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Proceedings of the TRIUMF Muon Physics/Facility Workshop, Vancouver, August 8-9, 1980 MacDonald, J. A.; Ng, J. N.; Strathdee, A. Feb 28, 1981

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TRI UMFPROCEEDINGSPR OCETRIUMF MUON PHYSICS/FACILITY WORKSHOPDINGPSDET  ISUSMO FHY/ AYFLEditors: J. A. MACDONALD J. N. NG A. STRATHDEEMESON FACILITY OF:UNIVERSITY OF ALBERTA SIMON FRASER UNIVERSITY UNIVERSITY OF VICTORIAUNIVERSITY OF BRITISH COLUMBIA TRI-81-1PROCEEDINGSPR OCETRIUMF MUON PHYSICS/FACILITY WORKSHOPDINGPSDET  ISUSMO FHY/ AYFLEditors: J. A. MACDONALD J. N. NG A. STRATHDEEPostal address: TRIUMF4004 Wesbrook Mall Vancouver, B.C. Canada V6T2A3 February 1981TRI-81-1PREFACEThe TRIUMF Muon Physics/Facility Workshop was held August 8-9, 1980 preceding the Second International Topical Meeting on Muon Spin Rotation (ySR2). The dual nature of the workshop title was intended to convey the two goals set by the organizers— to achieve an over­view of some of the physics to be done with muons and to derive recommendations for the planning of future muon facilities at TRIUMF. The organizing committee consisted of J.H. Brewer (University of British Columbia), D.A. Bryman, K.P. Jackson, J.A. Macdonald, J.N.Ng and J-M Poutissou (TRIUMF). The workshop was sponsored by TRIUMF and by the TRIUMF Users Executive Committee (TUEC) .The first of the two goals was admirably met thanks to the presentations of the six invited speakers and to the discussions and comments of the participants, set down here in these proceedings.Sadly, the workshop coincided within days of the death of one of its strongest supporters and a founding participant in the TRIUMF project. R.M. Pearce is remembered with respect and affection by all who knew him, including many of the participants of the workshop who wished to attend a memorial service which conflicted with the original programme. A revision of the workshop programme resulted in which a discussion of future facilities at TRIUMF was dropped.Nevertheless, the second goal of the workshop has been addressed by informal discussions among users since the workshop. These dis­cussions were summarized and recommendations approved at the TUEC Annual General Meeting in November 1980. A brief report on future facilities is also included with these proceedings.The organizers gratefully acknowledge the efforts of those participants who served as session chairmen and scientific secre­taries, in particular C. Oram and G. Azuelos who helped in the tran­scriptions of the discussions. Thanks are due also to Patricia Stewart who helped in preparing material for distribution prior to the workshop and especially to Lorraine Gray for her work in the preparation of these proceedings.J.A.M. J.N.N. A.S.! i iC O N T E N T SPageNuclear muon capture: An overviewJ. Deutsch .........................................................  1Majorana neutrinos and neutron oscillations: Low energytests of unification modelsR.E. Marshak, Riazuddin and R.N. Mohapatra ....................  10Muonic molecules and atomsH. Schneuwly ....................................................... 33Muon spin polarization PhenomenaT. Yamazaki ........................................................ hiMuon properties, tests of QED, "-oniums1E. Zavattini ....................................................... 62Muon decayA. Sirlin ..........................................................  81Future muon facilities at TRIUMFJ. Doornbos, J.A. Macdonald, J.N. Ng and J.H. Brewer ......... 93List of participants .................................   101vNUCLEAR MUON CAPTURE: AN OVERVIEWJ. DeutschInstitut de Physique Corpuscu1 aire, Universite Catholique de Louvain, Louvain-1a-Neuve , Belgium1. INTRODUCTIONThe aim of this talk will be to introduce discussions on possible muon capture experiments at TRIUMF. We shall emphasize some questions which appear to be of current interest. We begin by summarizing the main objectives pursued, that is, the why's of nuclear muon capture:1) Test of the basic muon-nucleon interaction:a) Determination of the mass and helicity of the muon neutrinob) Study of the muon-nucleon coupling2) Investigation of the general symmetry properties of the weak hadron current; for example, (G-invariance), time-reversal invariance, and con­servation of the isovector current (CVC).3) Probing of the nuclear many-body system:a) The meson field in the nucleus. Is there renormalization of the coupling due to the fact that the pion is not free?b) Deep inelastic muon capture in which the neutrino carries away little energy and a lot of energy is given to the nuclear system, leading to high energy nucleon emission.c) Muon-induced fission.These objectives are approached by studying both ordinary and radia­tive muon capture. The how's of nuclear muon capture are, then, to measure:1) Total capture rates, e.g. isotope effects, spin correlation effects wi th odd-nuclei.2) Partial capture rates to bound, followed by gamma- or beta-decay fis sion, or unbound final states, followed by particle emission.3) Correlations between two or three of the following quantities: the muon spin (S), spins of the initial and final nuclear states (Jj and Jf), the neutrino (or recoil) momentum (k = -krec) and, eventually, the energy direction or polarization of the radiative photon. We can consider:Double correlations:- S • J f  or Kv • J f  (polarization or alignment of the final nucleus ob­served through the characteristics of a subsequent beta- or gamma-ray)- S • J; (hyperfine effects, use of polarized targets)Triple correlations: e.g. (S x krec) • JfCorrelation experiments are all rather recent and there is much work to be done here.22. TEST OF THE BASIC MUON-NUCLEAR INTERACTION2.1 The muon neutrino: its massThere is a clear interest in a more precise determination of the muon- neutrino mass; this interest has received new emphasis by possible indica­tions for a finite electron-neutrino mass1 and neutrino oscillation.2The actual best upper limit on the mass of the muon neutrino ($600 keV) is derived from measurements on the two-body decay ir -*■ y + v, sensitive only to the square of the neutrino mass. It was noted that three-body final state reactions or decays, in the kinematical region of minimal neutrino energy, feature a linear dependence on the neutrino mass. Amongst other possible candidates (K[ PR + y + v , PR -> y + y + v) , the muon capture reactiony“ + 6Li -> 3H + 3H + vpresents the advantage of allowing a rather accurate energy measurement on the two companion particles in the final state.3 The sensitivity is rather high; the maximum allowed 3H energy (~44.4 MeV for mv = 0) decreases by about 60 keV if mv = 100 keV, a shift much greater than the resolution of actual solid-state particle detectors. Unfortunately the most inter­esting high-energy region of the 3H energy distribution is expected to be only very scarcely populated: ~1 0 -9/muon stop for both tritons above 42 MeV. One well-placed event may already yield significant restrictions on mv ; its probability of occurrence is, however, very low. A reliable extrapolation of the Dalitz plot from the region of higher rate which is easier to investigate requires, however, reliable information on the cap­ture mechanism and on the final state interaction. More theoretical study is needed to establish the amount of model dependence introduced by these i ngred ients .2.2 The muon neutrino: its helicityThe helicity of the neutrino emitted in the RRE •* y+ + v reaction was first inferred from rather moderately accurate measurements of the polari­zation of muons emitted by pions in flight.1* The interpretation requires, in principle, a knowledge of the decay kinematics. Another attempt, using a determination of the circular polarization of mu-mesic X-rays from nega­tive muons also emitted by pions in flight in the RRF -* y~ + v reaction,5 obtained also an indication for the helicity of the muon antineutrino; its interpretation requires not only the knowledge of the decay kinematics but in addition a sufficient understanding of the muon depolarization in the outer Bohr orbits. The kinematical depolarization factor can be deter­mined in both types of experiments by measuring the decay asymmetry of positive muons produced in "identical" kinematical conditions. Positive