MESON DECAYS AND STRUCTURE OF MUON

Progress of Theoretical Physics, VoL 21, No.6, June 1959

Meson Decays and Structure of M non

Mituo TAKETANI*, Yasuhisa KATAYAMA** Paulo LEAL FERREIRA, Gerhard Wilhelm BUND

and Paulo Roberto. de PAULA e SILVA

Instituto de Fisica Te6rica, Slio Paulo, BRASIL

(Received January 17, 1959)

799

Considering the peculiar position of the muon among the elementary particles, it seems reasonable to assume an inner structure for this particle. As it seems we are now reaching a new experimental stage in which the determination of the structure of the muon seems possible, we must consider theoretically how to determine it through the various decay processes. The analyses proposed here are mainly connected to pion and muon decay problems. Assuming the so-called universal Fermi interaction, with V and A couplings, by which the decay processes are realized, our considerations are restricted to the behaviours of the non-local form factor which arise from the muon structure. Also possibilities of the electron structure are discussed. The present analysis sensitively depends on the deviation of the asymmetry parameter. Its deviations from the original local value must be within about 6%, and a value larger than this cannot be explained without considerations of the electron structure and some other sources.

§ 1. Intl"od uetiou

Up to now any difference of properties between muon and electron, except the remarkable mass difference, has not been discovered. A little before the discovery of pion beta-decay,l) the contradiction between experiment and theory was strongly suggesting another possibility of distinguishing the muon from the electron, but the recent experimental values are becoming surprisingly close to the theoretical value, the difference between these results being within the experimental errors. Therefore, the problem of explaining why elementary particles so similar to muon and electron have very different masses, is surely one of the most fundamental problems In future.

Now, it seems to be very difficult to get any key for explaining the large mass value of the muon. *** The possibility of explaining the electron mass, through

* On leave from Department of Physics, Rikkyo University, Tokyo, Japan. ** On leave from Research Institute for Fundamental Physics, Kyoto University, Kyoto, Japan.

*** A. Salam and J. Tiomno (preprint) also start from the same opinion. They supposed the existence of a strong interaction of the muon, in order to explain its mass. Although this possibility must be analysed, it seems to be very unnatural to us. We thank them for sending us the manuscript before publication.

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800 Taketani, Katayama, Ferreira, Bund and Paula e Silva

the muon, is discussed by some of us elsewhere.2)

Concerning the decay rate of TC -) e + v and 1r -) /1 + v, as clarified by Ruderman and Finkelstein3

) and also by one of us (M. T.) and Tokyo group,4) through a

more detailed analysis, only the cases of Fermi interaction with A coupling for

Sakata's scheme* and of j)5 (jJv) interaction for direct Yukawa's scheme can explain

its small value such as given experimentally, From these results, one can under

stand the reason why the possibility of decreasing the ratio is due to the mass difference between electron and muon. And at that time, if one had postulated

very strictly the so-called universality of weak interactions, one would have predicted

the fact that S, T and P coupling cannot be the main terms in j9-decays. On the contrary, today, from the agreement between theory and experiment in TC-e and

j9-decays, one may conclude that V and A couplings are the main terms in the universal weak interaction between leptons and nucleons, and possibly also other

baryons. On the other hand, one could also hope, if the experimental results of

the pion decays' ratio would become more precise, to be in position for deciding

whether the structure of muon and electron are same or not. As is known, Lee, Yang5

) and other6) have considered a structure for the muon

in order to explain the experimental deviation of the Michel parameter from the theoretical value. Though the recent experimental values have become close to the

value given by the two-component neutrino theory, there still remains some pos

sibility of such attempts owing to the insufficient experimental situations.

The experimental evidences of pion beta-decay have made the circumstances

better. However, we still have some doubts about the agreement between experimental and theoretical ratios and we rather believe that the progress of the ex

perimental techniques has been such as to allow the possibility of obtaining information about the muon structure in a more definite way, even in the case in which the departure from the, local value is small; in other words, if the experimental

accuracy could be improved to another decimal order, then the problem would be

come a definite one.

When many attempts were made to forbid the pion beta-decay/) after the experiment of Anderson and Lattes,8) we were inclined to the opinion that the ratio

should at least be the upper limit value given by Anderson and Lattes and that

the departure from the theoretical value could be explained by taking into account

the muon structure. This opinion first came to us by noticing the fact that every

ratio of decay processes, for instance, K-::;.e+v/K-'7p+v, and K-'71r+p+v/K--71r + e + v, sensitively depend on the mass ratios of the produced particles. **

Although the present experimental situation has been much improved, such

* The scheme of pion decay through the nucleon pair was first proposed by S. Sakata, Phys.

Rev. 58 (1940), 83, and afterwards used through the famous triangle scheme by J. Tiomno and J. A. Wheeler, Rev. Mod. Phys. 21 (1949), 153. We then use the name of "Sakata's scheme" for this

mechanism, while the direct mechanism we call "Yukawa's scheme".

** A similar idea has recently been proposed by K. Fujii and K. Iwata, in press.



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Meson Decays and Structure of Muon 801

improvement does not seem enough to give the values of the pion-decay ratio and also of Michel and asymmetry parameters, in the muon decay, with sufficiently strict accuracy for determining the muon structure. The value of the muon anomalous magnetic moment would become another tool for such an atte:rnpt.9) But the experimental accuracy is also a little too rough to exclude the possibility of a muon spreading of about mv~l-..... 2 yukawa. *

The outstanding experimental skill displayed in the pion beta-decay business promises, however, to give better information in the near future. We wish, therefore, to proceed with the analyses of the muon structure within the possibilities of

future experimental values.

