IL NUOVO CIMENTO VoL. 43 A, N. 2 21 Gennaio 1978

Radiative Corrections to the Leptonie Decays of the Charged Heavy Leptons (*)C)-

A. ALl a n d Z. Z. AYDIN (***)

I I . Insti tut ]i~r Theoretische Physik der Universit~t - Hamburg

(ricevuto il 10 Ottobre 1977)

S u m m a r y . - We present numerical estimates for the radiative cor- rections to the decays Lm-~emvoVL,~.mv~vL in the context of a V ± A interaction. The radiative corrections change the Michel parameter ap- preciably from its bare V i A value, for the decay LT--> e=FVeVL, and have a measurable effect on the electron energy spectrum.

The ~ n o m a l o u s ~> ~e e v e n t s (~) ~nd t he i nc lu s ive m u o n p r o d u c t i o n in

e+e - e x p e r i m e n t s (2) r e p o r t e d b y t he S L A C - L B L group a n d D E S Y are be i ng

(*) To speed up publication, the authors of this paper have agreed to not receive the proofs for correction. (**) Work supported joint ly by the Alexander yon Humboldt Foundat ion and the Bundesministerium fiir Forschung und Technologic. (***) Permanend address: Universi ty of Ankara, Ankara, Turkey. (1) M.L . PERL, C+. S. ABRAMS, A. M. BOYARSKI, M. BREIDENBACH, D. D. BRIGGS, F. BULOS, W. CHINOWSKY, J. T. DAKIN, O. J. FELDMAN, C. ]~. FRIEDBERG, D. FRYBER- GER, G. GOLDHA~ER, G. HANSON, F. B. HEILE, B. JEAN-MARIE, J. A. KADYK, R. R. LARSON, A. M. LITKE, D. LUKE, B. A. LuLu, V. L~TH, D. LYON, ¢. C. MOREHOUSE, J. M. PATERSON, F. M. PIERRE, W. P. TUN, P. A. I~APIDIS, S. RICHTER, B. SADOULET, R. F. SCtIWITTERS, W. TANENBAUM, G. H. TRILLING, F. VANUCCI, J. S. WHITAK~R, F. C. WILKELMANN and J. E. WIss: Phys. Rev. J~ett., 35, 1489 (1975); J. BURMESTER, L. CRIEGEE, H. C. D:EHNE, K. DERIKUM, R. DEVENISH, G. FL~GGE, J. D. FOX, G. FRANKE, CH. GERKE, P. HARMS, G. HORLITZ, TH. KAKL, G. K~rES, 3[. RSSSLER, ]R. SCHMITZ, U. TIMM, H. WAHL, P. WALOSCHEK, G. C. WINT]~R, S. WOLFF and W. ZIM- MERMANN: Phys. I~ett., 68 B, 301 (1977). (2) G. J. FELDMAN, F. BULOS, D. L~JKE, G. S. ABRAMS, M. S. ALAM, A. M. BOYARSKI, M. BREII)ENBACH, J. DORFAN, C. E. FRIEDB:ERG, D. FRYBERGER, G. GOLDHABER, G. HANSON, F. B. HEILE, J. A. JAROS, J. A. KADYK, R. R. LARSEN, A. M. LITKE, V. L~TH, R. J. MADARAS, C. C. MOREHOUSE, H. K. NGUYEN, J. M. PATERSON, M. L.

270

RADIATIVE CORRECTIONS TO TIIE LEPTONIC DECAYS :ETC. 271

interpreted as coming from the production and decay of a pair of charged heavy leptons, L ÷. So far only the leptonie decay modes, namely