muons are known to have a decay asymmetry which approaches one (0.975 ±0.015),6 the value predicted by the conventional "V,A" theory of mu-decay7 corresponding to complete \}{\ = 1 helicity of the neutrinos involved.Recently, correlation experiments on the y- + 12Cg>s> -*■ v + 12Bg,s . capture reaction, using novel experimental methods, have yielded quantita­tive information concerning the neutrino helicity. The experiment, per­formed attheALS, CEA, Saclay and at SIN, measured the average polarization3Pav = <0p ' Jf>8 and the longitudinal polarization P|_ = <KV • Jf>9 of the l2B nucleus by the observation of its 3-decay asymmetry. These experi­ments were reviewed in Ref. 10. As a more accurate experiment, performed by the same authors, is currently under analysis, we do not quote here the results obtained in Refs. 8-10.11 We shall discuss only the implica­tions of the experiment.It may appear that, since both measurements (Pav and P[_) use tlie beta-decay asymmetry of 12B, they rely on the two-component neutrino theory (J(Ve = -1) in beta-decay. In reality, the beta-decay asymmetry from polarized nuclei can be treated as an experimentally determined quantity [e.g. AeXp = (1.01 ± 0.02)At^ , Ref. 12], and so the use of beta- decay does not require assumptions about the helicity of the electron neutrino. The muon asymmetry measurements already mentioned indicate the validity of the two-component neutrino theory for the muon neutrino with a high accuracy but without determining the sign of the helicity.7 We can then use the results obtained for Pav and P|_ to fix the sign of the neutrino helicities involved; the result obtained from Pav yields a (+1) helicity for the muon antineutrino emitted in ir--decay13 and the results obtained for P[_ yield (-1) helicity for the muon antineutrino emitted in ^"-capture with a precision (~%) characteristic of the p-decay experiment.In order to check these helicities directly, i.e. independently of the pure "V,A" assumption on muon-decay, one must note that Pav and P|_ do not depend only on these helicity values, but also on the dynamics of muon capture. A way to circumvent this dependence is offered, however, by the ra tio Pav/P|_> which, assuming {JCv= ~^ Hv) , can be measured readily and so allows one to elim­inate the effect of nuclear dynamics.10 This approach is expected to yield a total accuracy (statistics plus calibration) of c^lO%;10 improvements may possibly be obtained by some other methods we mention hereafter.An independent measurement of the muon neutrino helicity could be performed measuring, e.g. by Mott scattering,1  ^ the p+-polarization from ir+-decay at vest. Another, independent, determination of the muon anti­neutrino helicity could be obtained measuring the neutron polarization for neutrons emitted in muon capture from the singlet (F = 0)-state15 of muonic hydrogen (pp) . This method requires a high luminosity polarimeterand the development of suitable calibration methods.2.3 The muon-nucleon couplingSome details of this problem are addressed in the talk of E.Zavattini contained in these proceedings.16 Let us stress here only the importance of measuring directly the muon capture rate (Ay) from the triplet (F = 1) state of the (pp) system.17 In this context it would be important to test the residual polarization of the triplet system at suitably low densities by electron asymmetry measurements.173. SYMMETRY OF THE WEAK HADRON CURRENTFor easy reference, let us recall that muon capture between states|i> and |f> is governed by a matrix element containing<f |[Vx + Ax]|i> with Vx = Fv (q2)yx + FM (q2)apX qyAX = [FA (q2)yx - Fp (q2)qx]y5We have neglected form factors induced by second-class nucleon currents since beta-decay experiments have shown that we need not introduce these form factors, which are incompatible with the current gauge theories suc­cessful in other domains.3.1 Time-reversal invarianceThis invariance requires that the relative phase of all above- mentioned form factors should be 0 or 180°. This was tested by triple correlation experiments in beta-decay using transitions within isodoub­lets.18*19 It was noted, however, that some possible time-reversal-violat­ing form factors would be negligible in such transitions, and so other test cases should be investigated.20 Moreover, it would be interesting to obtain corresponding information for the (y,Vy) vertex with which we are concerned here.It has been stressed21 that time-reversal invariance implies a definite relation between two double correlations in muon capture [e.g.,Pav and Pi mentioned in the section on neutrino helicity measurements (Sec. 2.2)]. The sensitivity is, however, moderate and the results of Ref. 9 imply only cos 0 > 0.6 for the relative phase of the two amplitudes involved. (T-conservation: cos 0 = 1 ,  maximal T-violation: cos 0 = 0.)A direct observation of triple correlations would be a more sensitive test for time-reversal invariance. For a more detailed discussion of these possibilities refer to Refs. 22 and 2b.3.2 CVCThe hypothesis of CVC relates the weak and electromagnetic currents of hadronic systems. To be more specific, it predicts that <f|VjJi> and the isovector part of the corresponding electromagnetic transition are components of the same vector in isospace. As a consequence, the weak vector form factor F\/(q2) , and the weak magnetism form factor Ffl(q2) , should be predictable from the isovector form factors of the analogous electromagnetic transition.We shall not review here the evidence deduced from beta-decay which is very precise for Fy(q2 ~  0), but only moderately so for F^(q2 ~  0).The tests to be performed in muon capture (q2 «s m2) are clearly of inter­est because of the possible breakdown of trivial impulse approximation similarities between electromagnetic and weak currents due to the increas­ing role of mesonic effects at higher q2 . CVC should hold even in their presence.The test of the predictions on Fy(q2 «  m2) may be of less interest because we know that mesonic contributions to this form factor are small.A possibility of such a test was discussed, however, in Refs. 22 and 25.An interesting possibility to extract Ff-\(q2 «  m2) from the results of the above-mentioned correlation experiments and tne related capture rate has been noticed by different authors.10’26 However, in the result of these experiments F^(q2) enters in conjunction with F/\(q2) (Ref. 27), and an independent determination of F/\(q2 = m£) is required. Generally the recipe FA(q2 = m2) = F/\(0) [M(q2 = my)/M(0)] is used, where AC T0) is computed from the beta-decay ft-value and M(q2)/M(0) is the variation of the corresponding 180° electron scattering form factor. The assumption FA(q2)/FA(0) = M(q2)/M(0) is crucial in the argument,28 and the assess­ment of its limit of reliability clearly requires more theoreticalattention29 and is currently under investigation.30b. MUON CAPTURE AS A PROBE OF NUCLEAR PROPERTIES b .1 Mesons in nuclei: renormalization of the effective weak couplingsThis section deals with the axial weak current— more specifically, with the so-called induced pseudoscalar form factor Fp(q2). As this form factor arises (in a microscopic treatment of muon capture) from the coupling of the ( y , v )  vertex to a virtual pion, which is itself coupled to a nucleon, it is expected to be most sensitive to any alteration of thismesonic field by the nuclear medium. So the question is whether the in­duced pseudoscalar coupling of an isolated proton (as measured in hydrogen) is modified when this proton is imbedded in a nucleus.In practice the experiments measure the ratio of the form factorsCp = gp/gft - < |Fp(q2heIUM eAC Tu 2)|> • This ratio is also more easily inter­pretable in terms of an eventual renormalization of the coupling.The expectations for the renormalization of Cp in nuclear matter were discussed extensively by M. Ericson in Refs. 31 and 32. Briefly, the wave function <}>(x) of a pion coupled to an isolated nucleon with a strength g r is written, in obvious notation, byV2<J> (x) + m2<j)(x) = (gr/2m)v*o .In a nuclear medium this equation will be changed in two respects: 1) Due to the interaction of the pion with the other nucleons, characterized by their polarizabi1ity a, the left-hand side of the equation becomes V(l+a)V + m^ <j>(x) or— equivalently— in