t o t a l d e c a y p r o b a b i l i t y P in ~ - d e c a y ? W e n o t e t h a t t h e o n l y l a r g e p a r a m e t e r l i k e l y to a p p e a r in t h e c a l c u l a t i o n i s ¢o -= l n ( m ~ / m e) ~ 5.332. B a r r i n g e x c e p t i o n a l l y l a r g e n u m e r i c a l c o e f f i c i e n t s one e x p e c t s t h e c o r r e c t i o n s of o r d e r a 2 t o P to b e q u i t e s m a l l , s a y , of r e l a t i v e o r d e r <~ 10 . 4 u n l e s s t h e r e e x i s t c o n t r i b u t i o n s o r o r d e r a2w n w i t h a r e l a t i v e l y l a r g e e x p o n e n t n. T h e s e t e r m s e x h i b i t a n i n t e r e s t i n g f e a t u r e : t h e y d i v e r g e in t h e m a t h e m a t i c a l l i m i t m e ~ 0. T h r o u g h o u t t h i s p a p e r w e w i l l r e f e r to t h e s e t e r m s a s mass singularities, a l t h o u g h t h e y a r e c e r t a i n l y f i n i t e in t h e r e a l w o r l d ! It i s w e l l k n o w n t h a t t e r m s of o r d e r aw e x i s t in t h e r a d i a t i v e c o r r e c t i o n s to t h e e l e c t r o n s p e c t r u m in p - d e c a y . M o r e o v e r , one k n o w s t h a t s u c h m a s s s i n g u l a r i t i e s of o r d e r a c a n c e l e x a c t l y a f t e r i n t e g r a t i o n o v e r a l l e l e c t r o n e n e r g i e s * T h i s i s c l e a r l y e x h i b i t e d in eq. (3), w h e r e t h e a n a l y t i c a l e x p r e s s i o n f o r 51 d o e s not c o n t a i n m a s s s i n g u l a r i t i e s . O b s e r v e t h a t t h e e x i s t e n c e of t e r m s of o r d e r a2¢0n w i t h n = 4, s a y , w o u l d m a k e t h e c o r r e c t i o n s of o r d e r a 2 to P p o t e n t i a l l y a s l a r g e a s t h e c o r r e c t i o n s of o r d e r a g i v e n in eq. (3)! F o r t u n a t e l y , n a t u r e d o e s not a p p e a r to b e s o m a l i c i o u s . On t h e b a s i s of t h e t h e o r e m on c a n c e l l a t i o n of m a s s s i n g u l a r i t i e s d e r i v e d t o a l l o r d e r s in ( u n r e n o r m a l i z e d ) p e r t u r b a t i o n t h e o r y b y K i n o s h i t a [5] a n d b y L e e a n d N a u e n b e r g [6], we o b t a i n in s e c t . 2, b y m e a n s of a s i m p l e a r g u m e n t , t h a t p a r t of t h e r a d i a t i v e c o r r e c t i o n s of o r d e r a2 t o P w h i c h c o n t a i n m a s s s i n g u l a r i t i e s in an e x p a n s i o n in p o w e r s of t h e r e normalized (i.e. physical) charge. The result is: a w51 = __~co ( ~ ) 2 (Tr2 - ~ ) ~ - 3 . 5 x 10 - 5 52m . s . - 237r where the superscript
m . s . a n d t h e s u b s c r i p t 2 i n d i c a t e t h a t we h a v e only
* Mass singularities and their cancellations in the radiative c o r r e c t i o n s of o r d e r a r e d i s c u s s e d m sect. 3 of ref. [4].
(5)
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t h e c o n t r i b u t i o n s of o r d e r a2 c o n t a i n i n g m a s s s i n g u l a r i t i e s . T h e f a c t t h a t 5~n ' s " i n v o l v e s only co r a i s e d to t h e f i r s t p o w e r and t h e o b s e r v a t i o n t h a t eq. (5) in i n d e e d v e r y s m a l l f o r t h e a c t u a l v a l u e s of rnp a n d m e a r e p a r t i c u l a r l y c o m f o r t i n g . It s h o u l d be c l e a r l y u n d e r s t o o d t h a t t h i s r e s u l t a p p l i e s only to t h e t o t a l d e c a y p r o b a b i l i t y , i . e . t h e c o r r e c t i o n s to the i n v e r s e l i f e t i m e T~ 1. T h i s i n c l u d e s a s u m m a t i o n o v e r a l l f i n a l s t a t e s a c c e s s i b l e to o r d e r a2, 1.e. p - ~ e - ~ e V p , p - ~ e - ~ e V p y , p ~ e - ~ e V p y y and p - --" e - ~ e v t ~ e + e - . F o r p a r t i a l d e c a y r a t e s , f o r t h e c o r r e c t i o n s t o t h e e l e c t r o n s p e c t r u m in p - d e c a y , e t c . , t h e t h e o r e m on c a n c e l l a t i o n of m a s s s i n g u l a r i t i e s i s not a p p l i c a b l e and we e x p e c t t e r m s of o r d e r a2o:n w i t h n > 1. We e m p h a s i z e t h a t t h e s u b t l e c a n c e l l a t i o n s i m p l i e d by t h i s t h e o r e m i n v o l v e h a r d a s w e l l a s s o f t p h o t o n s and e+e - p a i r s * A n o t h e r q u e s t i o n one m a y l i k e to a s k i s : w h a t i s t h e e r r o r a r i s i n g f r o m th e n e g l e c t of t h e f i n i t e e l e c t r o n m a s s in eq. (3) ? T h i s q u e s t i o n i s c l e a r l y not on t h e s a m e f o o t i n g a s t h e p r e v i o u s one, b e c a u s e t h e r a d i a t i v e c o r r e c t i o n s of o r d e r a h a v e b e e n c o m p u t e d e x a c t l y and, t h e r e f o r e , c a n b e c o m p l e t e l y c o n t r o l l e d at p r e s e n t * * . We s t a t e o u r r e s u l t in t h e f o r m of t h e f o l l o w i n g t h e o r e m : in th e V - A t h e o r y , a n d to f i r s t o r d e r in a, t h e c o n t r i b u t i o n s to 51 a r i s i n g f r o m t h e f i n i t e e l e c t r o n m a s s a r e of o r d e r of m a g n i t u d e (a/n)(rne/mp)2~, (a/~)(rne/mp)2 a n d s m a l l e r $. B e c a u s e of t h e s m a l l n e s s of ( m e ~ m 9 2 we s e e t h a t t h e e f f e c t of t h e f i n i t e e l e c t r o n m a s s in t h e c a l c u l a t i o n of 61 can be s a f e l y n e g l e c t e d . T h e s e d i s c u s s i o n s l e a d u s to t h e c o n c l u s i o n t h a t eq. (3) i s v e r y l i k e l y a g o o d a p p r o x i m a t i o n to t h e r a d i a t i v e c o r r e c t i o n s to t h e t o t a l d e c a y p r o b a b i l ity an d t h a t , t h e r e f o r e , eq. (4) is t h e b e s t v a l u e at p r e s e n t f o r G p , in t h e f r a m e w o r k of t h e l o c a l V - A t h e o r y . In t h e l i g h t of t h e p r e v i o u s d i s c u s s i o n ,
* F o r example an approximate calculation for the total rate of /~- --*e-~eVtte+eusing the method of Eckstem and P r a t t [7] suggests that this partial decay rate contains t e r m s of o r d e r oL2w3. According to the theorem on mass smgulamtms these t e r m s must cancel s l m d a r contributions of o r d er oL2 to the total decay rate amsmg, for example, from the w r t u a l radmtlve corrections. Th~s is not s u r prosing on the basis of expem~nce gathered in the study of c o r r e c t i o n s of o r d e r ~. for example, to o r d e r ~ the virtual radiative c o r r e c t m n s plus soft photons contain t e r m s of o rd er qo) 2 which are cancelled by the contributions from hard real photons. ** See Behrends et al. [8] and Grotch [9]. The function D m eq. (19) of ref. [8] must be replaced by D+C where C is given in eq. (C.4) of ref. [4]. $ The mare point of theoretmal interest in this theorem is the following: m the charge retention order, the radmtlve e o r r e e t m n s to the a x i a l - v e c t o r amphtude m //-decay are obtained from the c o r r e c t i o n s to the vector amplitude by changing the sign of the electron mass. The general argument that leads to this conclusion was given m sect. 3 of Berman and Slrhn [10]. Since m the charge retention order there is no interference between the V and A interactmn in the calculatmn of the decay spectrum of unpolamzed muons, it follows that m the V-A theory the t e r m s odd m m e cancel m the c o r r e c t m n s to the spectrum. It is still n e c e s s a r y to check that the further integration over the electron momentum to obtain the c o r r e c t m n s to the integrated probablhty does not introduce contmbutions of o r d er C~(me/rntt)n with n < 2. This we have done by studying the exphclt expressions for the c o r r e c tmns to the spectrum as given m ref. [8].