(1)

have been established. Starting from the premise that these events are coming genuinely from charged heavy leptons, there has been a lot of theoretical activity to understand the nature of such heavy leptons (3), the structure of the weak current responsible for their decays and the nature of the neutrinos being emi t ted (~,5). The tests suggested consist of detailed comparison of the various electron (muon) energy-angle correlations with the data tha t will be

available in the near future. I n part icular it has been poin ted out (~) t ha t the electron and the rouen

energy spectra are sensitive to the value of the Michel pa rame te r ~. Al though the value of ~ does not determine uniquely the na ture of the weak interact ion

responsible for the processes (1), ~ value of ~ close to -~ would suggest a V - - A interaction, whereas ~--~ 0 would favour V + A. On the other hand, it is also known from the s tudy of the ~-decay (~) tha t the wdue of ff is sensitive to the radiat ive corrections. This then le;Ids to the necessity of including radi:ltive

corrections to the processes (1) and of comparing the electron (inuon) energy spectrum with the ones obtained after incorporat ing radiat ive corrections.

The relative impor tance of the radiat ive corrections to leptonic processes

such as (1) can be guessed from the order p}~rameter tha t ~.haracterizes such

PERL, I. I)ERUZZI, M. PICCOLo), F. M. PIERRE, T. P. PUN, P. RAPIDIS, B. RICIITEU, B. SADOULET, R. F. ~CH]VITTERS, W. TANENBAUM, G. H. TRILLING, F. VANNUCCI, J. S. ]VHIT=4KER and J. E. WISS: Phys. Rev. Lett., 38, 117, 576 (E) (1977); J. BUI~,~IESTEn, L. CRIEGEE, It. C. DEHNE, K. DERIK-UM, R. DEVENISII, (]. FLIi(~GE, .|. D. FO:<, G. FRANKE. (~II. GERKE, P. HARM% G. HORLITZ, Til. KAIIL, (~. KNIES, M. Ri)SSLER, R. SCtlMITZ, U. TIMM, H. WAHJ,, P. WALOSCEEK, G. C. WINTER, g. \VoLFi, ~ arid ~ . Z,~I- MERM~NN: Phys. Lett., 68B, 297 (1977). (3) A. ALl and T. C. YAN<;: Phys. Ret'. D, 14, 3052 (1976); (!. II. A~BRIaUT: Phys. Rev. D, 13. 2508 (1976). (a) Y. S. TSAI: Phys. Bee. D, 4 2821 (1971); J. D. BJORKEN and (!. tt. LEWELLYN SMITH: Phys. ]~ev. D, 7, 887 (1973); see M. L. PERL and P. RAPIDIS: SLAC-PUB-1496 (1974), for more references. (5) K. FUJIKAWA and K. KAWAMOTO: Phys. t~ev. D, 13, 2534 (1976); Phys. Rev. D, 14, 59 (1976); A. PAlS and S. B. TREIMAN: Phys. Tier. D, 14, 293 (1976); S. Y. PI and A. I. SANDA: Phys. Rev. Lett., 36, 1 (1976); Rockefeller Universi ty Report COO- 2232B-109; T. HAGIWARA, S. Y. PI and A. I. SANDA: Rockefeller Universi ty Report C00-2232B-105. (6) R. E. BEI-IRENDS, R. J. FINKELSTEIN and A. SIRLIN: Phys. Rev., 101, 866 (1956); T. KINOSHITA and A. SIRLIN: Phys. Tier., 107, 593 (1957); 113, 1652 (1959); S. M. BER- )IAN: Phys. Rev., 112, 267 (1958).

272 A. ALl ~nd z. z. AYDIN

corrections. The first-order radiat ive correction to the processes (1) is of order

, , :

instead of the fine-structure cons tant a. This number is 0.5 for t ~ e and mL ~ 2 GeV ~nd 0.06 for ~ ~ ~ and m~ ~ 2 GeV, the corresponding number for the ~-decay being 0.2. I t is~ a priori~ obvious tha t the effect of the radiative correction for the decay L ~ --> e%~vL would be substantial ly larger than tha t

for the ~-decay, while it will be re la t ively smaller for the dec~y L ~ --~ ~%~v L. The purpose of this note is to present a numerical es t imate of the radiative corrections to the Michel pa ramete r Q, within the context of a V T A inter- action. We also present the effect of the r~diative corrections on the electron

energy spec t rum relevant for the SLAC-LBL and D E S Y experiments (~). We s tar t b y recalling that the effective current (~ current H~miltoni~n

for the processes (1) leads to the following one-parameter Michel formula for

the electron (muon) energy dis tr ibut ion:

(2)

where

(3)

, 0 4

2Et

~ ~ , 0

a ~ m g o

384x~ 4

~ = e , ~,

for ( V - - A , V ÷ A ) ,

GF is the Fe rmi coupling constant ~ 1.05.10-5(m,) -2. a L and a R are introduced to let the V -- A, V -~ A interact ion a different overall normalization. The ra-

diative corrections to the processes (1) are then implemented by assuming a

universal e lectromagnetic interact ion H i .... ~ ~ , e ~ i y ~ y , iA~. i=e,~,L

The diagrams tha t contr ibute to the first order in ~ (,i.e. to order G~ ~ in the decay probabil i ty) are shown in fig. la) (vir tual-photon contribution) and

fig. lb) (bremsstrahlung contribution). The dec~y distribution including the radiat ive correction for the ~-decay was calculated in ref. (6) by assuming an effective current Q current interaction. We shall rely on their calculations

and present here (after some s t ra ightforward algebra) the relevant formulae for the V ± A interactions. The electron (muon) energy spectrum, including

RADIATIVE CORRECTIONS TO THE LEPTONIC DECAYS ETC. 2 7 ~

,4 I - L'- ~. L- ~, L- L- "v~ L-

vL',.

a) b)

Fig. 1 . - Feynman diagrams for the first-order radiative correction to the decay L~ ~ e÷v~,~. "The diagrams for L~:-~ ~TvT,~ are similar, a) Virtual-photon contribu- tion, b) bresstrahlung contribution.

the r ad i a t i ve correc t ions , t h e n r eads as

(4) dx -- 4~a~,Rx 2 3 ( 1 - - x ) ~- 2~ x - - 1 ~- ~-~/~,a(x)

where

(5)

]L(x) = (6 - - lx )R(x) -~ (6 - - 6x) l n x ~-

-~ ~ [(5 -~- 17x - - 34x~)(cot -~- In x) - - 22x -~ 34x ~] ,

]a(x) ~ 12(1 - - x)R(x) ~- 12 In x ~ 36(1 - - X)(~OA - - ~ot) -~

i - - X . X ~ ~ - - ~ [ 2 ( ] ~- 4 x - - 1 ~ " ) ( o ) t - ~ l n x ) - - 1 2 x ~ - 57x2],

(6)

- ) i x n - - ~ R(x) = '2 1 z~ _ 2 ~- (or ~- 2 In 1 _

- - h l x ( 2 h i x - - ] ) + ( 3 1 n x - - ] - - l ) ln (1

\ m=l '

- - x ) ,

(t = e, ~ ) ,

where m A is a cut-off , which appea r s for t he V ~ - A i n t e r a c t i o n a n d will be r ep laced b y the mass of t he W - b o s o n in a r eno rma l i zab l e t h e o r y . W e recal l

t he f ini teness of t he r a d i a t i v e co r rec t ion for t h e V - - A ease in a c u r r e n t (~) cur- r e n t t h e o r y ; a consequence of V - - A b e i n g i n v a r i a n t u n d e r t h e F ie rz t r ans fo r -

mat ion . I n t ab le I , we p r e se n t t he va lues of t h e func t ions ]r,R(x) a p p e a r i n g in eq. (5).

Co lumns 3 a n d 5 g ive t h e c o n t r i b u t i o n of t h e r a d i a t i v e cor rec t ion , as a

f rac t ion (~o) of t he ba re d e c a y p robab i l i t y , for t h e decays L ~ -~ eVvo,r a n d L ~ - + ~vvCvr, r espec t ive ly , for t h e V - - A case. The s a m e quan t i t i e s for t h e

V ~ A case are g iven in co lumns 7 a n d 9. W e obse rve t h a t t h e r a d i a t i v e

1S - I1 NuOVo Cimento A .