the propagator (1/q2 + n%) the mass m^ has to be replaced by an effective massmiT2  =  mTT +  V7T =  m£ / 0 + a )  »2) Due to the induced polarization of the medium around the emitting nucleon (screening), the coupling gr has to be changed to gr(l+5ot), where the parameter E, expresses the rather uncertain effect of short-range nucleon-nucleon correlations, the Pauli effect, etc. As we shall see, there is fortunately a way to test a independently of E,.The weak axial current of the nucleon can be written as31 A = -g/\<J + gpq(o*q), where gp can be expressed in terms of the pion decay constant f^ as gp = fTr/(q2+m2). In nuclear matter we expect the following modification of g/\ and g p :5Immediately one sees that the ratioCpff * ( » ? " < ' ) ■ ( V h y d r o g . ) ( < X M « 2* < )is independent of the parameter E, and measures directly the effective nucleon polarizabi1ity a through the effective mass m^2 = m^/(l+a). One notes also the q2 dependence of C p ^  and hence the interest of radiative6muon capture experiments, where one may hope to test this feature.The parameter a clearly depends on the internucleon distance: in thecentral region of "infinite" nuclei one expects31 1 + a = 0.25, i.e.Cp = 0.3 Cp. In real life the quenching is smaller in the most readily investigated light nuclei; the nuclear dimensions are small on the scale of the pion "polarization range" mirc2/fic and, most importantly, the par­tial transitions studied connect to low excited final states, which involve the valence nucleons located in the nuclear periphery.Any one of the correlation experiments mentioned in the Introduction (analysed in conjunction with a partial capture measurement33) can determine, with a rather slight model dependence, the ratio Cp.34 In practice, the only correlation experiments performed with sufficient accu­racy are the measurements of Pav and P|_ mentioned earlier.8-10 Subject to the corrections currently under investigation11 and to the large error bars for Cp on the nucleon,16 there does not seem to be a large renormali­zation of Cp in the muon capture transition 12C -y 12Bg.s .. Other correla­tion experiments performed on 31B (Ref. 35) and 28Si (Ref. 36) are notprecise enough for our purposes. It is clearly of interest to extend this work using the various experimental techniques mentioned in the Introduc­tion and discussed elsewhere.22*37The quenching expected in the 12C -* 12B g S> transition has not been evaluated as yet; it is also clear that a methodic experimental program extending to the medium-A nuclei would allow us to pin down the renormali­zation effect more reliably than the consideration of an isolated case which depends rather critically on the effective "transition radius" at hand.31 It would also be of interest to single out muon capture on the core protons. This capture constitutes a fraction of the giant resonance capture (which, as a collective oscillation, implies also the core nucleons), but the evaluation of this fractional parentage still needs theoretical clarification. An experimental search for possible signatures of the highly excited "deep-hole" states would also be of interest.38k .2 "Deep inelastic" muon captureThis topic which implies the observation of high-energy nucleons from muon capture will be discussed only briefly. High-energy direct nucleons (provided they are really direct ones and not issued from final state interactions of the direct neutron) should shed light on the high-momentum components of nucleons in nuclei and, for direct protons, on short-range proton-proton correlations.39 They allow us, moreover, to test ideas advanced stressing the similarity between these processes and pion cap­ture.40 The experimental situation is not clear41 and may require further experiments, especially in low-A nuclei where the final state can be uniquely identified.42*4.3 Muon induced fissionMuons incident on heavy nuclei constitute an electromagnetic probe of these systems (both static and dynamical) and can provide, moreover, a source of strong nuclear excitation upon capture.There has been interest in the last few years in attempting to extract information about the fission mechanism in actinides, particularly with regard to the characteristics and role of the deformed shape-7isomers.43”45 The experimental evidence45 for muon-induced excitation of these isomers is not yet conclusive; further experiments to verify the gamma back-decay to the ground state or to possibly observe the fission channel (which should be suppressed46 relatively in the presence of an orbital p”) are needed. Careful and systematic measurements of absolute fission yields and lifetimes47 may also be helpful when extended to other nuclei.Possible issues of further investigation are: a) is there a muon-decay induced (electro-) fission? b) what is the atomic physics basis of the preferential sticking of the muon on the heavy fission fragment?48REFERENCES AND NOTES1. V.A. Lubimov et al. , Phys. Lett. 9**B, 266 (1980).2. F. Reines et al. , Univ. of California, Irvine reprint (unpublished).3. B.R. Wienke and S.L. Meyer, Phys. Rev. C 3., 2179 (1971);S.L. Meyer and B. Zeidman, ANL preprint (unpublished).**. For example, M. Bardon et al. , Phys. Rev. Lett. 23 (1961) .5. T. Yamazaki et al. , Phys. Rev. Lett. 39_, 1**62 (1977).6. I.I. Gurevich et al. , Rep. 1AE 1297, 1•V • Kurchatova A t . En. Inst., Moscow (unpublished).7. For example, A.M. Sachs and A. Sirlin, in Muon Physios, vol. II, eds. V.W. Hughes and C.S. Wu (Academic, New York, 1975) and referencesc i ted.8. A. Possoz et al. , Phys. Lett. 70B, 365 (1977)•9. P. Truttmann et al. , Phys. Lett. 83B , b8 (1979)•10. L. Grenacs, Lectures given at the "Ettore Majorana", International School of Physics of Exotic Atoms, Erice (Pergamon, Oxford, 1980).11. A new experiment is also in progress at SIN to assess the corrections induced by the contribution of excited 12B levels.12. For example, L.V. Chirovsky et al. , Phys. Lett. 9**B, 127 (1980).13. The kinematical and cascade depolarization of the negative muon is measured using the experimental value of the y+-decay asymmetry and CPT-i nvar i ance.1**. For example, Kaphal et al. , Rev. Sci. Instrum. 5£, (1979) and refer­ences cited therein.15. For the discussion of other possible approaches see Ref. 37-16. E. Zavattini, these proceedings, p. 62.17. J. Brewer et al. , contribution to 8-IC0HEPANS, Vancouver, 1979 (unpubli shed).18. R. I . Ste i nberg et al. , Phys . Rev. Lett. 3_3, **1 (197*0 •19. F.P. Calaprice et al. , Phys. Rev. D 9_, 519 (197**) •20. C.W. Kim and H. Primakoff, Phys. Rev. 180, 1502 (1969) •21. J. Bernabeu, Polarization of correlation experiments in muon capture, in Proc. 1978 Spring School on Gauge Theories and Rare Decays, Zuoz (SIN, Vi 11i gen, 1978),22. J. Deutsch, in Nuclear and Particle Physics at Intermediate Energies, ed . J.B. Warren (Plenum, New York, 1975), p. 33-23. Z. Oziewicz, Acta Phys. Pol. 3\_, 501 (1967).2b. A. Possoz et al. , Phys. Lett. 87B , 35 (1979).25. J.P. Deutsch and L. Grenacs, Proc. *tth Intern. Conf. High-Energy Physics and Nuclear Structure, Dubna, 1971.26. Cl. Leroy and L. Palffy, Phys. Rev. D J_5, 92** (1977).827. + F^(q2) (Ev/2Mp) = F^(q2) + 0.05 F(^(q2) .28. Typically a v23  uncertainty in this assumption implies a ±5P% inaccu­racy on the determination of F/>^(q2 «  m2) in the case of the A = 12 system.29. J. Delorme, in Mesons in Nuclei, vol. I, e d . M. Rho and D. Wilkinson (North-Hol1 and, New York, 1979), p. 107.30. V. Haxton, private communication.31. M. Ericson, in Progress in Particle and Nuclear Physics, vol. 1 (Pergamon, Oxford, 1978), p. 67-32. M. Ericson, in Mesons in Nuclei, vol. Ill, e d . H. Rho and D. Wilkinson (North-Hol1 and, New York, 1979), p. 905.33. Note that the conjunction of two correlation experiments (e.g. Pav and P|_) allows one to avoid recourse to a capture rate measurement.34. The model dependence is strong, however, for simple capture rate measurements (both for ordinary and radiative muon capture) or for the determination of the radiative photon spectrum. For example,A. Wullschleger and F. Scheck, Nucl . Phys. A236, 325 (1979).35. J.P. Deutsch et al. , Phys. Lett. 29B, 145 (1969).36. G.H. Miller et al. , Phys. Rev. Lett. 29_, 1194 (1972);S. Ciechanowicz, Nucl. Phys. A267, 472 (1976).37. J.P. Deutsch, in Progress in Particle