M. Roos, A. Sirhn, Muon decay
299
we expect the t h e o r e t i c a l e r r o r in eq. (4) to be roughly of r e l a t i v e m a g n i tude £ 10-4, which s e e m s to be a r e a s o n a b l e e s t i m a t e of the neglected contributions of o r d e r a 2. Recently, 5 has been r e - e v a l u a t e d [ 11] by m e a n s of an exponentiation m e t h o d which includes s o m e contributions of o r d e r ~2 and higher and the c l a i m has been made that 5 ~ -2 × 10-3 r a t h e r than the value given in eq. (3). In sect. 3 we show by d i r e c t calculation that the exponentiation method as i m p l e m e n t e d in ref. [11] is ambiguous (i.e. the s a m e g e n e r a l a r g u m e n t s m a y be used to d e r i v e v e r y different a n s w e r s ) . F u r t h e r m o r e , we point out that this method leads, in its p r e s e n t f o r m , to contributions to the total decay r a t e of o r d e r ~2wn, 0 ~< n ~<4, in s h a r p d i s a g r e e m e n t with the conclusions obtained in sect. 2 f r o m the t h e o r e m on cancellation of m a s s singularities.
2. DETERMINATION O F 5~n's" Suppose that the total decay p r o b a b i l i t y for y - d e c a y is expanded in the form M2(A ) + . . . ] ,
(6)
w h e r e / 5 is the u n c o r r e c t e d decay probability, e o is the b a r e c h a r g e and A is an u l t r a v i o l e t c u t - o f f which will be r e m o v e d at a l a t e r stage of the a r g u ment. The r e a s o n why a cut-off is n e c e s s a r y in M 2 when the expansion is done in t e r m s of the b a r e c h a r g e is that in the contributions of fourth o r d e r t h e r e a r e many g r a p h s involving v a c u u m polarization. In the p u r e e l e c t r o d y n a m i c s of e l e c t r o n s and photons, as is well known, the d i v e r g e n c e s a s s o c i a t e d with the e l e c t r o n loops a r e , in effect, ' a b s o r b e d ' in the c h a r g e r e n o r m a l i z a t i o n of l o w e r - o r d e r d i a g r a m s . The s a m e is true in the case of p decay (see fig. 1). As we have not yet p e r f o r m e d the c h a r g e r e n o r m a l i z a tion in eq. (6) and M 1 is finite, a c u t - o f f is c l e a r l y needed in M 2. The t h e o r e m on the cancellation of m a s s s i n g u l a r i t i e s as applied to ~decay, a s s e r t s that each t e r m in the ' u n r e n o r m a l i z e d ' p e r t u r b a t i o n expan-
O.
Fig. 1. Two related diagrams of second and fourth order. The divergences associated with the e- and /~-loops in diagram {b) are 'absorbed' m the charge renormahzation of diagram (a}.
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M.Roos, A. Sirlin, Muon decay
s i o n of eq. (6) f o r the total d e c a y p r o b a b i l i t y is f r e e f r o m e l e c t r o n m a s s s i n g u l a r i t i e s [5, 6]. In p a r t i c u l a r , this is the c a s e f o r M2(A ). L e t us p e r f o r m now the c o n v e n t i o n a l c h a r g e r e n o r m a l i z a t i o n : e2
e2
2~
A
2a
A +
1
In eq. (7) we have only r e t a i n e d t h o s e t e r m s of o r d e r a 2 which exhibit the u l t r a v i o l e t d i v e r g e n c e and the m a s s s i n g u l a r i t y . I n s e r t i n g eq. (7) into eq. (6): P =/5{l+aM1
2a 2 + 3~ [ 2 1 n ( 3rn)
+ ¢ o l M I + a 2 M 2 ( A ) + "" "t "
(8)
It is well known that the p e r t u r b a t i o n e x p a n s i o n f o r p - d e c a y in p o w e r s of the r e n o r m a l i z e d c h a r g e is finite to all o r d e r s [10, 12]. T h e r e f o r e the t e r m (4a2/3~) In (A/rn 9 c a n c e l s the c u t - o f f d e p e n d e n c e of M2(A). Eq. (8) now b e comes 2a 2 p =/5 [ l + a M 1 + ~ _ WMl+C~2Mf+. " .] ,
(9)
where M f =M2(A) +~-ln
~
.
(10)
Note that M f is f r e e f r o m u l t r a v i o l e t d i v e r g e n c e and m a s s s i n g u l a r i t i e s . T h u s the only c o n t r i b u t i o n of o r d e r c~2 with m a s s s i n g u l a r i t y in eq. (9) is 5~n.s.
2a 2 wM 1 2a = 3~ =~-wS1
"
(11)
C o m b i n i n g eqs. (3) and (11) we obtain the r e s u l t of eq. (5). T h e following o b s e r v a t i o n s a r e in o r d e r : (i) Note that the m a s s s i n g u l a r i t y is i n t r o d u c e d b e c a u s e of our u s e of c o n v e n t i o n a l c h a r g e r e n o r m a l i z a t i o n (see eq. (7)). As p o i n t e d out in r e f . [6] it is p o s s i b l e , f o r t h e o r e t i c a l p u r p o s e s , to obtain a r e n o r m a l i z e d p e r t u r b a tion e x p a n s i o n f r e e f r o m m a s s s i n g u l a r i t y if one r e n o r m a l i z e s the p h o t o n p r o p a g a t o r by c h o o s i n g a s u b t r a c t i o n point q2 = M2 ¢ 0. In our c a s e , howe v e r , it is m o s t c o n v e n i e n t to p e r f o r m the c o n v e n t i o n a l c h a r g e r e n o r m a l i z a t i o n b e c a u s e we w i s h to identify a with the u s u a l f i n e - s t r u c t u r e c o n s t a n t . (ii) In e v a l u a t i n g eq. (11) we have a s s u m e d that the weak v e r t e x is not r e n o r m a l i z e d in the c a l c u l a t i o n . Such a p r o c e d u r e is the a p p r o p r i a t e one when Gp in eq. (1) s t a n d s f o r the ' b a r e ' c o u p l i n g of the weak i n t e r a c t i o n s , ' u n r e n o r m a l i z e d ' with r e s p e c t to e l e c t r o m a g n e t i s m . As is well known, a r e n o r m a l i z a t i o n of the weak v e r t e x is not n e c e s s a r y to obtain finite a n s w e r s , b e c a u s e the c o r r e s p o n d i n g d i v e r g e n c e s a r e a u t o m a t i c a l l y c a n c e l l e d by t h o s e a r i s i n g f r o m the wave 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 and m u o n fields [10, 12]. (iii) T h e m e t h o d u s e d in this s e c t i o n to d e t e r m i n e 5~n ' s " is, to a c o n s i d e r a b l e extent, a n a l o g o u s to K i n o s h i t a ' s d i s c u s s i o n of the c o n t r i b u t i o n s of o r d e r a2 and h i g h e r in the c a l c u l a t i o n of the p m a g n e t i c m o m e n t [13].
