VOLUME 2, NUMBER 4 PHYSICAI. RKVIKW LKTYKRS FEBRUARY 15, 1959 spin. one finds the rate for muon decay accompanied 4Horwitz, Miller, Murray, Schwartz, and Taft, Bull. by photons of energy greater than ao, . Am. Phys. Soc. Her. II, 3, 363 (1958). n mp 171 1 A(yo) = — I(ln — 6 ln —- 6 {1-y )- (1-y )o ~ ~ Sw iI~~ m 12] yo - RADIATIVE DECAY OF THE MUON* + S [L (1) L (yo)1- o [6+ {1-yo)'1{1-yo)»(1-yo) — Toichiro Kinoshita + (1-yo) (125+45 yo SSyoo+ 7 yoo), , (2) Laboratory of Nuclear Studies, Cornell University, Ithaca, New York, where and Alberto Sirlin Physics Department, Columbia University, New York, New York For yo«1, Eq. (2) coincides with the expression (Received January 26, 1959) obtained by integrating Eq. (25b) of reference 2 over all electron energies. The inner bremsstrahlung accompanying the Equation (2) shows that about 4.9% of muon de- muon decay (p e+ v+ v+y) has been studied pre- cay is accompanied by photons of energy greater viously, and its contribution to the radiative than 2me, the threshold energy for the produc- correction of the decay spectrum, the lifetime, ' ' tion of electron-positron pairs. Even for (uo-20m&, and the asymmetry has been analyzed in detail. the rate is still as large as 1.2%. Thus it seems Recently, some other aspects of this problem likely that the inner bremsstrahlung accompa- ~ 6 have been discussed by several authors. The nying the muon decay may be detected and inves- purposes of this note are to derive the spectrum tigated experimentally with techniques now avail- of the photon emitted in the decay of muons in able. the framework of the two-component theory of Equations (1) and (2) constitute a definite and the neutrino and to point out the possibility that clear-cut prediction of the two-component neu- this effect may be observed with present tech- trino theory and of the usual laws of quantum niques. electrodynamics as applied to the electron and The probability that a photon and an electron muon fields. In particular, they are independent are emitted in muon decay with any given ener- of the relative magnitudes of the V and A inter- gies and momenta has been given by Eq. (17b) of actions in the order (ep, ) {vv) or, what amounts to reference 2. To find the photon spectrum in the same thing, of the S and V interactions in the which we are now interested, we have only to order (ev)(vp). Thus, if the predictions of Eqs. integrate it with respect to the energy and angle (1) and (2) were to be contradicted by experi- of the electron, taking account of the conserva- ments, the consequences would be far reaching tion law of energy and momentum. The result is indeed. given by The question naturally arises as to whether measurements of the radiative decay rate can dN a Il p 17 (1-y)dy 2 ln — — - + ln (1-y) ~ new No 3m' give us information about the nature of the weak interaction in the more general framework of the four-component neutrino theory. This is 3 I - 1 x —-2 (1-y)o (1-y)(22-1Sy), (1) now being studied. y 12 One of us (A.S.) would like to thank Professor where No is the total probability of muon decay T. D. Lee for his comments and Dr. M. Bardon, without radiative corrections and y = 2&v/m„ is Dr. D. Berley, and Dr. J. Lee for useful con- the photon energy in units of its maximum value. versations. In deriving Eq. (1) the mass of the electron has been neglected in comparison with its energy whenever this approximation has not led to spu- Supported in part by the joint program of the Office of Naval Research and the U. S. Atomic Energy Com- rious divergences. This is certainly justified mission. when the condition (1-y)' » (2 ms/m&)* is fulfilled. A. Lenard, Phys. Rev. 90, 968 (1953). Integrating Eq. (1) further over the range of ~Behrends, Finkelstein, and Sirlin, Phys. Rev. 101, photon energy between = and = y =yo 2&so/m & y 1, 866 (1956). 177 VOLUME 2, NUMBER 4 PHYSICAL REVIEW LETTERS FEBRUARY 15, 1959 3T. Kinoshita and A. Sirlin, Phys. Rev. (to be pub- If by the rate of p e+ v+ v+ y one understands, the lished). total rate of radiative decays with photons of energy 4R. H. Pratt, Phys. Rev. 111, 649 (1958}. greater than a certain energy ~0, then their result is ~N. Tzoar and A. Klein, Nuovo cimento 8, 482 (1958). at variance with ours unless one arbitrarily chooses a In this paper, it is asserted that the relative rate for relatively large value of ~o. inner bremsstrahlung is of order 10 per muon decay. 6C. Fronsdal and B. Uberall (to be published). 178.