ccnd Nuclear Physics, vol. 1 (Pergamon, Oxford, 1978), p. 247.38. J.P. Deutsch, in Proc. 1972 Spring School on Weak Interactions and Nuclear Structure, Zuoz (SIN, Villigen, 1972).39- Bertero et al. , Nuovo Cimento 5^, 1379 (1967).40. J. Bernabeu et al. , Phys. Lett. 69B, 161 (1977);T.E. Ericson and J. Bernabeu, Phys. Lett. 70B, 170 (1977).41. T. Kozlowski et al. , communication 8-IC0HEPANS, Vancouver, 1979(unpubli shed).42. N. Mukhopadhyay, in Proc. 8-IC0HEPANS, Vancouver, 1979, e d . D.F. Measday and A.W. Thomas, Nucl. Phys. A335, 111 (1980), and discussions during conference.43. S.D. Bloom, Phys. Lett. 48B, 420 (1974).44. S.N. Kaplan et al. , Phys. Lett. 64B, 217 (1976).45. W.D. Fromm et al. , Nucl. Phys. A278~, 387 (1977).46. J. Blocki et al. , Phys. Lett. 42B, 415 (1972).47. For example, D. Chultem et al. , Nucl. Phys. A247, 452 (1975);S. Ahmad et al. , Phys. Lett. 92B, 83 (1980).48. Dz. Ganzorig et al. , Phys. Lett. 77B, 252 (1978);W.U. Schroder et al. , Phys. Rev. Lett. 43_, 672 (1979) •SUMMARY OF DISCUSSIONZAVATTINI suggested that because of the high ir intensity provided by the meson facilities one should address the problem of the structure function of the pion (for example, by studying the radiative pion decay in detail) And that these experiments were more feasible than the y“ + 5Li -* 3H + 3H9DEUTSCH answered that the two experiments were not in competition and that an attempt should be made at doing both.ZEHNDER pointed out that the limitation in the 6Li + y” experiment comes from the thickness of the target compatible with the energy and angular resolution required.DEUTSCH answered that one could get out of this problem by studying various thicknesses because only the sum of the two triton energies is requi red.ZEHNDER commented that 12C y capture rates are dominated by the weak mag­netism form factor contribution and that all other induced terms contribute only 9%. This implies that y capture is mainly testing the CVC hypothesis at h i gh q2 .DEUTSCH agreed but still thought y capture experiments should be pushed to a precision that would allow precise test of induced pseudoscalar terms, for example.MARSHAK asked about the prospect of improving the Vy mass measurement in the next few years.DEUTSCH replied that using present RR decay experiments at rest (Frosh's experiment at SIN) or in flight (Seiler's experiment at SIN) one cannot expect to reduce the upper limit to below 300 keV.ZIOCK commented that his current LAMPF experiment could not reduce the upper limit on the y neutrino mass to less than 0.5 MeV and that the empha­sis of the experiment was on the search for exotic neutrino masses as described in a paper by Shrock for which the problem of systematic cali­bration was not crucial.SHROCK explained in a comment how these exotic massive neutrinos would contribute to additional lines in the y momentum spectrum.ECKHAUSE asked about the status of the radiative y capture in 3He, in particular what was the residual polarization.DEUTSCH reported that recent tests done at SIN had shown that the residual polarization was low. Molecular effects are thought to be responsible for the strong depolarization.YAMAZAKI asked how well could we measure gp/g/\ in radiative y capture experiments. He argued that one should consider heavy nuclei such as 209Bi because of the possibility of applying nuclear polarization tech­niques to get an 80% polarization of the y“ in the ground state.DEUTSCH pointed out that for 12C and 160 a comprehensive review has been done by F. Sheck, but it would be very worth while to do radiative y cap­ture in heavy targets the way suggested.10MAJORANA NEUTRINOS AND NEUTRON OSCILLATIONS: LOW ENERGYTESTS OF UNIFICATION MODELS*R.E. Marshakt and Riazuddinf Department of Physics Virginia Polytechnic Institute and State University Blacksburg, VA 24061R.N. Mohapatra§Department of Physics City College of the City University of New York New York, NY 100311. INTRODUCTIONThe formu1 ation of the universal (V-A) charged weak current theory in 1957 1 assumed massless (left-handed) neutrinos, electron-muon universal­ity and a form of baryon-1epton symmetry couched in the permutation in­variance requirement: A -«-»■ y, n +-*- e, p «-*- v . The consequences of this"baryon-1epton symmetry principle" were worked out in a paper by Gamba, Marshak and Okubo2 and led to the concept of two groups: "weak isospin"Iw and "weak hypercharge" Yw , related through the "weak" Gell-Mann- Nishijima relation: Q. = l3W + Yw/2; it was observed that if one setYw = B-L+T (with B and L the baryon and 1epton numbers, respectively and T a "triality" quantum number), then Yw = B-L for weak isodoublets and Yw ^ B-L for weak isosinglets. The baryon-1epton symmetry principle has undergone a series of reformulations, first when the SU(3) group (with its three quarks and single Cabibbo angle) was introduced, 3 then when two distinct neutrinos (ve ^ v^) were identified4 (leading to the postulation of the charmed quark5), and finally within the framework of the six quark- s ix lepton model . 6From 1961 to 1968, Glashow, Salam and Weinberg7 developed what is now called the standard gauge model, based on the electroweak group SU(2)l x U(l) (with SU(2)l the left-handed weak isospin group and U(l) a weak hypercharge group). This electroweak group was gauged with four massless vector bosons (three associated with the left-handed weak isospin group and one associated with the weak hypercharge group) in such a way that, after the symmetry is broken by a doublet of Higgs bosons, the three left-handed W's of the weak interaction acquire masses that are determined by one parameter, sin20^ (0\^  is the Weinberg angle), and the photon re­tains its zero mass. The total neutral weak current is predicted to have the form p (I3L - sin20^ Q) [experimentally, p = O .985 ± 0.015, sin20^ =*The material in this paper has been presented by R.E.M. at the Neutrino '80 (Erice) Conf., the Seventh Trieste Conf. on Particle Physics (1980) and the Muon Physics Workshop (TRIUMF 1980); also, by R.N.M. at the XX Intern. Conf. on High Energy Physics (Madison 1980). tWork supported by Department of Energy.^0n leave from Physics Dept., Quaid-i-Azam University, Islamabad,Paki stan.§Work supported by National Science Foundation and CUNY-PSC-BHE Research Award No. RF-13^06.0.23 ± 0.02], a prediction that has been confirmed by all measurements up to the present. This minimal version of the standard gauge model of the electroweak interaction implicitly assumes zero mass (left-handed) neutrinos. The unification of weak and electromagnetic interactions is achieved by taking as given the parity conservation of the electromagnet­ic interaction and the maximal parity violation (left-handed character) of the weak interaction.Beginning in 197^, Mohapatra, Pati, Salam, and others8 have developed an alternative model of the electroweak interaction which restores parity to the status of a high energy symmetry of weak interactions, namely the group S U (2) |_ x onTyhs x nTihc/Es a This left-right symmetric model implic­itly assumes that the neutrinos possess finite mass and that, if subse­quent to spontaneous breakdown, my^ >> , the structure of the low energyneutral weak current interaction is indistinguishable from the standard model for (m^|_/my^)2 less than, say, 10%. Further, it has been shown by Marshak and Mohapatra9 that unlike the case of the SU (2) i_ x  U(l) model, the vector U(l) generator in the left-right symmetric model can be iden­tified with (B-L) symmetry. One implication of this observation is that the mass scale associated with spontaneous breakdown of parity can be associated with the breakdown of local (B-L) electroweak symmetry. To see this explicitly, we note that in the S U (2) (_ x  SU(2)r x  U(l)g_[_ model, electric charge is given by Q = I3L + I3R + (B-L)/2 where I|_ s  are the generators of the SU (2) |_ s  groups. The above relation implies that -AI3R = 1/2 A(B-L) in the energy region where SU(2)j_ x U (1)yw is a good symmetry (Yw is the weak hypercharge of the standard model). Furthermore, the left-right symmetric model enables us to study the phenomenological implications of local (B-L) breaking and the possible existence of inter­mediate mass scales without reference to grand unification models. Henceforth, when we speak of the left-right symmetric model, we shall mean the electroweak group SU(2)[_ x  SU(2)r x  U(1)b—L-When the SU(3)C (color group) is adjoined to the standard electro­weak group, one is led naturally to the grand unification group SU(5). If SU(3)C is adjoined to the left-right electroweak group, one can examine some interesting consequences of the "partial unification" group:SU (2) x SU(2)r x SU(V) (where B-L is the fourth color). One can also move on to the grand unification level, and there are at least two interesting options: the grand unification group S0(10) or the grandunification group [SU(2N)]t+ (N > 2 ) .  