Our analysis will proceed as follows: i) If the elementary particles have some structure, this structure will manI

fest itself through the interactions. We introduce then non-local interactions for

the muon and also for the electron. ii) As we can guess the muon structure through the form factor appearing

in the non-local interactions, we must analyse the way of determining it through the various decay processes in which the muon is present.

iii) The baryons also may have some structure related to the strong interactions. In order to separate these problems from the so-called applicability problems in the present field theory, we restrict ourselves to the pion decay ratio, the Michel and asymmetry parameters in the muon decay and kaon decay ratios, i. e., K -)e + J) / K -> P + J) and K -> 7r + p. + J) I I( -) 7r + e + J). Their absolute rates are not discussed.

iv) The analyses are devoted mainly to Sakata's scheme, in which decay processes are realized through the Fermi interaction with V and A couplings, in accordance with the recent experiments. The usual assumption of V - A combination/G) seems to be unnecessary.

Before a detailed analysis, we build up the formalism of non-local interactions expressing the muon structure (~2) and roughly estimate their effects on the pion decay ratio (§ 3) and the muon decay (~4). Such analysis is used to determine the muon structure (~5) and its validity, and effects on kaon decays are discussed (§ 6). The possibility of a structure for the electron and its relation to the muon structure is considered in the last chapter (§ 7) .

~ 2. Formulation of the muon structure

The interactions which represent the lepton structures are only the electromagnetic and weak ones. Restricting ourselves to the decay processes considered here, there is no contradiction with the assumption of Sakata's scheme, in which the decay processes are produced by Fermi interactions between four kinds of

* 1 yukawa= 10- 13 em. As this unit is closely related with the pion Compton length, it is suitable to use the name of Yukawa for it. See M. Taketani, Prog. Theor. Phys. Suppl. 3 (1956), 5.



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fermions. For beta-decay, muon-capture and muon-decay, this scheme has been

almost established since its proposal by Sakata. For the pion decays, there is no

evidence to decide whether they occur through Sakata's or through Yukawa's scheme

with the direct couplings. * Even if we restrict ourselves to the former scheme,

we cannot decide whether this scheme can be extended to include all baryons or not.**

Considering the possibilities of hyperon-lepton decays, we introduce tentatively

the following three kinds of Fermi interactions:

i) NI +!J.~ N 2 +v

ii) -LVI + e:-'? N2 + V

iii) /1.+v~e+v

where N I , N 2 , /1., e and v are baryons, muon, electron and neutrino, respectively.

We assume, moreover, in this scheme, the conservation of electric charge, baryon

number and also lepton number. *** The structures of the elementary particles turn these interactions to the non

local ones.l1) The general form factors appearing in the interactions may depend

not only on the space-time coordinates of the four kinds of fermions, but also on

their properties and also on the coupling types. In order to clarify the model proposed in this paper, we must restrict it by

the following assumptions, which will be introduced step by step.

Assumption I: The form factor are invariant against any translation of coordinates, as well as Lorentz transformation, and their local limits must exist.

The translational invariance restricts the form factor, which initially depends

on the four coordinates of each fermion, from four to three space-time degrees of freedom. Although the requirement of Lorentz invariance is too severe to construct a consistent theory/2) we can hope that the future non-local theory will include it in a consistent way and the deviations from the local theory will not be drastic enough to forbid us to maintain our usual images on the elementary particle physics.

Assumption II: The form factors do not depend on the type of coul>lings. If the form factors are mainly due to an intrinsic structure of the particles,

they will appear in the interaction without a strong dependence on the coupling types. Then the change of the type of coupling could be ascribed, approximately,

to the change of the numerical value of the coupling constant. After the proposal

of Lee and Yang concerning the phenomenological non-local effect on the muon

decay, some generalizations were proposed by Bludman and Klein and also by

Sirlin in order to include the dependence on the coupling types. But from these

generalizations no new results were obtained.

* A little before the discovery of pion beta-decay, some arguments were put forward on this

problem. See S. Ogawa, Prog. Theor. Phys. 13 (1955), 367.

** The discovery of hyperon-muon decay has been reported recently. **,,- We define p.-, e- and)) as particles and p.+, e+ and v as anti-particles.



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Taking these two assumptions into account the Fermi interaction Lagrangeans

can be written as follows: i) Case NI + 1.1. ;::. N2 + v. The form factor depends on the structures of N 1,

N 2 , fl and v.

~ ('

L](NIN 2 pv) = - J dXIdx2d.xsdx4F(XI' X2, Xs, x 4 ; ..LVIN 2flv)

· b J.l1(XI) Oi N2 (X2) ·P(X3)Oi(Ci +r5 C,:')V(X4). (1) i

ii) Case Nl + e;::' N2 + v. The form factor depends on the structures of N 1 ,

N 2 , e and v.

L,(N1 N 2 e))) = -1 dXldx2dxsdx4F(XI, X2, Xg, X4; N IN 2 ev)

· :8 N I(XI)OiN2 (X2) ·e(xS )Gi (Ci +75 C/»))(X4)' i

(2)

iii) Case fl + v;::' e + v. The form factor depends only on the lepton structure.