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M . R o o s , A . S i r l i n , Muon decay
3. REMARKS
ON T H E E X P O N E N T I A T I O N M E T H O D
T o f i r s t o r d e r in a, t h e e l e c t r o n s p e c t r u m in t h e d e c a y of an u n p o l a r i z e d muon can be written as d P = d/5
1 + ~~r- C 2 ( E ) + ~- C I ( E ) i n
,
(12)
w h e r e d/5 i s t h e u n c o r r e c t e d s p e c t r u m , E i s t h e e l e c t r o n e n e r g y a n d q2 = ( p p _ P e ) 2 = m 2 + m2e - 2 r a p E. T h e f u n c t i o n C I ( E ) i s g i v e n b y 1 (1+/3" I Cl(E ) = ~ in - 2 \1 - fl/
w h e r e fl = v / c i s t h e e l e c t r o n v e l o c i t y in u n i t s of c, a n d C2(E) i s a v e r y c o m p l i c a t e d f u n c t i o n * w h i c h s i m p l y c o n s i s t s of a l l t h e c o r r e c t i o n s of o r d e r a ( a r i s i n g f r o m b o t h v i r t u a l a n d r e a l p h o t o n s ) w h i c h a r e not c o n t a i n e d in t h e l a s t t e r m of eq. (12). It h a s b e e n k n o w n f o r a l o n g t i m e t h a t t h e l a s t t e r m in eq. (12) s h o w s a n a n o m a l o u s b e h a v i o u r in t h e i m m e d i a t e v i c i n i t y of t h e p o i n t q2 = 0, i . e . a t t h e v e r y t i p of t h e e l e c t r o n s p e c t r u m . A s q2 a p p r o a c h e s zero, this term becomes very large and negative and it clearly leads to a b r e a k d o w n of p e r t u r b a t i o n t h e o r y **. It h a s b e e n p o i n t e d out [4] t h a t , in p r a c t i c e , t h i s d o e s not c a u s e p r o b l e m s b e c a u s e t h e i n t e r v a l in w h i c h t h e unphysical behaviour occurs is enormously smaller than the present exp e r i m e n t a l r e s o l u t i o n **. If one i s i n t e r e s t e d in t h e s t u d y of t h e e l e c t r o n s p e c t r u m one m u s t c o n s i d e r t h e c o n v o l u t i o n of eq. (12) w i t h t h e r e s o l u t i o n d i s t r i b u t i o n a n d no p r o b l e m a r i s e s , a t l e a s t a t p r e s e n t . If one m e a s u r e s t h e l i f e t i m e w i t h o u t d e t e r m i n i n g t h e e l e c t r o n s p e c t r u m one c a n s i m p l y i n t e g r a t e eq. (12) o v e r a l l e l e c t r o n e n e r g i e s , a s t h e s i n g u l a r i t y of t h e l a s t t e r m i s of a l o g a r i t h m i c a n d , t h e r e f o r e , m i l d a n d i n t e g r a b l e c h a r a c t e r . T h u s , to o r d e r a, t h e c o r r e c t i o n s 61 to t h e t o t a l d e c a y p r o b a b i l i t y , o b t a i n e d b y i n t e g r a t i n g eq. (12) o v e r t h e e l e c t r o n e n e r g y , a r e f i n i t e a n d w e l l d e f i n e d . It i s s t i l l i n t e r e s t i n g to i n q u i r e w h a t i s t h e m e c h a n i s m t h a t r e m o v e s t h e a n o m a l o u s b e h a v i o u r a s q2 _~ 0 in t h e e l e c t r o n s p e c t r u m , in t h e i d e a l c a s e of p e r f e c t e l e c t r o n r e s o l u t i o n . A s p e r t u r b a t i o n t h e o r y b r e a k s down in t h i s l i m i t a n d a s t h e p h e n o m e n o n in q u e s t i o n i s c l e a r l y r e l a t e d to i n f r a r e d b e h a v i o u r , it i s e x t r e m e l y p l a u s i b l e t h a t t h e s o l u t i o n to t h i s p r o b l e m i n v o l v e s c o n s i d e r a t i o n of m u l t i p l e s o f t - p h o t o n e m i s s i o n in t h o s e c o n f i g u r a t i o n s in w h i c h t h e e l e c t r o n e n e r g y i s v e r y c l o s e to t h e t i p of t h e e l e c t r o n s p e c t r u m . In o r d e r t o i m p l e m e n t t h i s i d e a t h e f o l l o w i n g p r o c e d u r e w a s p r o p o s e d in r e f . [11]: one r e w r i t e s eq. (12) a s * The function C2(E) can be determined, for example, by comparing eq. (12) with the e x p r e s s i o n s of ref. [4]. ** The region of the s p e c t r u m m whmh this happens is, however, e x t r e m e l y small. F o r example the c o r r e c t i o n s of o r d e r ~ become of magnitude 2~ 1 roughly in the interval (Ema x - E ) / E m a x <~e -50, where Ema x ~ !2 m ;/ is the maximum posmble energy for the electron. The reason for the occurrence of the In q2 t e r m is that at q2 = 0 the e m i s s i o n of r e a l photons becomes impossible and, therefore, the infrared divergence associated with the virtual quanta is not cancelled.
M.Roos, A. Szrlm, Muon decay
302
dP=d/5
1 +~-CI(E ) in
m~me
~
,
a n d e x p o n e n t i a t e s t h e f i r s t f a c t o r , s o t h a t eq. (13) i s r e p l a c e d b y
mpme/Jl T h e u n p h y s i c a l b e h a v i o u r a s q2 __. 0 h a s b e e n r e m o v e d in eq. (14) and, in f a c t , t h e s p e c t r u m t e n d s now to z e r o a s q2 __. 0, a t h e o r e t i c a l l y d e s i r a b l e r e s u l t . H o w e v e r , n u m e r i c a l e v a l u a t i o n s u s i n g eq. (14) l e a d to r e s u l t s f o r 5 v e r y d i f f e r e n t f r o m eq. (3). A s t h e s e t w o e x p r e s s i o n s c o i n c i d e to o r d e r a, one m u s t c o n c l u d e t h a t eq. (14) l e a d s t o c o n t r i b u t i o n s of o r d e r a 2, s a y , of t h e s a m e m a g n i t u d e a s t h o s e of o r d e r a. T h i s a p p a r e n t l y m y s t e r i o u s a n d r e m a r k a b l e p h e n o m e n o n c a n r e a d i l y b e u n d e r s t o o d if one s t u d i e s t h e c o n t r i b u t i o n s of o r d e r a 2 a r i s i n g f r o m eq. (14) a s a n e x p a n s i o n in p o w e r s of w. After performing the integration over the electron energies using the exp l i c i t e x p r e s s i o n f o r C2(E ) we f i n d t h a t eq. (14) l e a d s to c o n t r i b u t i o n s t o P of o r d e r (a/~)2w n w i t h 0 --< n ~< 4. (Note t h a t w4 ~ 8.4 × 10 2, w h i c h i s c l o s e to 27r/a.) T h e s e o b s e r v a t i o n s e x p l a i n q u a l i t a t i v e l y why eq. (14) g i v e s r i s e to s u c h l a r g e c o r r e c t i o n s of o r d e r ( a / 2 ~ ) 2 . C l e a r l y t h i s b e h a v i o u r i s in s h a r p d i s a g r e e m e n t w i t h t h e c o n c l u s i o n s o b t a i n e d f r o m t h e t h e o r e m on m a s s s i n g u l a r i t i e s ( s e e s e c t . 2). I n d e p e n d e n t l y of t h i s t h e o r e m it i s not d i f f i c u l t to s h o w t h a t t h e e x p o n e n t i a t i o n m e t h o d a s i m p l e m e n t e d in r e f . [11] i s a m b i g u o u s . F o r e x a m p l e , we c a n r e w r i t e eq. (12) a s dP=d/5
f I + ~ Ca2 ( E ) +~. C I ( E )
ln
( m 2 +q2m 2)] '
(15)
with =
and exponentiate as follows: dP(2 ) = d/~ e x p
CI(E ) in
+y)~
1
77
C2(E)
.
(16)
E q . (16) a l s o r e m o v e s t h e a n o m a l o u s b e h a v i o u r of t h e s p e c t r u m a s q2 ~ 0 a n d i s i d e n t i c a l to eq. (14) to ~ r d e r a. H o w e v e r , d e t a i l e d e x a m i n a t i o n s h o w s t h a t e q s . (14) a n d (16) g i v e r i s e t o v e r y d i f f e r e n t c o n t r i b u t i o n s of o r d e r a2 . * In r e f . [11] the c l a i m is m a d e t h a t t h e a m b i g u i t y of the e x p o n e n t i a t i o n m e t h o d is
small, roughly of the o r d e r of 1 p a r t in 1'04. Our detailed result of eq. (17) obviously d i s a g r e e s with that statement. We note that eq. (6.11) of ref. [11], which was used as a rough estimate of the ambiguity, was greatly underestimated n u m e r m a l ly m eqs. (6.12) and (6.14) of that paper. It is easy to see why eqs. (14) and (16) give r i s e to such different c o r r e c t i o n s of o r d e r ~2: both the second and third t e r m s in the square bracket of eq. (12) contain t e r m s of o r d e r ~co2, while their sum contains at most contributions of o r d e r ot¢o; instead the second and third t e r m s of eq. (15) contain only t e r m s of o r d e r aW. When the exponentlatlon is c a r t i e d through, this leads to contributions of o r d e r ~2con with n --<4 in P(1) and n ~< 2 m P(2).
M.Roos, A. Sirlin, Muon decay
303
We find by explicit analytical calculation $ that P(1) p- P(2) = - 2 ( ~ ) 2 (0)4_ 6co3 + 13.8co 2_ 12.1co) + O ( a 3) = - 2 . 4 × 10 - 3 + . . .