274 A. ALl and z. z. AYDIN

TABLE I . -- Values o/ the /unction /L(x), ]a(x) appearing in eq. (6) o/ the text. C o l u m n s 3 a n d 5

g i v e t h e r e l a t i v e c o n t r i b u t i o n of t h e r a d i a t i v e c o r r e c t i o n s f o r t h e V- -A case . T h e s a m e q u a n t i t y

i s g i v e n i n c o l u m n s 7 a n d 9 fo r t h e V + A c a s e .

x L-> 2 ~ L (V- -A) L-> #v~v L (V- -A) L-+ eVo'n ( V + A ) L-~ ~v~vL ( V + A )

(~ /2~) /dx) ~ (~/2~)/~(x) ~ ~ / ~ ( x ) ~ ~ /~(x)

/~(x) 3 - - 2x 2 ~ / ~ ( x ) 3 - - 2x 4-~ ]~(x) 12~ 1 - - x 4-~/~(x) 12~ 1 - - z (%) (%) (%) (%)

0.1 1.475 52 .0 0 .070 2.5 0 .980 36.3 0 .097 3.6

0.2 0 .484 18.6 0 .060 2.3 0 .358 14.9 0 .086 3.8

0.3 0 .248 10.3 0 .040 1.6 0 .185 8.8 0 .069 3.3

0.4 0 .138 6.3 0 .025 1.2 0.101 5.6 0 .052 2.9

0.5 0 .072 3.6 0 .014 0.7 0 .050 3.3 0 .038 2.5

0.6 0 .027 1.5 0 .006 0.3 0 .016 1.3 0 .024 2.0

0.7 - - 0 .006 - - 0.3 - - 0 .002 - - 0.1 - - 0 .002 - - 0.6 0 .014 1.5

0.8 - - 0 .033 - - 2 .4 - - 0 .009 - - 0.6 - - 0 .017 - - 2.8 0 .006 1.0

0.9 - - 0 . 0 6 2 - - 5.2 - - 0 . 0 1 7 - - 1 . 4 - - 0 . 0 1 7 - - 5.8 0 .000 0.1

0 .95 - - 0 .083 - - 7.6 - - 0 .023 - - 2.1 - - 0 .013 - - 8.5 - - 0 .001 - - 0.6

0 .99 - - 0 .132 - - 13.2 - - 0 .037 - - 3.7 - - 0 .004 - - 14.3 - - 0 .001 - - 2.2

c o r r e c t i o n f o r b o t h t h e V - A a n d V ~ - A c a s e s i s s u b s t a n t i a l f o r t h e d e c a y

L ÷ - + e % o , L , a t t h e l o w e r a n d t h e u p p e r e n d o f t h e e l e c t r o n e n e r g y s p e c t r u m .

I n f ig . 2 w e p l o t t h e e l e c t r o n e n e r g y s p e c t r u m f o r b o t h t h e V ± A c a s e s .

4.0

-~ 1.o

I I o 0.5 1.o

x=2E./m L

F i g . 2. - E l e c t r o n e n e r g y d i s t r i b u t i o n a s a f u n c t i o n of x ~ 2 Ee/mL, f r o m t h e d e c a y

L - - + 3 - ~ v L i n t h e r e s t f r a m e of L - f o r t h e V T A i n t e r a c t i o n s . D a s h e d c u r v e s cor-

r e s p o n d t o t h e r a d i a t i v e t y c o r r e c t e d c a s e , s o l i d l i n e s w i t h o u t r a d i a t i v e c o r r e c t i o n s .

W e h a v e a s s u m e d 1.8 < roT. ~< 2.0 GeV, m,~ = 0.

I ~ A D I A T I V E COI~I~]~CTIONS TO T H E L]~PTO:NIC I ) ~ C A Y S ]~TC,

/, °

275

I I I I O.2 O.4 0.6 0.8

x=2Edm L 1.0

Fig. 3. - The func t i on 3 - - 2 x + (~/2z)fL(x) co r r e spond ing to t he r ad i a t ive cor- rec t ion for t he decay L~-~ e~v~,~ w i t h t h e V---A in t e r ae t i ou us a f u n c t i o n of x.