L/(e/,-vv) =-1 dXldx2dXsdx4F('XI' x 2 , Xs, x 4 ; efl)))))

· ::8 e (Xl) Gill. (X2) . J.,(Xs) Oi (Ci 75 C/) V(X4)' (3) i

Here N I , N 2 , p, e and)) are the baryons, muon, electron and neutrino fields, respectively, which depend on the x/s coordinates; the F's are the form factors. The summation is taken for the five coupling types, i. e., S, V, T, A, P, which correspond to the following operators;

0.8 =1, Oy=ra, 0. - z (.. "" ) 'I'------F c rar~-/~/a , 2V 2

OA= -Zrar5, Op=r5, (4)

Ci and C/ being the coupling constants, which are not necessarily equal for all cases.

Our next task is to separate these form factors so as to make them correspond to the structure of each particle, this being achieved by the introduction of the following assumptions.

Assurnption III: The form factors can be sejJarated in factors mainly

corresponding to the baryon pair (NI, N 2 ) and lepton jJair (f.1.v) or (ev).

This assumption can be accomplished by reducing the coordinate dependences of the form factors, such as

('

F(Xl, X2, Xs, X4; N IN2I"))) = J dX'G(XI, X2, x'; N 1 N 2)F(x', Xg, X4; flV), (5)

F(xl1 X2, X3, X4; N IN 2 ev) = 1 dx'G(x1 , X2, x'; N 1N 2)F(x', X3, x 4 ; ev), (6)



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F(XI' X2, Xs, X4; efl))))) = I dx' F(x', Xl, x 4 ; e)))F(x', X2, Xs; fl))). (7)

As it seems safe to assume that the form factor for the baryon pair is likely to be produced mostly in the strong coupling region, it seems that the separation ac

cording to this assumption is very reasonable. And the same reason may assure the next assumption.

Assumption IV: The structure, i. e. the form factor of baryon pair (Nl'

N 2 ) does not sensitively depend on the leptons.

\Ve can then take

G(Xl, X 2, x'; N1N 2) ~G(XI-X2; N1N 2) (~( X'- ~li~2). (8)

As the separation of the form factor of lepton into the (p-))) and (e)))' pairs

does not seem to be a suitable one, we must consider it more rigorously. But if

we expect the form factors to have the same footing through the three cases, the

previous assumption is not so unreasonable. The ways of the separation into two

lepton pairs are three, as proposed by Lee and Yang, but they are not independent,

independent cases being confined to (/rv) and (e-, v) or (tC v) and (e- v) (v means antineutrino). Then if the neutrino structure is not so important, the sepa

ration can be proposed as above without loss of generality.

Concerning the muon and electron structure, we must discuss the following three cases:

Assumption V--a: The muon structure is more dominant compared to the electron structure.

\Ve can substitute

F( ' . ),-..."J"(! )"( ) X , X3, X4, e v = 0 x - X3 0 Xs - X4 (9)

for eqs. (6) and (7).

Assumption V-b: The electron structure lS more dominant compared to

the muon structure.

In the opposite way to (9), we can take

(9')

for eqs. (6) and (7).

Assumption V-c. Neither the muon structure nor the electron one are negligible.

As explained in the introduction, there may exist some reasons for taking the assumption V-a. Detailed discussions are, therefore, devoted to this case. But as

the other possibilities remain, that is V-b and V-c, we shall discuss them later. Assuming the canonical formalism, and using eqs. (5), (6), (7), (8) and (9),

we have for the first order Hamiltonians:



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where

Meson Decays and Structure of Muon

H] CiIIN 2 /J.v) = (2rr) -2 J drl] dp2 dp[J- dp'll()4 (PI- P2+ pIJ. - jY'II)

o :::s NI (jY l ) 0,: N2 (P2) . P. (p[J-) Oi (Ci + 75 C/) v (P'II) i

· G[ - (Pl- P2) 2/ M12, M22/ M12]

o F([J-)[- (p[J-_plJ)2/m [J-2,-p,//m l/, _j)lJ2/m [J-2],

HJ(N1-L~2e v) = (2rr) -2 J dpIdp2dPe dP'v(j4(PI-P2+ jYe - j)'II)

· ::8 Nl (h) Oi JV2 (P2) . e (j)c) OJ (C; + 75 C/) v (P'v) i

('

l-I](e /11,;1/) = (2 rr)-2 ,\ dpedjYJLdj--;lJdp/ (~4(Pe- jY[J-+ P'II- jY/)

· ~ e (jY e) (),:f~ (jJ1J . ~ (PlJ) 0,: (Ci + 75 C/) v (p/) i

G (Xl- X2; NI N 2) = (2rr) -4 j' d j) eip(Q-.c21 G[ - p2/ M12, M2 2/ Mtl,

F (Xl, X2, X3; /1 v) = (2rr) -8 J dk dl ei1c(q-.I 2 )+iZ(1:2-;:3)

o f([J-,[-P/mIJ. 2, - (k-l)2/mr"'Z' -P/mIJ.2].

805

(10)

(11)

(12)

(13)

For the decay processes considered here, these first order Hamiltonians are sufficient. Moreover, in this approximation, as we consider the leptons to obey the free Dirac equations, i. e., - PI",2/m~JL2= 1, - p'II2/m[J-2=O, the form factor for muon becomes

which is essentially a one parameter function. We can get, therefore, from the decay processes, information about the behaviour of this function only in the - (jY[J- - P'v) 2/ m[J-2,> 0 region, just opposite to the case of electron-proton scattering.