(17)
H e r e P(1) a n d P(2) r e f e r to t h e t o t a l d e c a y p r o b a b i l i t y c a l c u l a t e d on t h e b a s i s of e q s . (14) a n d (16), r e s p e c t i v e l y , a n d we h a v e n e g l e c t e d t e r m s t h a t v a n i s h a s r n e / m ~ ~ O. W e n o t e t h a t t h e a m b i g u i t y in t h e a 2 t e r m s , a s m e a s u r e d b y eq. (17), i s of t h e s a m e o r d e r of m a g n i t u d e a s t h e c o r r e c t i o n s of o r d e r a in eq. (3)! $ A l t h o u g h we h a v e c h o s e n to i l l u s t r a t e t h e a m b i g u i t y b y m e a n s of a s i m p l e e x a m p l e , i t i s c l e a r t h a t t h e a r b i t r a r i n e s s of t h e m e t h o d , a s a p p l i e d to g - d e c a y , i s of a m o r e g e n e r a l k i n d . O b s e r v e , f o r i n s t a n c e , t h a t e x p o n e n t i a t i o n s s u c h a s g i v e n in e q s . (14) a n d (16) a f f e c t to s o m e e x t e n t t h e e n t i r e e n e r g y s p e c t r u m f r o m q2 = 0 to q2 = (rn~z_ me)2 a n d t h a t in t h e r e g i o n w h e r e q2 i s s i g n i f i c a n t l y d i f f e r e n t f r o m z e r o , t h e y i n v o l v e h a r d a s w e l l a s s o f t p h o t o n s . W e n o t e i n p a s s i n g t h a t c o n t r a r y t o eq. (14), eq. (16) l e a d s to c o n t r i b u t i o n s of o r d e r a Z t o P s i g n i f i c a n t l y s m a l l e r t h a n t h e t e r m s of o r d e r a, a p r o p e r t y w h i c h c o n f o r m s m o r e c l o s e l y to i n t u i t i v e e x p e c t a t i o n s $. In o u r o p i n i o n , h o w e v e r , n e i t h e r e q s . (14) n o r (16) s h o u l d be u s e d to c o m p u t e t h e c o r r e c t i o n s to P b e c a u s e b o t h c o n t r a d i c t in o r d e r a 2 t h e c o n c l u s i o n s o b t a i n e d f r o m t h e t h e o r e m on m a s s s i n g u l a r i t i e s . It i s i n d e e d v e r y p l a u s i b l e t h a t s o m e k i n d of e x p o n e n t i a t i o n i n v o l v i n g (a/~r)Cl(E) in q2 a c t u a l l y o c c u r s , b u t if t h i s i s s o t h e r e s h o u l d e x i s t a d d i t i o n a l c o n t r i b u t i o n s of o r d e r a 2 w h i c h c a n c e l t h e a n o m a l o u s m a s s s i n g u l a r i t i e s g e n e r a t e d to t h i s o r d e r b y t h e e x p o n e n t i a l . It s h o u l d b e c l e a r f r o m t h e a b o v e d i s c u s s i o n t h a t t h e i n c l u s i o n of a n a r b i t r a r i l y s e l e c t e d a n d p a r t i a l s e t of t e r m s of o r d e r a 2 a n d h i g h e r , i n v o l v i n g l a r g e p o w e r s of w a s in e q s . (14) a n d (16) i s h i g h l y m i s l e a d i n g in t h e e v a l u a t i o n of t h e c o r r e c t i o n s to P in / ~ - d e c a y . If one w i s h e s to i n c l u d e t e r m s of h i g h e r o r d e r it s e e m s n e c e s s a r y in t h i s c a s e e i t h e r t o u s e a g e n e r a l t h e o r e m s u c h a s t h e t h e o r e m on m a s s s i n g u l a r i t i e s , w h i c h a p p l i e s to t h e s u m t o t a l of a l l t h e c o n t r i b u t i o n s of a g i v e n o r d e r , o r o t h e r w i s e t o do a f a i r l y c o m p l e t e c a l c u l a t i o n of t h e s e c o r r e c t i o n s . T h e s e o b s e r v a t i o n s s u g g e s t a l s o t h e i m p o r t a n c e of e x p a n s i o n s of t h e f o r m ( a / ~ ) 2 Z n Cn con in t h e e v a l u a t i o n of t h e c o r r e c t i o n s of o r d e r a 2. C o r r e c t i o n s of s u c h o r d e r t o t h e e n e r g y a n g l e d i s t r i b u t i o n in ~ - d e c a y m a y b e i n d e e d v e r y u s e f u l f o r p r a c t i c a l a s well as theoretical reasons. In computing P ( 1 ) - P ( 2 ) we have included both the contributions of o r d e r ~2 in eqs. (14) and (16) which obtain in the Taylor expansion of the exponentials and the c r o s s t e r m s between the t e r m s of o r d e r ~ in the exponentials and the t e r m s (Ot/~)C2(E) and (q/Tr)C2(E). These c r o s s t e r m s a r e negative and large and dominate the p o s i tive contmbutions of the f i r s t type. This explains the negative sign in eq. (17). Should we a r b i t r a m l y d i s m i s s the c r o s s t e r m s , the eontmbutmns of o r d e r OL2 to either P(1) or P(2) a r e n e c e s s a r i l y positive: we find that the ambiguity in this case is still very large-but of opposite sign to that given m eq. (17).
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M . R o o s , A . Sirhn, Muon decay
We s h o u l d l i k e to t h a n k P r o f e s s o r T . K i n o s h i t a f o r h i s c o m m e n t s and int e r e s t in t h i s p r o b l e m .
REFERENCES [1] M.Roos et al., P a r t m l e Data Group, Phys. L e t t e r s 33B (1970) 1. [2] B. Taylor, W. P a r k e r and D. Langenberg, Revs. Mod. Phys. 41 (1969) 375; J. F. Hague, J . E . Rothberg, A. Schenk, D. L. Wllhams, R.W. Wllhams, K.K. Young and K . M . C r o w e , Phys. Rev. L e t t e r s 25 {1970) 628; R. De Voe, P . M . McIntyre, A. Magnon, D.Y. Stowell, R . A . Swanson and V. L. Telegdl, Phys. Rev. L e tt e r s 25 {1970) 1779; 26 (1971) 213, e r r a t a . [3] S . M . B e r m a n , Phys. Rev. 112 (1958) 267. [4] T.Klnoshlta and A . S l r h n , Phys. Rev. 113 (1959) 1652. [5] T.Kmoshlta, J. Math. Phys. 3 (1962) 650; Lectures in theoretmal physics, Umv ers l t y of Colorado (Interscience P u b h s h e r s , New York, 1961). [6] T.D. Lee and M. Nauenberg, Phys. Rev. 133B {1964) 1549. [7] S . G . E c k s t e m and R . H . P r a t t , Ann. of Phys. 8 (1959) 297. [8] R . E . B e h r e n d s , R . J . F m k e l s t e i n and A . S ir l i n , Phys. Rev. 101 (1955) 866. [9] H.Grotch, Phys. Rev. 168 (1968) 1872. [10] S.M. Berman and A. S~rhn, Ann. of Phys. 20 (1962) 20. [11] L.Matsson, Nucl. Phys. B12 (1969) 647. [12] Y a . A . S m o r o d l n s k n and Ho T s o - H s m , Zh. Eksper. Teor. Flz. 38 (1960} 1007; J E T P (Sov. Phys.) 11 (1960) 724; G. P r e p a r a t a and W . W e l s b e r g e r , Phys. Rev. 175 (1968) 1965; S . L . A d l e r , Phys. Rev. 177 (1969) 2426. [13] T.Kmoshlta, Nuovo Clmento 51B (1967) 140.
N u c l e a r P h y s m s B29 (1971) 296-304.
N o r t h - H o l l a n d Publishing Company
REMARKS ON THE RADIATIVE CORRECTIONS OF ORDER a 2 TO MUON DECAY AND THE DETERMINATION OF G~ Matts ROOS * CERN, Geneva, Switzerland A l b e r t o S I R L I N ** Department of Physics, New York Umvers,ty, New York, USA 1Received 17 M a r c h 1971 A b s t r a c t We d i s c u s s s e v e r a l t h e o r e t m a l a s p e c t s of the study of the r a d i a t i v e c o r r e c t m n s to the total decay p r o b a b i l i t y P in p - d e c a y . On the basis of the t h e o r e m on c a n c e l l a t i o n of m a s s s m g u l a r i t m s , we obtain by a s i m p l e a r g u m e n t , that p a r t of the r a d i a t i v e c o r r e c t i o n s of o r d e r a 2 to P which contains m a s s s i n g u l a r i t i e s (i.e. t e r m s whmh d i v e r g e in the m a t h e m a t m a l l i m i t m e --~ 0) in an expansion m p o w e r s of the r e n o r m a h z e d charge. We also state a t h e o r e m on the effect of the finite e l e c t r o n m a s s on the r a d i a t i v e c o r r e c t i o n s of o r d e r a. We then d i s c u s s an exponentiation method r e c e n t l y a p p h e d to study the c o r r e c t i o n s to P. We show by d i r e c t calculation that the method, as p r e s e n t l y i m p l e m e n t e d , is ambiguous and that it l e a d s , to o r d e r a 2 , to a v i o l a t m n of the conclusions obtained on the basis of the t h e o r e m on m a s s s i n g u l a r i t i e s . Finally we give what we c o n s i d e r to be the best value p r e s e n t l y available for Gtt m the f r a m e w o r k of the l o c a l V-A t h e o r y , and b r m f l y d i s c u s s the e x p e c t e d o r d e r of magnitude of the t h e o r e t m a l e r r o r .