Tile shnpe of the r~diutive correct ion suggests t ha t , over ~ l~rge r~ngc of x, d l ' /dx could still be t~pproximnted b y ~ line:~r formul~ (2) wi th an effective Michel p,~rameter ()e~" TO il lustr~te this poin t we h,~ve p lo t t ed in fig. 3 the

quan t i ty (*)

(7) s(~) = i - ~ ( ~ / ~ i ~ ( o ) 3 - 2x + ~I~(,~,) ,

corresponding to the decay dis t r ibut ion of L - ~ e-9o,~ for the V - - A case. As is cle:~r f rom fig. 3, ~S~(x) is well descr ibed b y ,~ s t ruight line in the rnnge

(*) Here FL(0) is def ined as 1

FL(O ) = 2fX~/L(X) dx . o

The h m e t i o n s S(x) and St(x) are chosen so as to sa t i s fy t he n o r ma l i z a t i o n cond i t ion 1 1

fS(x)x2dx = . f ~ I ( X ) X 2 d x = 1 .

o o

For the V + A case, the co r r e spond ing func t i o n is

12 ( l - x ) [1 + ~ ]~(x)] S(x) = ~ + (~/2~)F~(o) ~ i Z - x j '

where 1

FR(O) = 2 f x2 /R(x) dx . 0

276 A. ALI and z. z. AYDIN

0.3 < x < 0.95. F i t t i n g th i s l ine t o t h e M i c h e l f o r m u l a

(8) S~(x) 12(] x ) + = - - 8 ~ . ( ~ x - - 1 ) ,

o n e f inds Qe~ = 0.66. Th is shows t h a t t h e Miche l p a r a m e t e r for L ~ -+ e~o~.

c h a n g e s b y 12 % f r o m i ts u n c o r r e c t e d V - A v a l u e in th i s r ange . T h e s a m e

fit to L ~ -> e ~ w i t h t h e V + A i n t e r a c t i o n leads to a v a l u e of ~o~ = - - 0.13

t o be c o n t r a s t e d w i t h ~ = 0 w i t h o u t r a d i a t i v e co r rec t ion . A l t e r n a t i v e l y , one

can d e t e r m i n e e~, b y f i t t i ng S(x) w i t h S~(x) b y us ing t h e m e t h o d of l eas t squares .

T h e v a l u e of ~off so d e t e r m i n e d is v e r y s e n s i t i v e to t h e l ower l i m i t of x. T h e

v a l u e s of ~ f for t h e v a r i o u s r a n g e s of x a r e p r e s e n t e d in t a b l e I I . S ince t h e r e

TABLE I I . - Va~es o/ the e/]ective Michel parameter obtained by the method o/ least squares ]or the various ranges of x = 2Et/m L.

Range of x L-+ evov L L-+ ~v~L

q ( V - - A ) q( V + A) q(V--A) e(V + A)

0.1 < x < 0.95 0.52 - - 0.34 0.71 - - 0.07

0.2 < x ~ 0.95 0.63 - - 0.18 0.72 - - 0.07

0 . 3 ~ x ~ 0 . 9 5 0.66 - -0 .13 0.72 - -0 .07

0.1 ~ x ~ 0.9 0.49 - - 0.38 0.72 - - 0.07

0 . 2 < x ~ 0 . 9 0.62 - -0 .21 0.73 - -0 .07

0.3 ~ x ~ 0.9 0.66 - - 0.15 0.73 - - 0.07

Bare values 0.75 0 0.75 0

wi l l a l w a y s be an e x p e r i m e n t a l cu t on t h e e l e c t r o n (muon) m o m e n t u m , a n d

t h e r e is a l w a y s a f i n i t e - e n e r g y r e so lu t i on , w e h a v e n o t e v a l u a t e d Q~f~ for t h e en t i r e

t h e o r e t i c a l r a n g e 0 < x < 1 (*).