PilOn decays

The present evperimental evidencesl) ,13) on both pion decays, i. e., rr----fe+v and

rr -) /1 + v are in agreement with the usual theoretical predictions, which take into account similar mechanisms for both cases, regardless of whether direct interaction

exists between pion and leptons or not. Both the direct (Yukawa's) scheme and the indirect (Sakata's) scheme give the same formula for the ratio of these decays,

(15)



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The coefficient a expresses the detailed contributions for the ratio, due to: i) non-derivative coupling and derivative coupling effects for Yukawa's scheme and

ii) P and A coupling effects of Fermi interaction for Sakata's scheme. This term can be written as

(16)

where

r p·) = F L~ m 21m 2'1 { (r; ± 07') I 1 / m. (f± to,) } ± \lL) '" lL . • T / ,L. - , (17)

if we start with

H(rrP.v) = (2 rr)-21 dp" dP1Ldp\J;;4 (!J",--jJlL+P,;) F(lL) [ -p",2/Jn p•2l

, ¢(p",)p.(jJp.Hi r5(f+r5f') +ir5(irp,,) (Y+r5Q') ] v (P\J) (18)

for Yukawa's scheme, where ¢ (jJ7r.) is pion wave function with momentum p", and

f, f' and y, y' are the non-derivative and derivative couplings constants respectively.

Similar expression is given for the electron case.

On the other hand, for the case of Sakata's scheme, we obtain

I (tL)-F [', 2/" 2] '" {(C ±C ') /I( 2/M2 " 21M2) ± - (JL) rtl,,, / nl lL , . 2_1 A A ,[:1 rn,,, I 1, In", I J. '[2 "V\,.Nz

(19)

and the similar expression for the electron case, where the summations are taken

on all baryon pairs lV1 , N2 compatible with strangeness conservation, and the func

tions A, P are the contributions from each pair through A, P couplings respec

tively.

The possibilities of getting small values for the decay ratio as indicated experimentally are either i) the case in which the non-derivative coupling in Yukawa's

scheme, or P coupling in Sakata's scherlle, is negligible compared to the other couplings or ii) strong cancellations occur for the electron case in Sakata's scheme.

At any rate, it is necessary to retain the derivative coupling or the A coupling of

Fermi interaction for the pion decay processes.

Although the present information give no decision as to the mechanism of pion

decays and, consequently, cannot exclude the second possibility, it seems to be more

suitable to take Sakata's scheme with the first possibility.

If the pion decays are realized through Sakata's scheme, they are closely related

with the ;9-decay phenomena. Although all recent experiments on j9-decay have

been in favour of V and A couplings, it is still very difficult to exclude S, T, P couplings completely.

The theory with only V and A couplings, however, have two good advantages

for the pion decay. As we can discuss the ratio without knowing much about any

strong interaction phenomena, the ratio may give some information about the weak



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interactions and the structure of leptons. And we must not worry about the competition between 1T---)e+v+r and rr-~p+v processes, as before, when all experiments were indicating the Sand T couplings.

The universality of Fermi interactions with the restricted form IS not yet established rigorously, but it seems to be natural and simple to take this standpoint. We assume then:

Assumption VI: All Fermi interactions of the V, A type are considered

universal. * With this assumption, (15) becomes

R=1.3 ·10-4 [FCe) (m"" 2/m(2) / FCI-L) (m?t2/mIL2) J2

and with the assumption V-a,

R= 1.3 . 10-4 [F(IL) (m",,2/m\L2) J- 2•

(20)

(21)

The muon structure reveals itself through the last factor, i. e., the value of the form factor at a point m"" 2/m I-L2= 1.7. Therefore, we can get some information

about the forn1 factor fronl

(22)

Although the experimental value of the ratio Rex]) is not yet definitely established, we may at least take**

that IS,

(23) ***

As far as V and A couplings are concerned, there IS no difficulity for pion

radiative decay 1T~>e+v+r. The ratio of this decay to pion-muon decay, R" becomes

(24)

Taking the theoretical ratio14)

* Strict universal Fermi interactions with V and A couplings have the form of V -A. Other

possibilities have only a restricted meaning. For instance, the statement that V and A couplings

depends on the orders of operators. The proposal here considered is restricted to the charge retaintion

order chosen in the previous chapter. Even if we start with V and A couplings for (FOp.) (j)"O/J.I) i. e. charge retaintion order, this gives not only V and A couplings, but also Sand P couplings for (eO))) (-yOI p.) i. e. charge exchange order.

** This value was given by T. Fazzini et al. loco cit., and it is almost equal to the maximum value given by H. L. Anderson and C. M. G. Lattes, loco cit.

*** The recent values reported are all surprisingly close to 1.3 X 10-4• The maximum value is CERN's (1.4±O.2) .10-4• Although the experimental errors are still large and the way of correcting

them incomplete, we may safely put the minimum limit for f to O.8~O.9.



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and considering the condition for the form factor (23), the theoretical value IS not inconsistent with the experimental value15

)

R, exp:S 8 .10-6•

§ 4. J\'[uon decay

The muon decay is one of the best examples for the consideration of the

existing weak interaction theories. No strong interaction intervenes in this process

and the radiative effects are at most of the order of a few per cent, the field reac

tions of the weak interactions being negligible. Muon decay may then give some

information about the muon structure, within the uncertainty of the radiative effects. The calculations of the radiative effects worked out by Kinoshita and Sirlin16

) have

indicated that the corrections to Michel parameter p is about 6%, while the correction to the asymmetry parameter is almost zero. Therefore the asymmetry para

meter seems to be more sensitive with respect to the muon structure than pis. The energy and angular distribution of electron is given by

dN = 7n!Lv,a dSd dx x 2 [{Ki:- -xK;r +i. (I-x)K-} 3.26 7.: 4

=f i. cos(j {-Ki + (2-x) K 2+ + 1/;' (I-x) K-} ]