1. ~ T R O D U C T I O N A s i s w e l l k n o w n , in t h e f r a m e w o r k of t h e l o c a l V - A t h e o r y , t h e w e a k i n t e r a c t i o n c o u p l i n g c o n s t a n t G p i s d e t e r m i n e d f r o m t h e k n o w l e d g e of t h e m u o n l i f e t i m e rt~ b y m e a n s of t h e f o r m u l a 1 rtz
192~ 3
P
rn/~
where m p and m e stand for the muon and electron masses and 6 is the rad i a t i v e c o r r e c t i o n t o t h e t o t a l d e c a y p r o b a b i l i t y . In eq. (1) w e h a v e n o t i n c l u d e d n e g l i g i b l e c o n t r i b u t i o n s of o r d e r [ ( m e / m p)4 In (rn t z / r n e ) ] a n d h i g h e r . * On leave of absence f r o m the D e p a r t m e n t of N u c l e a r P h y s i c s , U m v e r s i t y of H e l sink1, Helsinki, Finland. ** R e s e a r c h s u p p o r t e d in p a r t by the National Science Foundation.
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In u n i t s of ~ = c = 1, t h e m o s t r e c e n t v a l u e s of Tp [1], mp [2] a n d t h e f u n d a m e n t a l c o n s t a n t s [2] d e t e r m i n e 1
G ~ = (1.4324 4- 0 . 0 0 0 3 ) ( 1 + 5 ) - ~ x 10 . 4 9 e r g . c m 3 1
= (1.1635 4- 0.0002)(1 + 5 ) - ~ x 10 - 5 G e V - 2 .
(2)
T o f i r s t o r d e r in a, a n d n e g l e c t i n g s m a l l c o n t r i b u t i o n s w h i c h v a n i s h in t h e m a t h e m a t i c a l l i m i t m e / m ~ ~ 0, 5 i s g i v e n b y [3, 4] 5 = 51-
~
(~r2
~-4.2x10
.3
(3)
w h e r e t h e s u b s c r i p t 1 i n d i c a t e s t h a t we h a v e o n l y i n c l u d e d t h e c o r r e c t i o n s of f i r s t o r d e r in a. U s i n g t h i s v a l u e t h e c o r r e c t e d G/z b e c o m e s G # = 1.4354 x 10 . 4 9 e r g . c m 3 = 1.1660 X 10 "5 G e V - 2 .
(4)
A q u e s t i o n of c o n s i d e r a b l e t h e o r e t i c a l a n d p r a c t i c a l i n t e r e s t i s t h e f o l l o w i n g : w h a t i s t h e o r d e r of m a g n i t u d e of t h e c o r r e c t i o n s of o r d e r a 2 to t h e t o t a l d e c a y p r o b a b i l i t y P in ~ - d e c a y ? W e n o t e t h a t t h e o n l y l a r g e p a r a m e t e r l i k e l y to a p p e a r in t h e c a l c u l a t i o n i s ¢o -= l n ( m ~ / m e) ~ 5.332. B a r r i n g e x c e p t i o n a l l y l a r g e n u m e r i c a l c o e f f i c i e n t s one e x p e c t s t h e c o r r e c t i o n s of o r d e r a 2 t o P to b e q u i t e s m a l l , s a y , of r e l a t i v e o r d e r <~ 10 . 4 u n l e s s t h e r e e x i s t c o n t r i b u t i o n s o r o r d e r a2w n w i t h a r e l a t i v e l y l a r g e e x p o n e n t n. T h e s e t e r m s e x h i b i t a n i n t e r e s t i n g f e a t u r e : t h e y d i v e r g e in t h e m a t h e m a t i c a l l i m i t m e ~ 0. T h r o u g h o u t t h i s p a p e r w e w i l l r e f e r to t h e s e t e r m s a s mass singularities, a l t h o u g h t h e y a r e c e r t a i n l y f i n i t e in t h e r e a l w o r l d ! It i s w e l l k n o w n t h a t t e r m s of o r d e r aw e x i s t in t h e r a d i a t i v e c o r r e c t i o n s to t h e e l e c t r o n s p e c t r u m in p - d e c a y . M o r e o v e r , one k n o w s t h a t s u c h m a s s s i n g u l a r i t i e s of o r d e r a c a n c e l e x a c t l y a f t e r i n t e g r a t i o n o v e r a l l e l e c t r o n e n e r g i e s * T h i s i s c l e a r l y e x h i b i t e d in eq. (3), w h e r e t h e a n a l y t i c a l e x p r e s s i o n f o r 51 d o e s not c o n t a i n m a s s s i n g u l a r i t i e s . O b s e r v e t h a t t h e e x i s t e n c e of t e r m s of o r d e r a2¢0n w i t h n = 4, s a y , w o u l d m a k e t h e c o r r e c t i o n s of o r d e r a 2 to P p o t e n t i a l l y a s l a r g e a s t h e c o r r e c t i o n s of o r d e r a g i v e n in eq. (3)! F o r t u n a t e l y , n a t u r e d o e s not a p p e a r to b e s o m a l i c i o u s . On t h e b a s i s of t h e t h e o r e m on c a n c e l l a t i o n of m a s s s i n g u l a r i t i e s d e r i v e d t o a l l o r d e r s in ( u n r e n o r m a l i z e d ) p e r t u r b a t i o n t h e o r y b y K i n o s h i t a [5] a n d b y L e e a n d N a u e n b e r g [6], we o b t a i n in s e c t . 2, b y m e a n s of a s i m p l e a r g u m e n t , t h a t p a r t of t h e r a d i a t i v e c o r r e c t i o n s of o r d e r a2 t o P w h i c h c o n t a i n m a s s s i n g u l a r i t i e s in an e x p a n s i o n in p o w e r s of t h e r e normalized (i.e. physical) charge. The result is: a w51 = __~co ( ~ ) 2 (Tr2 - ~ ) ~ - 3 . 5 x 10 - 5 52m . s . - 237r where the superscript
m . s . a n d t h e s u b s c r i p t 2 i n d i c a t e t h a t we h a v e only
* Mass singularities and their cancellations in the radiative c o r r e c t i o n s of o r d e r a r e d i s c u s s e d m sect. 3 of ref. [4].