It is evident that as accelerators attain higher and higher energies, these various types of unification models can be subjected to more and more searching tests. However, in the present talk, we wish to summarize the present status of the 1ow en­ergy tests of the electroweak, "partial unification", and grand unifica­tion models alluded to above. By low energy, we really mean energies that are below, say, several hundred MeV. We shall focus our attention on the following topics: finite mass neutrinos, neutrinoless double3-decay, rare muon processes, and neutron oscillations, as tests of the various types of unification models.2. FINITE MASS NEUTRINOSNeutrino oscillations, if they occur, provide evidence on neutrino mass; the reverse is not true. The present experimental limits on the neutrino mass are given in Table I.1112Table I. Limits on neutrino masses.Neutrino type Mass Type of evidenceve 30 ± 10 eV Shape of 3H S_spectrum9a~  1 eV Neutrino oscillations9 1^vy <0.57 MeV Muon range in iT+ ->-y+ + V yVT <250 MeV R decayA. Electroweak ModelsIt appears from Table I that the neutrino may have a finite massand, if so, one would have to understand a finite but small mass of theneutrino (i.e. much smaller than the mass of its associated lepton). We examine this problem within the context of electroweak symmetry models and also in partial and grand unified models. Broadly speaking, such models at the electroweak level fall into two classes: the standardmodel SU|_ (2) x  U(l) and the left-right symmetric model SU (2) |_ x  SU(2)R v  U (1)b-L • These models admit various possibilities for the nature of the neutrino and its mass. As we shall see, the neutrinoless double 3-decay can make a distinction between these possibilities.Below we summarize the different possibilities regarding the nature of the neutrino and its mass.1. SU(2)L x U (1) model:(a) Put vR in a singlet representation. Using the usual Higgs doublet <)>(l/2, 1) with <<)>> = (^) , the neutrino acquires a Dirac massgiven by the first term of the following expression:hi (v"L i|_)(o)vR + h -c - + h2 (vL eL)(°)eR + h.c. (1)But the mass of ve here is arbitrary and there does not appear to be a natural way to make it small, i.e. much smaller than me .(b) We start with the usual lepton doublet |^_ = ( V*- J but in addi­tion to the Higgs doublet <j>, we postulate a Higgs \ eL/ triplet A (1 ,2) with the vacuum expectation value:< * > - ( " ) •  <2>Then the neutrino can acquire a Majorana mass:h3 C i T£<A> i|j|_ + h.c. (3a)so that:mv = h3v. (3b)A also contributes to vector boson masses and one obtains:13P =mWm| cos 2 0 ^=  1 -4v2Also, from the second term of Eq. (l):m~ = h 9 X .Since p is very nearly 1, v 2 /X 2  << 1 and thus mv could be quite small fact, since one expects h 3 /h 2  ^ 1 * one obtains:w(5)I n(6 )Using the present experimental limit on (1 - p), namely, 0.02, we have:mv < 35 keV.This is not a very useful limit and is, for example, much larger than the present limit on mVe (i.e. 60 eV). To sum up, in this case the charged weak current is still pure (V-A) but the neutrino is a Majorana particle and has acquired a finite mass through the violation of lepton number conservation [cf. Eq. (3a)] and thus can give rise to neutrinoless double 3-decay. But as we shall see, its contribution to this decay is too small to be detectable.2. Left-right symmetric model onl Tyh t ons Tyh t nW /r/Tih Tul k pw k ph d(a) Lepton doublets are (~ indicate that mass eigenstates of v and N may be different)(7)The Higgs structure is:4(1,2, 1/2, 0 h g8 Ti- 9- yh g s T9- i- yh<$><A|_>where:<AR> = ( 8 )<< K << VNeutrinos can have Majorana masses and this case has been considered in detail by Mohapatra and Senjanovic . 1 0  The physical Majorana neutrinos v and N have masses:m v  -hj_K*hovmN = h3v (9a)14and the mixing angle between ,l and 1s (the fields before symmetry break­ing) is:~  << 1 • (9b)Here hj, h2 and h 3 are the respective Yukawa coupling constants of the $, ¥  = T2 $*T2, Ar Higgs particles with the leptons. On the other hand:so that (for hi £ h2)mW R = 9V, me = h2K (10)2mmv ^ a — (11a)ny RmN " J  "Wr (llb)lyl 4 a ,c^Rwhere a = g/h3. Also it is important to note that (as long as hi < h2) :mv mN * me» ITI < —  • (12)mNIn this case not only are the neutrinos Majorana, but after V|_ and 1 s have acquired masses through lepton number nonconservation, the (e,v) charged current coupled with the lighter W|_ boson develops an additional small left-handed component (as long as the mixing between W|_ and 0 s is neglected):= ije- Yy (1 +y5) v + 6e" Yy (1+Y5)N I • (13a)Similarly, the right-handed current for (e,N) has an additional smallright-handed component:JyR = ije Yy(l_Y5)N + 6e Yy (l-Y5)v} , (13b)where 6 is the mixing between v and N. The neutrinoless double 3-decay can occur through the amplitude n and finite masses of v and N. As we shall see, it will get most of its contribution from the massive neutrino N which is chiefly responsible for the right-handed current.The assignment (7) for the leptons appears to be unique if the U(l)generator is identified with (B-L). Below we consider a case where it isnot so and the neutrinos are Dirac particles.(b) The lepton doublets are (E° is a neutral Dirac lepton)11:(14)while S( is a singlet. This assignment would not be possible if the weak hypercharge associated with the U(l) generator is identified with (B-L).15We postulate the Higgs structure as:L (1 / 2» 0, +1) <<f'L> = (°)<f>R(0 , 1/2, +1) «j)R> = (x ')*(1/2, 1/2, 0) <$> = (K °,) (15a)where: 0 Kk  ,  k  '  < <  A  < <  A '  .  ( 1 5 b )Here the neutrinos can have Dirac masses and this case has been consid­ered by Fayyazuddin and Riazuddin.12 In this case (in the limit6 8 k , 8  ' '  < A  .m W L ~ Y g x » mW R ~ Y 9 X ' • (16)The important point is that the "neutrino" E° gets most of its (Dirac) mass from:he(EjR eR) i x2 «}>R> E[ + h.c. (17)and one obtains for the physical neutrinos v and E° (with k ' << k for s impli c i ty) :mvhf 2 yh 3 A 'mEO = b3A ' . (18)The mixing angle between v and E° is given by:With h 3 1  h2 :u me m\> - b — —  ,mW R1mE0 - b mW R >(2°>where b = (g/2h3) . Also it may be noted thatmv mE°  ^mI> M  * ’ (2,)Although the mass relations here are similar to case 2(a), the neutrinos are Dirac particles. Thus in spite of the fact that v has mass, the neutrinoless double 0-decay would not occur in this case.We may summarize the situation in Table II:16Table II. Neutrino Hasses in Electroweak Models.Group Nature of Neutri no Mass of NeutrinoLeptonNumberSU(2)L x U(l) Di rac Arbi trary conservedMajorana m  < (' ’ P \ l/2 \ / me< 35 keV (1 - p = 0.02)violatedSU (2)L x )nTyhs x n TehW.8 Majorana2mgmu - a -rr~ v Wrv, N (1ight)(heavy)Pf, mN £ m| mv £ 2.5 eV for mN £ 100 GeVviolatedSU(2)L x SU(2hs x U (1) Di rac2mim* - b ^v, E°(1ight) (heavy)m v m^o S m| conservedB. Models with Partial and Grand UnificationIn the previous section, implications of electroweak unification for neutrino mass were discussed for two cases: (l) pure left-handed SU (2) |_ x U(l) model; (2) left-right symmetric SU(2)|_ x SU(2)r x  u(l) models. Also, both cases of Dirac as well as Majorana neutrinos have been discussed.It is the aim of this section to extend this discussion for the case of intermediate and grand unification groups which lead to the above electro­weak unification groups (with Majorana neutrinos) at low energies. We will first discuss the SU(5) model as the grand unification model that embeds the SU (2) [_ x U(l) group, and then we discuss the partial unifica­tion model SU(2)l x SU(2)r v  SU(4r) and two classes of grand unificationmodels that embed this group: one class based on the S0(10) group and the second class based on groups of type [SU(2N)]lt, (N * 2).1. SU(5) model:As is well known, the fermions of each generation are assigned to (5*)- and {10}-dimensional representations of the group (denoted by and ^ij|_). In the minimal version of the group that was originally pro­posed by Georgi and Glashow , 13 the Higgs multiplets were chosen to belong to {5}- and {24}-dimensional representations, denoted for our purpose by <j>i and Hi (i, j = 1, ... 