(*) The value of ~ef~ so determined is insensitive to the variat ion in the heavy-lepton mass in the range 1 . 8 G e V < m L < 2 . 0 G e V , relevant for the SLAC-LBL and DESY experiments. In any experimental determination of the electron spectrum coming from L~ decay, one has to impose an experimental cut on the bremsstrahlung photon energies. There is, however, a relation between the maximum photon energy and the energy of the electron, namely E~a~= AxmL/2, where A x = (mL/2)(1--x~aXimum). The value of the Michel parameter , and hence the effect of the radiat ive correction on the electron momentum spectrum, for any experimental cut on the bremsstrahlung photon energy, can be determined from our table I by choosing the corresponding x m ~ l ~ . For instance x ~ 0.95, 0.9 correspond to the photon energy cut E~ ~ = 50, 100 MeV, respectively. Table I I also shows tha t the Michel parameter does not depend sensitively on the upper bound of x (or equivalent ly on the cut on the energy of the bremsstrahlung photon) so long as 1 - - ~ is sufficiently bigger than 2(AEJmL), where A E is the energy resolution of the electron momentum.

R A D I A T I V E C O R R E C T I O N S TO T t I E L N P T O N I C I)]gCAY$ ETC. 277

There are several equ iva len t ways of incorpora t ing the effect of the ra- diat ive corrections for the processes (1), in order to enable a compar ison wi th the expe r imen ta l results . Suppor t ed b y the a r g u m e n t of the preceding para- graph, one can adop t the a t t i t ude t h a t the e x p e r i m e n t a l curves have to be compared wi th the theore t ica l d is t r ibut ions ob ta ined b y using the one-para- m e t e r Michel formula , however , wi th an effect ive ~, which takes into account the effect of the rad ia t ive correct ion adequa te ly . I t is k n o w n t h a t the electron (muon) energy dis t r ibut ion for a h e a v y l ep ton in mo t ion can be eva lua t ed analyt ical ly . One can t hen calculate in a s t r a igh t fo rward way the corrected energy spec t rum replacing o b y ~o~f in these fo rmulae to be compared di rec t ly wi th t he e x p e r i m e n t a l result . W e reproduce he re t he ana ly t i c fo rmula to cal- culate the electron (muon) ene rgy s pec t rum for a h e a v y lep ton in mot ion, val id for bo th the V q - A in terac t ions :

d F 2 [ 4x~ (3 + 3 f l - 4 x t ) - (9) d x t - - f l 1 ( l + f l ) a

,~ge 1 ( 1 4 x ~ + f l p ( 9 + 9 f l - - 1 6 x t ) J ] f m . . . . ~ - t ~ , , . . . . . ,

(lit) d l ' 32x'~ I -d,~'--t = i l --- f l~)~ (3 - - 2 e ) (

( 1 fl'-') + '_, .~ \1 -5

1 - - f l f o r O~:7,rt~-: -

where

and

K t ,r~ - - (t ,a)

/) ---- V/l - - ~tm~/lV 2 , W

Tile electron energy s p e c t r u m for tile decay L ~- =~ e%~, z is p lo t t ed ill fig. 4 for fi = 0.8 bo th with and wi thou t the r ad ia t ive correct ion. We have used 0o , - - 0 . 5 2 , - - 0.3~ for the V - - A, V @ A interact ions , respec t ive ly , to indica te the ex t r eme effect of the n~diative corrections. The effect for bo th the V :F A is significant. However , the two in terac t ions are still dis t inguishable.