(5)
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t h e c o n t r i b u t i o n s of o r d e r a2 c o n t a i n i n g m a s s s i n g u l a r i t i e s . T h e f a c t t h a t 5~n ' s " i n v o l v e s only co r a i s e d to t h e f i r s t p o w e r and t h e o b s e r v a t i o n t h a t eq. (5) in i n d e e d v e r y s m a l l f o r t h e a c t u a l v a l u e s of rnp a n d m e a r e p a r t i c u l a r l y c o m f o r t i n g . It s h o u l d be c l e a r l y u n d e r s t o o d t h a t t h i s r e s u l t a p p l i e s only to t h e t o t a l d e c a y p r o b a b i l i t y , i . e . t h e c o r r e c t i o n s to the i n v e r s e l i f e t i m e T~ 1. T h i s i n c l u d e s a s u m m a t i o n o v e r a l l f i n a l s t a t e s a c c e s s i b l e to o r d e r a2, 1.e. p - ~ e - ~ e V p , p - ~ e - ~ e V p y , p ~ e - ~ e V p y y and p - --" e - ~ e v t ~ e + e - . F o r p a r t i a l d e c a y r a t e s , f o r t h e c o r r e c t i o n s t o t h e e l e c t r o n s p e c t r u m in p - d e c a y , e t c . , t h e t h e o r e m on c a n c e l l a t i o n of m a s s s i n g u l a r i t i e s i s not a p p l i c a b l e and we e x p e c t t e r m s of o r d e r a2o:n w i t h n > 1. We e m p h a s i z e t h a t t h e s u b t l e c a n c e l l a t i o n s i m p l i e d by t h i s t h e o r e m i n v o l v e h a r d a s w e l l a s s o f t p h o t o n s and e+e - p a i r s * A n o t h e r q u e s t i o n one m a y l i k e to a s k i s : w h a t i s t h e e r r o r a r i s i n g f r o m th e n e g l e c t of t h e f i n i t e e l e c t r o n m a s s in eq. (3) ? T h i s q u e s t i o n i s c l e a r l y not on t h e s a m e f o o t i n g a s t h e p r e v i o u s one, b e c a u s e t h e r a d i a t i v e c o r r e c t i o n s of o r d e r a h a v e b e e n c o m p u t e d e x a c t l y and, t h e r e f o r e , c a n b e c o m p l e t e l y c o n t r o l l e d at p r e s e n t * * . We s t a t e o u r r e s u l t in t h e f o r m of t h e f o l l o w i n g t h e o r e m : in th e V - A t h e o r y , a n d to f i r s t o r d e r in a, t h e c o n t r i b u t i o n s to 51 a r i s i n g f r o m t h e f i n i t e e l e c t r o n m a s s a r e of o r d e r of m a g n i t u d e (a/n)(rne/mp)2~, (a/~)(rne/mp)2 a n d s m a l l e r $. B e c a u s e of t h e s m a l l n e s s of ( m e ~ m 9 2 we s e e t h a t t h e e f f e c t of t h e f i n i t e e l e c t r o n m a s s in t h e c a l c u l a t i o n of 61 can be s a f e l y n e g l e c t e d . T h e s e d i s c u s s i o n s l e a d u s to t h e c o n c l u s i o n t h a t eq. (3) i s v e r y l i k e l y a g o o d a p p r o x i m a t i o n to t h e r a d i a t i v e c o r r e c t i o n s to t h e t o t a l d e c a y p r o b a b i l ity an d t h a t , t h e r e f o r e , eq. (4) is t h e b e s t v a l u e at p r e s e n t f o r G p , in t h e f r a m e w o r k of t h e l o c a l V - A t h e o r y . In t h e l i g h t of t h e p r e v i o u s d i s c u s s i o n ,
* F o r example an approximate calculation for the total rate of /~- --*e-~eVtte+eusing the method of Eckstem and P r a t t [7] suggests that this partial decay rate contains t e r m s of o r d e r oL2w3. According to the theorem on mass smgulamtms these t e r m s must cancel s l m d a r contributions of o r d er oL2 to the total decay rate amsmg, for example, from the w r t u a l radmtlve corrections. Th~s is not s u r prosing on the basis of expem~nce gathered in the study of c o r r e c t i o n s of o r d e r ~. for example, to o r d e r ~ the virtual radiative c o r r e c t m n s plus soft photons contain t e r m s of o rd er qo) 2 which are cancelled by the contributions from hard real photons. ** See Behrends et al. [8] and Grotch [9]. The function D m eq. (19) of ref. [8] must be replaced by D+C where C is given in eq. (C.4) of ref. [4]. $ The mare point of theoretmal interest in this theorem is the following: m the charge retention order, the radmtlve e o r r e e t m n s to the a x i a l - v e c t o r amphtude m //-decay are obtained from the c o r r e c t i o n s to the vector amplitude by changing the sign of the electron mass. The general argument that leads to this conclusion was given m sect. 3 of Berman and Slrhn [10]. Since m the charge retention order there is no interference between the V and A interactmn in the calculatmn of the decay spectrum of unpolamzed muons, it follows that m the V-A theory the t e r m s odd m m e cancel m the c o r r e c t m n s to the spectrum. It is still n e c e s s a r y to check that the further integration over the electron momentum to obtain the c o r r e c t m n s to the integrated probablhty does not introduce contmbutions of o r d er C~(me/rntt)n with n < 2. This we have done by studying the exphclt expressions for the c o r r e c tmns to the spectrum as given m ref. [8].
M. Roos, A. Sirhn, Muon decay
299
we expect the t h e o r e t i c a l e r r o r in eq. (4) to be roughly of r e l a t i v e m a g n i tude £ 10-4, which s e e m s to be a r e a s o n a b l e e s t i m a t e of the neglected contributions of o r d e r a 2. Recently, 5 has been r e - e v a l u a t e d [ 11] by m e a n s of an exponentiation m e t h o d which includes s o m e contributions of o r d e r ~2 and higher and the c l a i m has been made that 5 ~ -2 × 10-3 r a t h e r than the value given in eq. (3). In sect. 3 we show by d i r e c t calculation that the exponentiation method as i m p l e m e n t e d in ref. [11] is ambiguous (i.e. the s a m e g e n e r a l a r g u m e n t s m a y be used to d e r i v e v e r y different a n s w e r s ) . F u r t h e r m o r e , we point out that this method leads, in its p r e s e n t f o r m , to contributions to the total decay r a t e of o r d e r ~2wn, 0 ~< n ~<4, in s h a r p d i s a g r e e m e n t with the conclusions obtained in sect. 2 f r o m the t h e o r e m on cancellation of m a s s singularities.
2. DETERMINATION O F 5~n's" Suppose that the total decay p r o b a b i l i t y for y - d e c a y is expanded in the form M2(A ) + . . . ] ,
(6)
w h e r e / 5 is the u n c o r r e c t e d decay probability, e o is the b a r e c h a r g e and A is an u l t r a v i o l e t c u t - o f f which will be r e m o v e d at a l a t e r stage of the a r g u ment. The r e a s o n why a cut-off is n e c e s s a r y in M 2 when the expansion is done in t e r m s of the b a r e c h a r g e is that in the contributions of fourth o r d e r t h e r e a r e many g r a p h s involving v a c u u m polarization. In the p u r e e l e c t r o d y n a m i c s of e l e c t r o n s and photons, as is well known, the d i v e r g e n c e s a s s o c i a t e d with the e l e c t r o n loops a r e , in effect, ' a b s o r b e d ' in the c h a r g e r e n o r m a l i z a t i o n of l o w e r - o r d e r d i a g r a m s . The s a m e is true in the case of p decay (see fig. 1). As we have not yet p e r f o r m e d the c h a r g e r e n o r m a l i z a tion in eq. (6) and M 1 is finite, a c u t - o f f is c l e a r l y needed in M 2. The t h e o r e m on the cancellation of m a s s s i n g u l a r i t i e s as applied to ~decay, a s s e r t s that each t e r m in the ' u n r e n o r m a l i z e d ' p e r t u r b a t i o n expan-
O.
Fig. 1. Two related diagrams of second and fourth order. The divergences associated with the e- and /~-loops in diagram {b) are 'absorbed' m the charge renormahzation of diagram (a}.
300
M.Roos, A. Sirlin, Muon decay
s i o n of eq. (6) f o r the total d e c a y p r o b a b i l i t y is f r e e f r o m e l e c t r o n m a s s s i n g u l a r i t i e s [5, 6]. In p a r t i c u l a r , this is the c a s e f o r M2(A ). L e t us p e r f o r m now the c o n v e n t i o n a l c h a r g e r e n o r m a l i z a t i o n : e2
e2
2~
A
2a
A +
1
In eq. (7) we have only r e t a i n e d t h o s e t e r m s of o r d e r a 2 which exhibit the u l t r a v i o l e t d i v e r g e n c e and the m a s s s i n g u l a r i t y . I n s e r t i n g eq. (7) into eq. (6): P =/5{l+aM1
2a 2 + 3~ [ 2 1 n ( 3rn)
+ ¢ o l M I + a 2 M 2 ( A ) + "" "t "
(8)
It is well known that the p e r t u r b a t i o n e x p a n s i o n f o r p - d e c a y in p o w e r s of the r e n o r m a l i z e d c h a r g e is finite to all o r d e r s [10, 12]. T h e r e f o r e the t e r m (4a2/3~) In (A/rn 9 c a n c e l s the c u t - o f f d e p e n d e n c e of M2(A). Eq. (8) now b e comes 2a 2 p =/5 [ l + a M 1 + ~ _ WMl+C~2Mf+. " .] ,
(9)
where M f =M2(A) +~-ln
~
.
(10)
Note that M f is f r e e f r o m u l t r a v i o l e t d i v e r g e n c e and m a s s s i n g u l a r i t i e s . T h u s the only c o n t r i b u t i o n of o r d e r c~2 with m a s s s i n g u l a r i t y in eq. (9) is 5~n.s.