5) • In this minimal model, due to the absence ofm i T one gets mv  =  0 exactly. However, it is possible to get a mass forthe neutrino by including a {15)-dimensiona1 Higgs multiplet (denoted bySij)*11+ To see how this comes about, note that, in the presence of S j j ,one gets the following Yukawa coupling:<^s = hs i|J C~ 1 j Si j + h.c. (22)It is easy to see that <^5 5>vac, 7* 0; as a result, the neutrino (v|_ = ^ 5)17acquires a Majorana mass proportional to hs<S55>. Next, we would like to remark that if we impose.a discrete symmetry on the model i//; -+ iiji;,ipjj -> -iipjj, (J)j -> Hj -> -Hj and S-- -> - S ; j , the Higgs potential ofthe system can be written as:It is possible to show that for a range of the scalar couplings A, a, 3 and y, there exists a minimum of this potential for which15 we have:where m ^  and my stand for the Wj_ and the lepto-quark X masses, respec­tively, and a is an arbitrary parameter expected to be of order 1. It is therefore clear that for this case <S55> ~  10"10 GeV, leading to an extremely small Majorana mass for the neutrino, i.e. mv «  hs • <Ss5> ~  10“ 3 eV (if hs «  10-2). Thus for the SU(5) model, a "natural" value for mv is zero; however, if the model is extended to include a { 15)-dimensiona1 Higgs multiplet, it would give mv 10”3 eV.2. Partial unification model based on SU (2) |_ x SU(2)r x  Sll(4'):This kind of partial unification model is an extension16 of the model of Ref. 9 that takes very seriously quark-lepton symmetry. To present the fermion assignment for this model, we define:We then choose i/j[_ to transform as (1/2, 0, 4) and \jjR to transform as (0, 1/2, 4) under the gauge group. The minimal choice of Higgs multiplets for the model are17: <f> = (1/2, 1/2, 0) ; A[_ = (1, 0, 10) and gs k (0, 1, 10) (denoted by a|j , a for SU(2)|_ or onTyhs index and i, j for SU(4') index).V (<j>, H, S) = - -  iin Tr H2 + —  (Tr H2)2 + —  Tr H^ 2 "  4 2- j  Tr(S+ S) + j  XS * (Tr S+S) + As Tr(S+SS+ S) + a : (4)+4>) Tr H2 + a2 4>+H2cJ) + Bi Tr S+ S Tr H2 + 32 T r S+H2S + y <j>+HSj>+ . (23)<H> «  M0 «mygand<S55> «  a • ¥ iM o ’(24)(25)18The first stage of symmetry breakdown from SU (2) |_ x_SU(2)r x SU(V) to SU (2) i_ x  U(l) x  SU(3)C is achieved by setting < A | ^ 2> = v / 0. The final stage is achieved by <<J>> = (§ £/) / 0. The Yukawa ’ coup 1 i ngs of the pre­vious section are then easily generalized to this case leading to Majorana neutrinos with the light neutrino mass being given by mv ~  a m|/mWR as before. Let us now proceed to discuss the grand unification models that incorporate the partial unification model SU (2) [_ x  SU(2)r 2  SU(V).3. S0(10) grand unification:The simplest grand unification model that embeds the SU (2) |_ x  SU(2)r P SU(4F) group is the S0(10) group . 18 There exist various patterns of symmetry-breaking for this group. Since our purpose is to recover SU(2)lx SU(2)r x SU(V) as an intermediate unification group, we will let the S0(10) symmetry be broken by Higgs multiplets belonging successively to representations {210} or {5M ,  {^5}, {126} and {10}, with the following hierarchy of mass scales:<<I>5L> or <<f>210> >> <(f)i+5> > <<f>126> >> 10> ■ (26)The various stages of symmetry-breaking are given by18a:/ . <<t>5i+> or «j>210>=(mu/g)S0(10)— — -------- — ----------»-SU(2)L xx SU(2)r x SU (y ')-«!>„>>=(mc/g)-SU (2) L x SU(2hs xX UB- l O )  x SU(3)c^ 2 ^ - ^ ^ > S U ( 2 ) l  x x U(l) x S U O ) 0* ^ 2-- -m-W-Lr-/-9- > U(l)eni x SU(3)C .The mass scale relevant for our purpose is therefore the one arising from <4>i26> ^ As is well known, the low energy value of sin20^ constrains the patterns of mass hierarchies in grand unified models. For the case of the S0(10) model, this has been analyzed by several authors.19 For this most general case, one obtains:s in2e^ /(m\</L) = ---11 a(mwL)3 TT5 mu — Zn m3 mu-  — £ n  -------mWL 8 mWR1 mu—  £n —2 mc8 a(mwL)3 aS (mWi_) ~11 a(mwL)3 IT3 £nmum^L - JinmumWR (27)One finds from these equations that a low mass mv/R is obtained only if mu = me and higher than 1019 GeV. If we choose mu —  1019 GeV, a value of mWR —  107 can be accommodated within the present uncertainties in the sin20w measurement. It is conceivable that with this value of m ^ ,  a suitable choice of the {126} Higgs boson self-coupling could yield an amplitude for the n ->• n transition comparable to that expected with the partial unification group.194. Grand unification models based on [ SU (2N)]^ :These models have been analyzed recently,20 in order to obtain low unifying mass scales. In such models, if N = 3, it is possible to obtain a unifying mass scale « 1 0 6 GeV without any conflict with the presently observed value of sin20w. Such models could therefore accommodate an «  1 TeV.3. NEUTRINOLESS DOUBLE g-DECAYNeutrinoless double g-decay becomes an interesting test of the various unification models, as we shall see below. As is well known, the neutrinoless double g-decay [(gg)0 ] can occur if the "intermediate" neutrino is a Majorana particle and one or both of the following condi­tions are fulfilled: (i) The (e, ve) current has the form i e YitO+Ys) + n (l~Y5)]ve> i•e . contains a component of "wrong helicity"; (ii) The neutrino has a finite mass.The ( g g ) 0  decay has been analyzed in terms of the amplitude n, the lepton number non-conserving parameter. Even though this decay occurs mainly through the finite mass of the neutrino, the contribution of the latter is expressed in terms of an "equivalent" n by Halprin, Minkowski, Primakoff and Rosen.21 This has the advantage that the uncertainty due to nuclear matrix elements is eliminated. The essential point of the analysis is that for (i) and mv negligible, the Coulomb potential appears while for case (ii), n negligible but the neutrino mass large, the poten­tial is a 6-function. Thus the matrix elements in the latter case are essentially determined by the modulus square of the wave function at the origin, | (0) | 2, for the two-nucleon system or the quark system dependingupon whether one assumes that the basic ( g g ) 0  process involves nucleons or quarks (see below). The present limit on n, consistent with ( g g ) 0  decay rates of i+8Ca -> 1+8Ti, 78Ge -> 78Se and 82Se 82Kr, is21a:|n| £  5 x 10_tt . (28)This value of n corresponds to mv ^ 1 keV for the lighter neutrino and implies a half-life for ( g g ) 0  decay of 82Se, for example:R Tvvhb z  1-1 > iolt±2 years 1/2 n2 y> 4 x  1020±2 years (29)where 10±2 reflects the uncertainty in the nuclear part of the matrix e 1ements.21^The observation of ( g g ) 0  decay requires a Majorana neutrino and thus excludes the cases 1(a) and 2(b) of Sec. 2. Moreover, if one sets n = 0, as is the situation for the case 1(b), and takes, for example, mVe to be 1 eV, 10 eV and 60 eV (its present upper limit), the half-lives for( g g ) 0  decay would respectively be about 106 , 1014 and 3 x 102 larger thanthe half-life (29) (since the dependence of n on mv is linear). Thus in this case ( g g ) o  decay would be hard to detect.In a model where there are two Majorana neutrinos, as in case 2(a),20where mv is very small (a few times eV),(33)o decay would get most of its contri­bution from the heavy neutrino N which has predominantly right-handed couplings.This case we now discuss in some detail.The basic process at the "quark level" is shown in Fig. 1. If u and d quarks are respectively replaced by n and p, then the analysis of Halprin et al . 