III virtue of Hamiltonian (12), where

and

x= !Pel/!Pemax!; a= C,,2 + C,,'2+ C}+Cj2,

i.a= -2 (CA'C,,+CA C/)

1

Kt=3'\ duui[f(xu) ±f(x-xu)], I(-~Kl=K2' o

(25)

(26)

In order to see the role of the form factor, we assume it to be so smooth

that its derivatives higher than three can be neglected. We have then

N In Va ') 2['{ . / x2(5-4x) (/ d - !L d S~ dx x (3 - 2x) j (x 2) +--- -,-, ---fl! X 2)

3 . 26 jT4 _ / 40 '

+!.~(12--- xlV' (x/2) +:~f!l' (x/2) )} =t=). cos (j { (1- 2x) f(x/2)

+_~2(3=_~.1:) fl! (x/2) +~(~-: xt (I' (x/2) +--:;C~f!!1 (x/2) }] (27) 40 21. 40

for electron spectrum and

3 _ 3 A 1 + (I-i.)A2/3 11 P ==- -- 0 - - ---- --------------.----' 4 j 32 A o-(I-i.)A2/24 ,



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Meson Decays and Structure of Muon

L1~ ==1._ ~ =L __ 131 *(1 ~!)_2I?2/3_ 10 Bo- (1-i.)B2/20

809

(28)

for the deviations of Michel and asymmetry parameters from the local values, where

Ao=f-~§-f' +}Z-f" _--'!l-f'" 8 18 1440'

(29)

and

Bo =f-~ f' + --~---f" - 1 l''' 5 5 21"

B1 =f'-~-~-f" +~li~of"" B2 = f'---~ f" +2~lf""

the values of all functions, f's, being taken at the point m fL2/m,/= 1.

The remarkable consequences deduced from the above formulae are the follow-mg:

i) The Michel parameter depends on the coupling constants through ). in virtue of the form factor, in opposition to the local V and A theory. But if we expand it in terms of the deviation from the V - A theory, i. e., I-A, we get

(30)

As we can estimate the first correction to be of the order of a few percent, the contribution of the deviation from the V - A theory then becomes negligible.

ii) The correction to the asymmetry parameter is proportional to A, as

~ ~i. [( 1 + 1~ ~~) + 2~0- (1-i.)~~ -j;:]. (31)

Then ~ is sensitive to the initial value A, and gives a good support to the V -A theory, but not to the two-component neutrino theory.17)

iii) U sing for F efL) the usual functions

1[1 + E2 P/mfL2]-r,

F(fL) ( - k2/ mfL2) = [1 + E2 P/2mfL 2]~ __ 2,

exp[- E2/?,2/m p•2],

with small values for E2, we have*

([1- E2 X]-2, Yukawa

f(x) = 1 [1- E' x/2]-" Exponential (32)

~ exp (E2 x), Gauss

(33)

* Similar conclusion for p was taken by Lee and Yang, loco cit. The tendency of ~<A IS

different from Sirlin's calculation, loco cit.



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The last conclusion seems to be inconsistent with the experimental tendencies

which favours Lip?: 0 and LI ~ > o. The possibilities of removal of this descrepancy

are the following: the first is to include the radiative corrections which may cancel

out LIp, that is, making LIp ~ o. The next possibility is to take into account the

higher derivative terms, like f" and f"! in order to realize the usual tendencies. As the former possibility seems to be too accidental, we should consider the latter. If this procedure would be correct, the form factor could be restricted by more

detail. The next chapters are devoted to this procedure.

§ 5. Analyses of the form factor

We discuss here the way of determining the form factor from data gIVen by

the decay processes raised before. If we restrict the behavior of the form factor to the energy region between 0 and m'JO the information available are: i) condition

of local limit, Ii) Michel parameter, iii) asymmetry parameter and iv) 7r->e+))/7r-> p+)) ratio.

The experimental data about the last three points are not yet definite and the

values reported recently may change to some extent. The data presently available

are the following:

('exp: 0.68 ± 0.02,18) 0.67 ± 0.05/9) 0.74 ± 0.02,20>1) 0.79 ± 0.04. 20ll )

¢exJl: 0.87±0.12,21) 0.909±0.144,22) 0.95:::1::0.06,201» 0.97±0.05,20b) 0.89±006.24)

Rexp .104: > 0.4,1) 0.9:::1: 0.4/3

) 1.24 ± 0.08,20(:) 1.3 ± 0.2,20d) 1.4 ± 0.2.20e)

Judging from these experimental circumstances, we are obliged to see how the behavior of the form factor changes if we make the sufficient allowances for these

three data, as the aim of this paper is to predict the possibility of the muon struc

ture when the experiments would give the definite values for them. It seems

sufficient to take, for the allowance regions, the domains

0.65 ::S p::S 0.85, 0.85::S ~ ::S 1.15, 0.5·10-4 ::S R::; 1.6.10-4•

In order to simplify the interpretation, the V - A combination which makes ). = 1 is chosen in the following analyses, but this is done only for the sake of

convenience and sometimes this choice may become unsuitable for the interpretation

of the results as discussed afterwards.