2a 2 wM 1 2a = 3~ =~-wS1
"
(11)
C o m b i n i n g eqs. (3) and (11) we obtain the r e s u l t of eq. (5). T h e following o b s e r v a t i o n s a r e in o r d e r : (i) Note that the m a s s s i n g u l a r i t y is i n t r o d u c e d b e c a u s e of our u s e of c o n v e n t i o n a l c h a r g e r e n o r m a l i z a t i o n (see eq. (7)). As p o i n t e d out in r e f . [6] it is p o s s i b l e , f o r t h e o r e t i c a l p u r p o s e s , to obtain a r e n o r m a l i z e d p e r t u r b a tion e x p a n s i o n f r e e f r o m m a s s s i n g u l a r i t y if one r e n o r m a l i z e s the p h o t o n p r o p a g a t o r by c h o o s i n g a s u b t r a c t i o n point q2 = M2 ¢ 0. In our c a s e , howe v e r , it is m o s t c o n v e n i e n t to p e r f o r m the c o n v e n t i o n a l c h a r g e r e n o r m a l i z a t i o n b e c a u s e we w i s h to identify a with the u s u a l f i n e - s t r u c t u r e c o n s t a n t . (ii) In e v a l u a t i n g eq. (11) we have a s s u m e d that the weak v e r t e x is not r e n o r m a l i z e d in the c a l c u l a t i o n . Such a p r o c e d u r e is the a p p r o p r i a t e one when Gp in eq. (1) s t a n d s f o r the ' b a r e ' c o u p l i n g of the weak i n t e r a c t i o n s , ' u n r e n o r m a l i z e d ' with r e s p e c t to e l e c t r o m a g n e t i s m . As is well known, a r e n o r m a l i z a t i o n of the weak v e r t e x is not n e c e s s a r y to obtain finite a n s w e r s , b e c a u s e the c o r r e s p o n d i n g d i v e r g e n c e s a r e a u t o m a t i c a l l y c a n c e l l e d by t h o s e a r i s i n g f r o m the wave 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 and m u o n fields [10, 12]. (iii) T h e m e t h o d u s e d in this s e c t i o n to d e t e r m i n e 5~n ' s " is, to a c o n s i d e r a b l e extent, a n a l o g o u s to K i n o s h i t a ' s d i s c u s s i o n of the c o n t r i b u t i o n s of o r d e r a2 and h i g h e r in the c a l c u l a t i o n of the p m a g n e t i c m o m e n t [13].
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3. REMARKS
ON T H E E X P O N E N T I A T I O N M E T H O D
T o f i r s t o r d e r in a, t h e e l e c t r o n s p e c t r u m in t h e d e c a y of an u n p o l a r i z e d muon can be written as d P = d/5
1 + ~~r- C 2 ( E ) + ~- C I ( E ) i n
,
(12)
w h e r e d/5 i s t h e u n c o r r e c t e d s p e c t r u m , E i s t h e e l e c t r o n e n e r g y a n d q2 = ( p p _ P e ) 2 = m 2 + m2e - 2 r a p E. T h e f u n c t i o n C I ( E ) i s g i v e n b y 1 (1+/3" I Cl(E ) = ~ in - 2 \1 - fl/
w h e r e fl = v / c i s t h e e l e c t r o n v e l o c i t y in u n i t s of c, a n d C2(E) i s a v e r y c o m p l i c a t e d f u n c t i o n * w h i c h s i m p l y c o n s i s t s of a l l t h e c o r r e c t i o n s of o r d e r a ( a r i s i n g f r o m b o t h v i r t u a l a n d r e a l p h o t o n s ) w h i c h a r e not c o n t a i n e d in t h e l a s t t e r m of eq. (12). It h a s b e e n k n o w n f o r a l o n g t i m e t h a t t h e l a s t t e r m in eq. (12) s h o w s a n a n o m a l o u s b e h a v i o u r in t h e i m m e d i a t e v i c i n i t y of t h e p o i n t q2 = 0, i . e . a t t h e v e r y t i p of t h e e l e c t r o n s p e c t r u m . A s q2 a p p r o a c h e s zero, this term becomes very large and negative and it clearly leads to a b r e a k d o w n of p e r t u r b a t i o n t h e o r y **. It h a s b e e n p o i n t e d out [4] t h a t , in p r a c t i c e , t h i s d o e s not c a u s e p r o b l e m s b e c a u s e t h e i n t e r v a l in w h i c h t h e unphysical behaviour occurs is enormously smaller than the present exp e r i m e n t a l r e s o l u t i o n **. If one i s i n t e r e s t e d in t h e s t u d y of t h e e l e c t r o n s p e c t r u m one m u s t c o n s i d e r t h e c o n v o l u t i o n of eq. (12) w i t h t h e r e s o l u t i o n d i s t r i b u t i o n a n d no p r o b l e m a r i s e s , a t l e a s t a t p r e s e n t . If one m e a s u r e s t h e l i f e t i m e w i t h o u t d e t e r m i n i n g t h e e l e c t r o n s p e c t r u m one c a n s i m p l y i n t e g r a t e eq. (12) o v e r a l l e l e c t r o n e n e r g i e s , a s t h e s i n g u l a r i t y of t h e l a s t t e r m i s of a l o g a r i t h m i c a n d , t h e r e f o r e , m i l d a n d i n t e g r a b l e c h a r a c t e r . T h u s , to o r d e r a, t h e c o r r e c t i o n s 61 to t h e t o t a l d e c a y p r o b a b i l i t y , o b t a i n e d b y i n t e g r a t i n g eq. (12) o v e r t h e e l e c t r o n e n e r g y , a r e f i n i t e a n d w e l l d e f i n e d . It i s s t i l l i n t e r e s t i n g to i n q u i r e w h a t i s t h e m e c h a n i s m t h a t r e m o v e s t h e a n o m a l o u s b e h a v i o u r a s q2 _~ 0 in t h e e l e c t r o n s p e c t r u m , in t h e i d e a l c a s e of p e r f e c t e l e c t r o n r e s o l u t i o n . A s p e r t u r b a t i o n t h e o r y b r e a k s down in t h i s l i m i t a n d a s t h e p h e n o m e n o n in q u e s t i o n i s c l e a r l y r e l a t e d to i n f r a r e d b e h a v i o u r , it i s e x t r e m e l y p l a u s i b l e t h a t t h e s o l u t i o n to t h i s p r o b l e m i n v o l v e s c o n s i d e r a t i o n of m u l t i p l e s o f t - p h o t o n e m i s s i o n in t h o s e c o n f i g u r a t i o n s in w h i c h t h e e l e c t r o n e n e r g y i s v e r y c l o s e to t h e t i p of t h e e l e c t r o n s p e c t r u m . In o r d e r t o i m p l e m e n t t h i s i d e a t h e f o l l o w i n g p r o c e d u r e w a s p r o p o s e d in r e f . [11]: one r e w r i t e s eq. (12) a s * The function C2(E) can be determined, for example, by comparing eq. (12) with the e x p r e s s i o n s of ref. [4]. ** The region of the s p e c t r u m m whmh this happens is, however, e x t r e m e l y small. F o r example the c o r r e c t i o n s of o r d e r ~ become of magnitude 2~ 1 roughly in the interval (Ema x - E ) / E m a x <~e -50, where Ema x ~ !2 m ;/ is the maximum posmble energy for the electron. The reason for the occurrence of the In q2 t e r m is that at q2 = 0 the e m i s s i o n of r e a l photons becomes impossible and, therefore, the infrared divergence associated with the virtual quanta is not cancelled.
M.Roos, A. Szrlm, Muon decay
302
dP=d/5
1 +~-CI(E ) in
m~me
~
,
a n d e x p o n e n t i a t e s t h e f i r s t f a c t o r , s o t h a t eq. (13) i s r e p l a c e d b y
mpme/Jl T h e u n p h y s i c a l b e h a v i o u r a s q2 __. 0 h a s b e e n r e m o v e d in eq. (14) and, in f a c t , t h e s p e c t r u m t e n d s now to z e r o a s q2 __. 0, a t h e o r e t i c a l l y d e s i r a b l e r e s u l t . H o w e v e r , n u m e r i c a l e v a l u a t i o n s u s i n g eq. (14) l e a d to r e s u l t s f o r 5 v e r y d i f f e r e n t f r o m eq. (3). A s t h e s e t w o e x p r e s s i o n s c o i n c i d e to o r d e r a, one m u s t c o n c l u d e t h a t eq. (14) l e a d s t o c o n t r i b u t i o n s of o r d e r a 2, s a y , of t h e s a m e m a g n i t u d e a s t h o s e of o r d e r a. T h i s a p p a r e n t l y m y s t e r i o u s a n d r e m a r k a b l e p h e n o m e n o n c a n r e a d i l y b e u n d e r s t o o d if one s t u d i e s t h e c o n t r i b u t i o n s of o r d e r a 2 a r i s i n g f r o m eq. (14) a s a n e x p a n s i o n in p o w e r s of w. After performing the integration over the electron energies using the exp l i c i t e x p r e s s i o n f o r C2(E ) we f i n d t h a t eq. (14) l e a d s to c o n t r i b u t i o n s t o P of o r d e r (a/~)2w n w i t h 0 --< n ~< 4. (Note t h a t w4 ~ 8.4 × 10 2, w h i c h i s c l o s e to 27r/a.) T h e s e o b s e r v a t i o n s e x p l a i n q u a l i t a t i v e l y why eq. (14) g i v e s r i s e to s u c h l a r g e c o r r e c t i o n s of o r d e r ( a / 2 ~ ) 2 . C l e a r l y t h i s b e h a v i o u r i s in s h a r p d i s a g r e e m e n t w i t h t h e c o n c l u s i o n s o b t a i n e d f r o m t h e t h e o r e m on m a s s s i n g u l a r i t i e s ( s e e s e c t . 2). I n d e p e n d e n t l y of t h i s t h e o r e m it i s not d i f f i c u l t to s h o w t h a t t h e e x p o n e n t i a t i o n m e t h o d a s i m p l e m e n t e d in r e f . [11] i s a m b i g u o u s . F o r e x a m p l e , we c a n r e w r i t e eq. (12) a s dP=d/5
f I + ~ Ca2 ( E ) +~. C I ( E )
ln
( m 2 +q2m 2)] '
(15)
with =
and exponentiate as follows: dP(2 ) = d/~ e x p
CI(E ) in
+y)~
1
77
C2(E)
.