21 for A ^  100 (33)o nuclei gives for the (right- handed) heavy neutrino N [case 2(a)] the equivalent210: Fig. 1. Neutrinoless doubledecay at quark level.n -IT? ]—  (3-5) 10* 1 GeV, (30)i.e.n < (3 .5 )  10"59  9for m^ - 100 GeV and (mwi/m^) 5 1/10. For the value of n in Eq. (30), (33)0 would require a half-life measurement of order 8 x 1022±2 years and may be difficult to detect. However, there is the additional possibility of enhancement in this case. If the basic process for (33)o 's shown in Fig. 1, then it can give rise to transitions at the one-baryon level of the form, N*- p e“e~, N*° ->- N*++ e“e“ , N*~ -*■ N*+ e“e", where N*(1236) is the I = 3/2, jP = 3+/2 resonance and the nucleon and N* are both s-wave three-quark systems. In this case, the analysis of Halprin et al. 2 1  gives for A «  100 (33)o nuclei (where Pn* is the probability for the "virtual" existence of N* in the nucleus):/mvJi \2 1n = ( -5M  —  PN* (1.6) 103 GeV. (31)\ m% )  mNFor PN* ~  10-2 (Ref. 21), and (mwL/mwR) $ 1/10, mN * 100 GeV:n < 1.6 x 10" 3.There is, however, no way to estimate Pn* reliably and Pn* = 10-2 may be an overestimate. If we take Pn* conservatively to be 10-2 and further correct for the two-nucleon overlap function which we may take22 as 10_2 , then Eq. (31) would give:n - 1.6 x io"\ (32)Such a value of n should be detectable in (33)o decay since this would correspond to a half-life of A x 1021±2 years. It is therefore very interesting to look for (33)0 decay within the framework of 2 (a).To sum up, a measurement of a half-life for (33 )0 decay in the range1020 to 102i+ years would have an important bearing in distinguishing thevarious cases discussed in Sec. 2 and placing a limit on mN for case 2(a) if the limit on ( m ^ / m ^ )  is known from other considerations. Possible21candidates for (BB)0 decay are 48Ca, 76Ge, 82Se, 128Te, 130Te, 136Xe,150Nd (Ref. 22a).4. RARE MUON PROCESSESAnother relatively low energy test of unification models is to study rare muon processes, particularly those involving intergeneration mixing of the neutrinos. These processes can be divided into two basic classes, according to whether they violate muon number only or both the muon and lepton numbers:(A) (i) ^  -> e4 + yK[ -* p± e±(i i) p“ + A(Z) + e" + A(Z)(B) p“ + A(Z) ■> e+ + A(Z+2)The former (A) can occur independently of whether the neutrinos in the "intermediate state" are Dirac or Majorana while the latter (B) occurs only if the neutrinos are Majorana. In each case there is a mixing between electron and muon type neutrinos. Thus the occurrence of (B) would sig­nal the Majorana nature of the neutrino, although the experiment may be difficult.At this time, we shall limit ourselves to the p -* e + y process.22*3 For this purpose, let us define the mass eigenstates of the neutrinos as:V} = cos 0 ve + sin 0 Vy\>2 = -sin 0 ve + cos 0 Vy, (33)where the v's refer to the light neutrinos. If there are heavy neutrinos (as in the left-right symmetric models), then we can also write:Ni = cos 0' Ne + sin 0' NyNo = -sin 0' Ne + cos 0' Nu . (34)The branching ratios for p -* e + y in the two cases, respectively, are given by23:BLr (p -* e + y)'tota3a32ttsin 0 cos2  2 - Imv2 “ mv 1 2-----mWL(35)22BR = sin 0' cos2mN22 mN !WL(36)Equation (35) holds for the standard SU(2) i x U(l) modeland (i s much 60 eV, mvfor mv/L ~  80 GeV v2) ~  (1 eV)2 , one has Bj_ < 3 x ]0-tt9 (sin 0 cos 0)2 , whichv 2too sma11 to be measurable. Even for the limiting values mve ~ p 0.6 MeV, B|_ < ^.5 x 10-26. Equations (35) and (36) hold for the left-right symmetric models where the v's are predominantly left- handed and light while the N's are predominantly right-handed and heavy,but have masses mW|/mWR ~  1/10.sma11erm^2 - mNithan m^R. In 104 QeV2 :this case, Eq. (36) gives, forBr «  k x 1CT8 (sin 0' cos 0')2 - (37)No real prediction for Q' exists but we could argue from the cosmological limits on the v e , V y  masses and the left-right symmetrical model relation [Eq. (12)] that 0 ’ ~  \/me/my; with this value of 0', Eq. (37) yields the not uninteresting branching ratio:v s ~  3 x 10"13 + 2 x 10“ 10. (38)We thus find that the heavy neutrino predicted by the left-right symmetric model brings within the realm of detection a rare weak process that is hopeless to measure if only the light neutrino exists (as predicted by the standard model).The above analysis can be carried over directly to the case of the rare process x y(e) + y if y is replaced by x and e by p or e.5. NEUTRON OSCILLATIONSThe final low energy test of unification models that we should like to consider is the determination of whether the AB = 2 nucleon transition is competitive with proton decay - a (B-L) - conserving transition. Again, it appears that the standard and left-right electroweak models may lead to sharply divergent predictions concerning baryon number non-conserving processes when they are augmented by the color group SU(3)C - We have already pointed out that the group SU (2) |_ x U(l) x SU(3)C leads directly to the grand unified group SU(5) without an intermediate stage (and an intermediate mass scale). On the other hand, the group SU(2)|_ x  SU(2)r 2  U (1)b-L x SU(3)C passes "naturally" through an interesting intermediate stage SU(2)|_ x  SU(2)r 2  SU(V) (which we have called the "partial unifi­cation group") - and an intermediate mass scale - and then on to a grand unified group like S0(10) or SU(2N)tf(N ^ 2) in which the "partial unifi­cation group" is embedded.A. Partial Unification GroupOf special interest for the question of AB = 2 nucleon transitions (e.g. n -> h giving rise to "neutron oscillations") is the "partial unifi­cation group" which predicts neutron oscillations but not proton decay.In particular, two of the authors (REM and RNM24) have shown that the23six-quark diagram (see Fig. 2) yields a rough estimate of the strength of neutron oscillations as follows:X h3<AR> , .An-n —  g > (39)m?where X is the scalar self-coupling of the Higgs boson g s T9- 1, 10), h is the strength of fermion-Yukawa coupling, and is the mass of the Higgs boson g s aPlausible values for these parameters are:A ~  a2 , h ~  a, Mgs I —  103 GeV, m/^ ~  10^ Fig. 2. Six-quark diagram GeV. With these values, we obtain a for neutron oscillations,lifetime for free neutron oscillations,tn-»n —  105-6 sec, consistent with a lifetime ~ lo30-32 years for a AB = 2 transition within a nucleus (see below). The AB = 2 nuclear transition (i.e. two nucleons transforming into several pions) might then be within accessible range of the experiments searching for proton decay.25 It should be noted that the same "partial unification group" that leads to a AB = 2 transition also prohibits proton decay (because of the existence of a hidden discrete symmetry9). It is not expected that the forbidden­ness of proton decay will hold at the grand unified level but the obser­vation of AB = 2 nuclear transitions in any way competitive with proton decay (or the detection of neutron oscillations with the appropriate mix­ing time) would provide a clear indication for the existence of a "partial unification" model with an intermediate mass scale of about 10° GeV.B. AB = 2 Transitions in Grand Unified ModelsHaving shown that partial unification models of type SU (2) (_ x SU(2hs onT—h lead naturally to AB = 2 transitions due to (B-L) being a local symmetry, an immediate question arises as to whether this phenomenon persists in any grand unified model that describes weak, electromagnetic as well as strong interactions using a single coupling constant. Here, we report on investigations of this question in the context of two grand unified models: 1) an extended SU(5) model26’27; 2) a model based on the (SU(4)]l+ grand unification group.20 Barring certain "exotic" possibili­ties, in the extended SU(5) model a large AB = 2 transition amplitude appears to conflict with lower bounds on proton lifetime. On the other hand, the [SU(4)]1+ model, of which the "partial unification group" is a subgroup, can yield a significant AB = 2 transition amplitude. Thus at the very least, observation of a AB = 2 transition would have profound impact on the future direction of grand unification.1. Extended SU(5) model:As is well known, the minimal set of Higgs multiplets required for the desired breakdown of local SU(5) symmetry is a pair of (5)" and {24}- plet Higgs multiplets.13 Several other phenomenological considerations often require the inclusion of several {45 }-dimensiona1 Higgs multiplets in the model.28 It is easy to check that in the presence of a { 5 or {45}- and {24}-plet of Higgs multiplets, the B-L quantum number remainsX24an exact symmetry of the model subsequent to spontaneous breakdown of SU(5) symmetry.29 Without further modification of the Higgs structure of the model, therefore, a (B-L)-v iol at i ng transition such as n -«-* n and Ni + N2 -* PR­) will be forbidden.It has been suggested26*27 that inclusion of an SU(5) symmetric (15)- plet Higgs boson causes a breakdown of the A(B-L) = 0 selection rule and can thereby lead to a AB = 2 transition. This is true but the phenomeno­logical requirements on this extended SU(5) model are such that the AB = 2 transition is greatly suppressed even in the presence of a {15)-plet.30 To see this, we simply consider the.Higgs multiplets presented in the earlier section, i.e. <{> j , Sj.