If we use the following definitions of the deviations for the case of V - A combination

(34)

the above mentioned regions correspond to

- 0.10::; LIp::; + 0.10, - 0.15::; LI~::; + 0.15, - 0.20:S Llr::; + 1.6. (35)

It may be sufficient to assume that the form used here is effective at most up to its third-order derivative and there is no data from which we can get infor-



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mation beyond this restriction. The validity of our approximation then depends on whether the form factor behaves so smoothly that its derivatives higher than the

third can be negligible: Assumption VII: The form factor here proposed should be effective at

most up to its third-order derivative. If one accepts this assumption, one can get the following equations from the

four conditions stated before.

where

1 f(O) -.cf· (l-X+0.50Y -0.17Z),

Lip = - (0.09X - 0.05 Y + O.OIZ) I (1-0.63X + 0.21 Y - 0.05Z),

Li;= + (0.10X-0.06Y +0.02Z) I (1-0.60X+0.20Y -0.05Z),

Lir= + (1.69X-0.26Y +0.22Z) I (1-X+0.50Y -0.17Z),

f f(l), X f' (1) /t(l), Y =.P' (1) If(1) , Z f"' (1) II(1).

(36)

Then if we take the conditions (35), we can know the allowance regions of X, Y, Z. Though the approximation used here depends on the assumption VII, one must bear in mind that they only give the rough tendencies of the form factor.

In order to judge the general tendencies, we select the suitable values distributed within these regions and indicate them in Figs. 1, 2, and 3.

vVe retain Lip = 0 throughout all Figures, because the devi::ttion of the Michel parameter cannot be fixed without definite knowledge about the radiative corrections.

The "reasonable" curves thus obtained are three curves corresponding to the value Lip=O and Ll;=O with the three different values of Llr(lb, 2b, 3b). They

f( -- k2/mlL 2)

Llp==O.OO Lls=-0.13

1

0 I 1

f~ I Llp=O.OO

Lls=-O.13

1

0 I 1

I 1.7

I 17

a=-O.67 Llp=O.OO "(=1.50 Ll~=O.OO

I

1 I

1.7 Fig. 1. Ll r=O.OO

a=-219 Llo=O 00 b=+1:04 tJs=O:OO

...

I 1

I 1.7

Fig. 2. Ll r=0.40

a=O.OO Llp=O.OO "(=0.00 Lls=0.05

a=O.OO .dp=O.OO b=O.OO Lls=0.05

I 1

I 1

I

1.7

I 1.7



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.:1.0=0.00

.:1~=-O.l3

.

a=-2.21 .:1.0=0.00 b=+1.30 LI~=O.OO

1 l::-----+---;;;-. ----• 0

. . .

a=O.OO Llp=O.OO b=O.OO LI~=0.05

eeo (loe

O~--------------------~--------------------~--------------------~ I 1

I 1.7

I 1

I 1.7

Fig. 3. Li r = 1.60

I 1

I 1.7

Behaviors of square of the form factor I( -k2/mr,;2) =F2( -1?2/mIJo2), corresponding the

deviations of three values Lir, Lip and Li';, defined bs eqs. (34). The ordinate axis corresponds

to the values of square of the form factor and the abscissas the values of - k2 in unit of

m',.t2. The solid lines are phenomenological ones determined from our analysis, and the dotted

lines are approximated analytical curves. The straight line is the standard local limit.

show that there is some possibility for changing the pion-decays ratio without

destroying the muon decay spectrum predicted theoretically. We could approximate

them by choosing either the exponential, Yukawa or Gauss type for the form factor

within the experimental errors. But to retain the least deviations from Lip =--= 0, and

Li~ = 0, it is more suitable to take the Gauss type, that is, we take

1,

f(x) = exp [0.20 xJ, exp [0.56 x J,

which correspond to

1.3.10-4,

for R= 0.9 ·10-\

0.5 ·10-\

(37)

(38)

the corresponding spreads* m space-time being 0, 1.0 m~l ~--:-1.4 yukawa, 1.7 m~l ~:

2.4 yukawa, respectively.

All other curves can be interpreted as some deviations from these standard

curves. As the tendencies of their deviations are common to each Figure, they can be approximated by the following formula with some correction factors Z (x),

(39)

where E2 takes the values 0, 0.20 and 0.56, corresponding to Fig. 1, 2 and 3 re

spectively. All curves appearing in Fig. 3 can be realized by taking

* We define the spread as (r2)=-6dF(IL)(-k2/m2IJ)/d(--k2/m211h2=o, but this does not mean

that the muon has such a spread.



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f3 (x) ~: [1 +ax+bx2] exp [0.56 x] , (40)

the exceptions being the cases for LI~ <: +0.06, which will be discussed later. We also plot these approximate curves, dotted with the coefficients a and b which were chosen so as to make minimize the deviations from the Lip and LI~ values taken from the beginning.

The use of similar correction factors, as applied to Fig. 3, becomes not good for certain part of the curves in Fig. 2 which resembles the_ behavior of a cubic function. Judging from the nature of the approximations used here, it may be more convenient to approximate with quadratic functions, because we may expect all third derivatives to be very small. Then we also take

j~(x) ~: [1 ax+bx2] exp[0.20x] (41)

for all curves in Fig. 2. Since the standard curve III Fig. 1 corresponds to the local case, i. e. the

straight line, the other curves are very sensItIve to the values of Lip and Li~. The rough behavior of all curves appear as sine-like. The functions which best fit them are

(42)

Although they are very curious, we indicate them III Fig. 1 by the dotted :lines.