(16)
E q . (16) a l s o r e m o v e s t h e a n o m a l o u s b e h a v i o u r of t h e s p e c t r u m a s q2 ~ 0 a n d i s i d e n t i c a l to eq. (14) to ~ r d e r a. H o w e v e r , d e t a i l e d e x a m i n a t i o n s h o w s t h a t e q s . (14) a n d (16) g i v e r i s e t o v e r y d i f f e r e n t c o n t r i b u t i o n s of o r d e r a2 . * In r e f . [11] the c l a i m is m a d e t h a t t h e a m b i g u i t y of the e x p o n e n t i a t i o n m e t h o d is
small, roughly of the o r d e r of 1 p a r t in 1'04. Our detailed result of eq. (17) obviously d i s a g r e e s with that statement. We note that eq. (6.11) of ref. [11], which was used as a rough estimate of the ambiguity, was greatly underestimated n u m e r m a l ly m eqs. (6.12) and (6.14) of that paper. It is easy to see why eqs. (14) and (16) give r i s e to such different c o r r e c t i o n s of o r d e r ~2: both the second and third t e r m s in the square bracket of eq. (12) contain t e r m s of o r d e r ~co2, while their sum contains at most contributions of o r d e r ot¢o; instead the second and third t e r m s of eq. (15) contain only t e r m s of o r d e r aW. When the exponentlatlon is c a r t i e d through, this leads to contributions of o r d e r ~2con with n --<4 in P(1) and n ~< 2 m P(2).
M.Roos, A. Sirlin, Muon decay
303
We find by explicit analytical calculation $ that P(1) p- P(2) = - 2 ( ~ ) 2 (0)4_ 6co3 + 13.8co 2_ 12.1co) + O ( a 3) = - 2 . 4 × 10 - 3 + . . .
(17)
H e r e P(1) a n d P(2) r e f e r to t h e t o t a l d e c a y p r o b a b i l i t y c a l c u l a t e d on t h e b a s i s of e q s . (14) a n d (16), r e s p e c t i v e l y , a n d we h a v e n e g l e c t e d t e r m s t h a t v a n i s h a s r n e / m ~ ~ O. W e n o t e t h a t t h e a m b i g u i t y in t h e a 2 t e r m s , a s m e a s u r e d b y eq. (17), i s of t h e s a m e o r d e r of m a g n i t u d e a s t h e c o r r e c t i o n s of o r d e r a in eq. (3)! $ A l t h o u g h we h a v e c h o s e n to i l l u s t r a t e t h e a m b i g u i t y b y m e a n s of a s i m p l e e x a m p l e , i t i s c l e a r t h a t t h e a r b i t r a r i n e s s of t h e m e t h o d , a s a p p l i e d to g - d e c a y , i s of a m o r e g e n e r a l k i n d . O b s e r v e , f o r i n s t a n c e , t h a t e x p o n e n t i a t i o n s s u c h a s g i v e n in e q s . (14) a n d (16) a f f e c t to s o m e e x t e n t t h e e n t i r e e n e r g y s p e c t r u m f r o m q2 = 0 to q2 = (rn~z_ me)2 a n d t h a t in t h e r e g i o n w h e r e q2 i s s i g n i f i c a n t l y d i f f e r e n t f r o m z e r o , t h e y i n v o l v e h a r d a s w e l l a s s o f t p h o t o n s . W e n o t e i n p a s s i n g t h a t c o n t r a r y t o eq. (14), eq. (16) l e a d s to c o n t r i b u t i o n s of o r d e r a Z t o P s i g n i f i c a n t l y s m a l l e r t h a n t h e t e r m s of o r d e r a, a p r o p e r t y w h i c h c o n f o r m s m o r e c l o s e l y to i n t u i t i v e e x p e c t a t i o n s $. In o u r o p i n i o n , h o w e v e r , n e i t h e r e q s . (14) n o r (16) s h o u l d be u s e d to c o m p u t e t h e c o r r e c t i o n s to P b e c a u s e b o t h c o n t r a d i c t in o r d e r a 2 t h e c o n c l u s i o n s o b t a i n e d f r o m t h e t h e o r e m on m a s s s i n g u l a r i t i e s . It i s i n d e e d v e r y p l a u s i b l e t h a t s o m e k i n d of e x p o n e n t i a t i o n i n v o l v i n g (a/~r)Cl(E) in q2 a c t u a l l y o c c u r s , b u t if t h i s i s s o t h e r e s h o u l d e x i s t a d d i t i o n a l c o n t r i b u t i o n s of o r d e r a 2 w h i c h c a n c e l t h e a n o m a l o u s m a s s s i n g u l a r i t i e s g e n e r a t e d to t h i s o r d e r b y t h e e x p o n e n t i a l . It s h o u l d b e c l e a r f r o m t h e a b o v e d i s c u s s i o n t h a t t h e i n c l u s i o n of a n a r b i t r a r i l y s e l e c t e d a n d p a r t i a l s e t of t e r m s of o r d e r a 2 a n d h i g h e r , i n v o l v i n g l a r g e p o w e r s of w a s in e q s . (14) a n d (16) i s h i g h l y m i s l e a d i n g in t h e e v a l u a t i o n of t h e c o r r e c t i o n s to P in / ~ - d e c a y . If one w i s h e s to i n c l u d e t e r m s of h i g h e r o r d e r it s e e m s n e c e s s a r y in t h i s c a s e e i t h e r t o u s e a g e n e r a l t h e o r e m s u c h a s t h e t h e o r e m on m a s s s i n g u l a r i t i e s , w h i c h a p p l i e s to t h e s u m t o t a l of a l l t h e c o n t r i b u t i o n s of a g i v e n o r d e r , o r o t h e r w i s e t o do a f a i r l y c o m p l e t e c a l c u l a t i o n of t h e s e c o r r e c t i o n s . T h e s e o b s e r v a t i o n s s u g g e s t a l s o t h e i m p o r t a n c e of e x p a n s i o n s of t h e f o r m ( a / ~ ) 2 Z n Cn con in t h e e v a l u a t i o n of t h e c o r r e c t i o n s of o r d e r a 2. C o r r e c t i o n s of s u c h o r d e r t o t h e e n e r g y a n g l e d i s t r i b u t i o n in ~ - d e c a y m a y b e i n d e e d v e r y u s e f u l f o r p r a c t i c a l a s well as theoretical reasons. In computing P ( 1 ) - P ( 2 ) we have included both the contributions of o r d e r ~2 in eqs. (14) and (16) which obtain in the Taylor expansion of the exponentials and the c r o s s t e r m s between the t e r m s of o r d e r ~ in the exponentials and the t e r m s (Ot/~)C2(E) and (q/Tr)C2(E). These c r o s s t e r m s a r e negative and large and dominate the p o s i tive contmbutions of the f i r s t type. This explains the negative sign in eq. (17). Should we a r b i t r a m l y d i s m i s s the c r o s s t e r m s , the eontmbutmns of o r d e r OL2 to either P(1) or P(2) a r e n e c e s s a r i l y positive: we find that the ambiguity in this case is still very large-but of opposite sign to that given m eq. (17).
304
M . R o o s , A . Sirhn, Muon decay
We s h o u l d l i k e to t h a n k P r o f e s s o r T . K i n o s h i t a f o r h i s c o m m e n t s and int e r e s t in t h i s p r o b l e m .
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