Al te rna t ive ly , one can adop t the a t t i t u d e of Kinosh i t a and Sirlin if), b y incorporat ing the rad ia t ive correct ions th rough modif icat ions of the coupl ing constants GL, G R in t roduced in (2). One t h e n obta ins a formula , in which the Michel p a r a m e t e r r emains unchanged , and the decay dis t r ibut ion is modif ied mul t ip l iea t ive ly b y a funct ion depending on ~ and x. Fo r the detai ls we refer to ref. if) and mere ly quote the r e l evan t fo rmula here (using ~v-~ = I and

278

ev+~ = 0) :

i d F (11) / ' d x

{ 4x~(1 + h~(x)) ( S - 2x)

12x" (1 -~ hR(x) - - A3)(1 - - x)

A . A L I and Z. Z . A Y ] ) I N

( for V - - A ) ,

( for V + A ) ,

I \

V+A

I x x

I l I I

0 0.2 0£ 0.6 0.8

Fig. 4 . - E lec t ron energy spec t rum as a func t ion of x ~ Ee /Ebeam f rom the decay of L - i n flight, w i th f l = 0.8. Solid curves for the uncor rec ted spec t rum, dashed ones for t he r ad i a t i ve ly cor rec ted spec t rum, a) V- -A in t e rac t ion ~ = 4, b) V4 -A in ter - ac t ion £ = 0, c) V- -A i n t e r ac t ion ~ = 0.52, d) V 4 - A in te rac t ion 0 ~ - - 0 . 3 4 . W e have assumed 1.8 ~< m L ~ 2.0 GeV and mv~ ~ 0.

w h e r e

(1.o)

hL(x) - - 2~ 3 - - 2x

l~(x) h~(x) = 12~ 1-- x'

1

A1 = ~ j 3 - - 2 x /~ . (x )dx ,

0

1

A~ 2~ f x 3 ---- --z~ ~ / L ( x ) d x ,

o

1

Aa --z~-- ~- f x~ Ja(x) d x .

0

R A D I A T I V E CORRECTIONS TO THE L E P T O H I C DECAYS ETC. 2 7 9

The values of hr(x), hR(x ) can be read f rom table I , those of Ai can be ob- tained by analyt ic integrat ion of the expression appear ing above and are pre- sented in table I I I .

Table I I I . - Values o] the parameters A l introduced in eq. (11) o] the text.

L --~ eve~ L L --> ~ L

A 1 0.027 0.002

A 2 --0.035 --0.010

A 3 --0.022 --0.022

Finally, we would like to com men t on the connect ion of our results with the ones obtainable in renormalizable gauge field theory models. Our remarks

are based on the conclusion of ref. (7), where radiat ive corrections to tz-deeay in gauge models are calculated and compared with the effective current @ @cur ren t theory predictions. A s t ra ightforward extension of the ~-decay to the I,-dec~y shows tha t in a SU2@U~ nlodel of the Weinberg-Salam type (s),

with (L,,L) being sequenti 'd leptons, the Michel pa rame te r ~ is unchanged from its current@9 current theory predict ion up to terms of order ~(m2/m~). I n fact most gauge theory models will reproduce the effective current @ current

prediction for ~o up to terms of order ~(lepton mass)2/m~v . The reason is tha t the Born terms, the bremsst rahlung diagrams and the re levant v i r tmd-photon

diagrams in such models can only differ by terms of order (lepton mass)/m w. Since typical ly m~/m w ~ 1/20, all such te rms arc undetectable .

To conclude, we remark tha t our calculations suggest tha t the radiat ive

corrections to the decay L - ~-~ e-~ovj~ will modify the Michel pa ramete r and the electron energy spect rum appreciably from its bare V - ~ A values. The ra- diative corrections leave the bare V ~ A values pract ical ly unchanged for the decay.

We thank G. K~A)IEI¢ for reading the manuscr ip t and for discussions.

(~) T. W. APPELQUIST, J. R. PRIMACK and H. R. QuI~H: Phys. Rev. D, 10, 2998 (1973). (8) A. SALAd: in Elementary Particle Theory: Relativistic Groups and Analitieity (Nobel Symposium, No. 8), edited by N. SVARTrlOLM (Stockholm, 1968), p. 367; S. WEII~- BERG: Phys. Bey. I~ett., 19, 1264 (1967).

2 8 0 A . A L I and Z. Z . A Y D I N

@ R I A S S U N T O (*)

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(*) T'raduzione a cura della Redazione.

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