As indicated in Figs. 1, 2, and 3, the approximation used becomes poor in the :region of LI~ > o. One can read from the Figs. that the variations of curves are sensitive to the variations of LI~, specially in its positive side (but not to LIp). In order to understand the circumstances, we add three curves corresponding to ,,1 ~ = 0.05. We also point out the region of validity of our approximation in Fig. 3. From this region, it is safe to restrict the deviations Lf ~ into

LI ~::S 0.06. (43)

We are then obliged to conclude that the large deviations as LI~ ~ 0.13 (which corresponds to the value ~ = 0.87) cannot be realized by the above considerations. Even if the experiment about the asymmetry parameter is very difficult, we hope that more experimental data must be accumulated. should consider some other reasons. The following consideration is one of the possibilities: The statement is to change the C,,j Cv ratio from -1 as in the /3-decay. From (27), we have ~ ~ i. [1- Bi10Bo]. Supposing the correction Bi10Bo would be the maximum value, i. e. 0.06 from (43), and ~=0.87, then we would get A ~ 0.93. With use of the definition of A and (taking) the two-component neutrino theory, CA=Ci and Cv=Cv', this gives

CAICv~-1.5

which IS not inconsistent with cAlev ~ -1.2 for {i-decay. If we accept the usual



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prescription of the renormalization theory, CA/Cv of J9-interaction is less than the corresponding quantity for p - e interaction, because the axial vector coupling of nucleon pair will become less than in the case of p. - e interaction, owing to the

renormalization effect (CA reIJ.::S CA omen.) of the strong interaction.

§ 6. Kaon decays

In considering the kaon decay problems, we apply the same line of thought.

Comparing it to the pion and muon decays, the present experimental knowledge about kaon decays is rather crude. Although we cannot expect any detailed infor,"

mation about the muon structure, it may be worthwile to know the asymptotic behaviour of the form factor previously discussed, as any small variation of the form factor in the region from 0 to m.,-c will be amplified in the region where the kaon decays are considered, i. e. from nl£(- m.,-c to mk. The experimental values for the kaon decays, if their accuracies would be improved, could then give a good criterion for the analysis of the muon structure.

The considerations are here restricted to the K -> e + v / K -> p. + v and the K-> 7r+p+v/K-)7r+e+v ratios, as a similar argument used in the case of pion decays, concerning the strong interaction effects applied here.

i) K->e+v to K->p+v ratio Taking Sakata's scheme of Fermi interactions with V and A couplings, we

have

(44)

which is similar to the pion case. If we extrapolate the form factor obtained in the previous chapter, its maximum

value becomes about 102 (or 105) for the case of R=0.9 X 10-1 (or R=0.5 X 10-4

).

That is, we have

10-7 (or 10-10) < R < 10-5

"""" £(2""",, ,

which is not inconsistent with the present experimental knowledge. ii) K->7r+p+v to !(->JT+e+v ratio

(45)

Assuming Sakata's scheme with Fermi interaction, the matrix element for muon

decay is given by If the such large deviation of the asymmetry parameter would be correct, we

M(~) = 2.:.i [G (p If, P'lt; M1 M 2) P R<t + Gf (1) K, P'lt; Ml M 2) i>'1C <tJ . J(~) lVl,.N2

(46)

where J(~) = [p /(0 P,,, 0J-1/2 ~ ( - P\I) r <t (Cli + 1'5 C/) P (PrJ,

the summation is taken over all baryon pairs (N1' N 2) which have the total



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strangeness equal to -1, and the unknown strong pair effects appearing through G and C'. The total probability per unit time is

p ---1 dP/ dp""O. f· [mJ((2P{L0 j)VO + mf((p71:°- M ) ) IHI2+7nIL2(M - 1)%°) IC'1 2

- 2m{L2l>vO Re (C' 1-1) J, (47)

where

PVo=mk-P%o-P{L°, M= (mIl+m71:2-m,/)/2mx (pion maximum energy),

H=C+C', G= ;S C(PI(, P71:; M 1 M 2) , G'=::8 C'(PJ(, P71:; M1 M 2) , ~,~ ~,~

f=F(~)[1 +2mI((M-j)'1cO) /m{L2]. (48)

Since we cannot get the exact value for the ratio, the following assumptions are introduced:

i) Concerning the form factor, we suppose it to be a monotonic function in

the region 1:::S 1 + 2m R(M - p." 0) / m/ :S 12, and we replace it by the average value. ii) C and G' are almost independent of Pot. We have then* :

where

As the perturbation calculation gives

(50)

we can estimate 7)'::::' 1/2. The ratio becomes

(51)

If we take the maximum value of f(6) as ....... 3 (or ,"",30) for R=0.9·10-4 (or R= 0.5 .10-4

), we have

(52)

This result is still consistent with the experimental value Rexp'::::: 1.16=:8:~! .23)

(The case of R= 0.5.10-4 gives too large a value, so that this case must be exduded).

§ 7. Possibility of the electron structure

In the previous chapters, we have assumed the existence of a structure for the muon and discussed the way how it could be determined from the meson decay

* Our result is a little different from the result of R. F. Streater and J. C. Taylor, Nucl. Phys. '7 (1958), 276. Their result is 0.92-0.42 ReCl;) +0.0811,l12 for the second factor of (49), a modification which replaces (51) by RKa:::::' 0.73 f(6).



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816 Ta,~etan£, Katayama, Ferreira, Bund and Paula e Silva

processes. Since the problem is not, strictly speaking, a definite one, still depend

ing on the future experimental results, it may give, at least, a possible answer to the questions raised. We cannot of course, deny the rather probable case of an

agreement between the experimental and the usual theoretical local values. On this point, we can consider the tvvo standpoints: either the muon structure may be too small to be experimentally detected, or the muon structure may be masked by some other effects, for example, radiative corrections or by the electron structure.

Although the assumption V-a which takes into account only the muon structure

seems to be preferable as stated in the introduction, we cannot reject the opposite

possibility, in which the electron structure plays the principal role (assumption V-b). This last possibility, is somewhat akin to the classical picture of Lorentz. And we

also must consider the third possibility given by assumption V-co For the V-b case, the form factor appearing in muon decay must be replaced

by Fee) (-k2jrne2) instead of FelL) (-1?2jm lL2) , in virtue of (9 /). For the ratio of

the plOn decays, we have

(53)

As the effect of the form factor is opposite, if we take the Gauss type of the form

factor as the case of V-a, we have

(54)

In the case in which we consider the electron structure, the way to realize the

same experimental possibilities as before, i. e.,

(55)

1S to introduce a stronger form factor, such as

(56)

But the effect of this form factor for the muon decay will be opposite to the

former case. For the V-c case, the effect on the pion decay ratio becomes

F(e) (m'If, 2jrn e2) j F(lL) (m'If,2jmfL2) ,

while for the muon decay, the effects will appear through

F(e) ( - k2j JJl e2

) • F(fL) (-Pj miL 2) .

Summarizing the above effects, through their rough estimates, we may conclude

as follows: 1) If the pion decay ratio would become larger than the usual theoretical

value, one should consider two possibilities: either the smooth (simple) electron structure or the singular complex muon structure would be dominant. The contri

butions to the muon decay would be opposite from one use to the other.



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2) If the pion decays ratio would become in agreement with the usual theoretical value, one should also consider two possibilities: either the both structures are negligibly small or both of them are almost the same. The spectrum of the muon decay could decide which of them is correct, because, in the latter possibility, the deviations of the parameter might be doubled.

3) If the pion decay ratio would become smaller than the usual theoretical one, two possibilities should again be considered: either the smooth (simple) muon structure or the singular complex muon structure would be dominant. The contributions to the muon decay would be opposite from each other.

A{~knowledgment

The authors express their deep gratitude to Prof. L Lopes and Mr. J. Leal Ferreira for their stimulating discussions. We are also most indebted to Mr. D. Rodrigues de Oliveira for his invaluable collaboration in the numerical computations.

References

1) T. Fazzini, G. Fidecaro, A. W. Merrison, H. Paul and A. V. Tollestrup, Phys. Rev. Letters 1 (1958), 247.

2) Y. Katayama, M. Taketani and J. Leal Ferreira, Prog. Theor. Phys. 21 (1959), 818. 3) M. Ruderman and R. Finkelstein, Phys. Rev. 76 (1949), 1458. 4) S. Nakamura, K. Ono, M. Sasaki and M. Taketani, Prog. Theor. Phys. 5 (1950), 244. 5) T. D. Lee and C. N. Yang, Phys. Rev. 108 (1957), 1611. 6) S. Bludman and A. Klein, ibid 109 (1958), 550; A. Sirlin, ibid. 111 (1958), 337. 7) J. Taylor, Phys. Rev. 110 (1958), 1216; H. Umezawa and H. Miyazawa, Nuovo Cimento

9 (1958), 586; C. G. Oliveira and J. Tiomno, Notas de Fisica (C. B. P. F., IV, 5, 1958) ; P. Prakash, in press; R. P. Feynman, Report in "Proceedings of the 1958 Conference on High Energy Physics", Geneva.

8) H. L. Anderson and C. M. G. Lattes, N uovo Cimento 4 (1957), 1356. 9) V. Berestetskii, O. Krokhin and A. Khlebvilov, J. E. T. P. 3 (1956), 761; T. Coffin, R. L.

Garwin, S. Penman, L. M. Lederman and A. M. Sachs, Phys. Rev. 109 (1958), 973. We express our thanks to Prof. S. Machida for his discussions on this problem.

10) R. P. Feynman and M. Gell-Mann, Phys. Rev. 109 (1958), 193; E. C. G. Sudarshan and R. E. Marshak, Phys. Rev. 109 (1958), 1860.

11) H. Yukawa, Phys. Rev. 91 (1953), 415, 416. 12) See, Proc. of the International Conference of Theoretical Physics, Kyoto, Japan. (1953),

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Rev. Letter 1 (1958), 249. 14) S. B. Treiman and H. W. Wyld, Phys. Rev. 101 (1956), 1552. 15) J. M. Cassels, M. Rigby, A. M. Wetherell and J. R. Wormald, Proc. Phys. Soc. 70 (1957),729. 16) T. Kinoshita and A. Sirlin, Phys. Rev. 106 (1957), 1110; 107 (1957), 593; S. M. Berman,

ibid. 112 (1958), 267. 17) L. B. Okun and Y. M. Sehter, Nuovo Cimento 10 (1958), 359. 18) Rosenson, Phys. Rev. 109 (1958), 958. 19) Crowe, Bull. A. P. S. 2 (1957), 206. 20) Reports at Meeting of Weak interaction, Gatlinburg, 1958.

a), c) Berkeley group, b) Columbia group, d) Chicago group, e) Cern group. 21) Wilkinson, Nuovo Cimento 6 (1957), 516. 22) R. A. Swanson, Phys. Rev. Letter 1 (1958), 276. 23) G. Alexander, R. H. W. Johnston and C. O'Ceallaigh, Nuovo Cimento 6 (1957), 478. 24) G. Lynth, J. Orear and S. Rosendorff, Phys. Rev. Letter 1 (1958), 471.



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MESON DECAYS AND STRUCTURE OF MUON

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