- and H-. The super-renormal i zabl e scalar coupling and the Yukawa couplings that cause (B-L)-symmetry breakdown are:8o k gmJ ) rH <J>; 4>j , (40)Ly = h ^ j c - ^ j S j j  . (41)As noted earlier, <4>5> / 0 leads to <s55> ^ 0. The indirectly observed equality of = mz cos ©W to w * th i n a few per cent leads to the con­straint that:(42)since S j j contains a Higgs triplet under the SU (2) |_ x U(l) group. A stronger constraint, however, follows from the theoretical considerations given below.The Feynman graph that leads to AB = 2 transitions in this model is given in Fig. 3- The resulting strength of the AB = 2 transition ampli­tude can be given as:AAB=2 ~V 1,tV h5m^m2O 5(*3)Furthermore, we note that there exist Higgs- mediated AB = 1 type proton-decay amplitudes whose strength is given by:AAB=l -  +5 , h5hs • XsM<4>5>2 m2 (44)5 SIt follows from Eq. (44) and the present lower limit on the proton lifetime that:h2< 10-30 (GeV)andFig. 3- Six-quark diagram in extended SU(5) model.25(45)Equations (43) and (45) imply that:(46)Using typical values for h5, <cf>5> and m 5, one obtains a characteristic mixing time for n <-> n of about 1027 years or an unobservably long lif time for AB = 2 transitions.312. An [SU(4)]Lf model for AB = 2 transitionsThe purpose of this subsection is to describe a grand unification model based on the gauge group [SU(4)]i+ group where by straightforward extension of the analysis in Ref. 16 one shows that a sizable AB = 2 transition amplitude may exist.We start by giving the fermion representations under this group. If we denote:then = (4, 1,4,1) and = (1,4, 1,4) under the [SU(4)]lf group (decom­posed as SU(4)|_ x SU(4hs x SU(4')|_ x SU(4­hs ha The Higgs bosons relevant for our purpose w M 1 be chosen to belong to the gl = (10, 1, 10, 1) and gs 5 (1, 10, 1, 10) representations of the group. They will be denoted by Al-gL-pq, A^ab;pq where a,b = 1,...4 are flavor indices and p,q = 1 ,.. . 4 are color indices. The invariant Yukawa and Higgs couplings are:As in the SU (2) |_ x SU(2)r x SU(4') case discussed in Ref. 16, breakdown of parity and B-L is achieved via the following vacuum expectation values:ip = (47)Ly = h [ ^ a>pC-1^Lb>qALb)Pq + L ^  s B + h.c.,R(49)This gives rise to a AB = 2 transition amplitude of order:Ah3V sin 20u C gvky z ..D....... K (50)26Note the presence of the Cabibbo suppression factor sin 26u in this case. An appropriate choice of m/\R, etc. can lead to gUNWk2 ~  10-30(GeV)-5 as in Ref. 16. We will not discuss the rest of the symmetry breaking in this case and refer the reader to Ref. 32 for further details and other con­sequences of such a model.C . Phenomenology of Neutron OscillationsSince the observation of a AB = 2 transition may have such profound implications for unification models, we summarize briefly the experimen­tal possibilities for detecting the AB = 2 transition either in a nucleus (in which case two nucleons will be transformed into several pions) or in the free state (in which case one looks for the conversion n -*■ n). For this purpose, we define an effective Hamiltonian for nucleons that induces a AB = 2 transi tion:X m = 5m rJ C-1 n + h.c. (51)A characteristic oscillation time for n «->■ n can be defined in terms of 6m as follows:Ln-m — -p-  — -1-?— ye a r s .  ( 52 )6m \6m in  GeV/  7Next we look for the lower limit on tn-ni from the upper limit on nuclear stability, namely 1030 years.25 To do that, observe that the same Hamiltonian in Eq. (51) that causes n n oscillations also converts two nucleons into pions, i.e. Ni + N2 -* it's; this can cause nuclear instabil­ity. One can relate these two AB = 2 processes through the formula33:(6m)2 «  (rNj + x y E PR 1 s) M . (53)From the knowledge that TR x i E 1y RR­) h. 1 * 103° years, and choosing Mof the order of several GeV, one finds:6m ~ 10"21 eV (5*0corresponding to tn-*n * 106 sec. As we shall see, a mixing time of this order should be well within present experimental capabilities.The proposed experiment consists of allowing a neutron beam of inten­sity l(n) to travel freely through space; after a time of flight x, the intensity of h impinging on a detector (and thereby releasing 2 GeV of energy per n in the form of pions) is given by (for 6m x << 1):l(n) ^  — jj—  (6m x)2 . (55)Equation (55) must be modified in the presence of an external magnetic field B (due to the non-zero magnetic moment, un> of the neutron); it becomes 31+:2i(h) ^  I (n) ("2~b") sin2 (yn B x) (56)27if B ^  1/2 gauss (earth's magnetic field), 2i_in Bearth —  10“ 11 eV and the ratio [l(h)/l(n)] would be ~ 10— 19, too small to be measured at the present time. However, by the simple device of shielding the earth's field, one can make the experiment feasible. Indeed, one can essentially recapture Eq. (55) by imposing the condition yn Bx << 1. When yn Bx << 1 holds, Eq. (56) simplifies to:I (n) *  ^  (6m x)2 . (57)Equation (57) can be used for thermal neutrons (for which a typical x is 10-2 sec) when B 5 2 x 10~3 Bea r t h and for "cold" neutrons (for which Tcold ~ 1 0 _1 sec), when B ~ 2 x 10-LfBeart^. The shielding requirements on the earth's magnetic field are not difficult to realize and experi­ments are under way or being planned with both thermal neutrons and with cold neutrons on existing reactors.35 Hopefully, these experiments can detect the baryon nonconserving process n + ii if ce 108 sec or evenlarger (~1010 sec).35If AB = 2 transitions take place, the above considerations indicate that the search for "free" neutron oscillations in a shielded earth's magnetic field can, in principle, yield a greater sensitivity than the search for "bound" AB = 2 transitions, i.e. through the process Nj + N2 -*■ it's in a nucleus. However, it would also be desirable to search for AB = 2 transitions in the experimental setups designed to detect pro­ton decay25 since so little effort is required to extend the search from the (B-L)-conserving process (i.e. proton decay) to the AB = 2 process (i.e. multipion emission with total released energy ~ 2  GeV).REFERENCES AND NOTES1. E.C.G. Sudarshan and R.E. Marshak, Proc. of the Padua-Venice Conf.on Mesons and Newly Discovered Particles, 1957, reprinted in P.K.Kabir, Development of Weak Interaction Theory (Gordon and Breach,London, 1963);R.P. Feynman and M. Gell-Mann, Phys. Rev. 109, 193 (1958);J.J. Sakurai, Nuovo Cimento 7_, 649 (1958).2. A. Gamba, R.E. Marshak and S. Okubo, Proc. Nat. Acad, of Sci. 45,881 (1959); we note that T=0, ±1 in the SU ( 2 ) |_ x  U(l) model (Ref. 7).3. M. Gell-Mann, Phys. Lett. 8, 214 (1964);G. Zweig, CERN preprint 8l¥2/Th 401 (1964);N. Cabibbo, Phys. Rev. Lett. K), 531 (1963).4. G. Danby et al. , Phys. Rev. Lett. %  36 (1962);J.L. Bienlein et al. , Phys. Rev. Lett. J_3, 80 (1964).5. B.J. Bjorken and S.L. Glashow, Phys. Lett. JJ_, 255 (1964).6. M. Kobayashi and K. Maskawa, Prog. Theor. Phys. b3_, 652 (1973)-7. S. Weinberg, Phys. Rev. Lett. Jj9, 1264 (1967);A. Salam, in "Elementary Particle Theory" ed. N. Svartholm, (Almquistand Wiksell, Stockholm, 1968);S.L. Glashow, Nucl. Phys. 22, 579 (1961).8. J.C. Pati and A. Salam, Phys. Rev. DIO, 275 (1974);R.N. Mohapatra and J.C. Pati, Phys. Rev. Dll, 566, 2559 (1975);G. Senjanovic and R.N. Mohapatra, Phys. Rev. DI 2 , 1502 (1975)-9. R.E. Marshak and R.N. Mohapatra, Phys. Lett. 91B, 222 (1980);28R.N. Mohapatra and R.E. Marshak, Phys. Rev. Lett. 1316 (1980).9a. V.A. Lyubimov, E.G. Novikov, V.Z. Nozik, E.T. Tretyakov and V.S. Kosik, Inst, of Theor. and Exper. Physics preprint (Moscow, 1980).9b. F. Reines, H.W. Sobel and E. Pasierb, Univ. of Calif, at Irvine,Phys. Rev. Lett. 45, 1307 (1980).10. R.N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44, 912 (1980) and Fermilab preprint (1980).11. Riazuddin and Fayyazuddin, Phys. Lett. 90B, 401 (1980).12. Riazuddin and Fayyazuddin, Phys. Lett. 96B, 331 (1980).13- H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32^ , 439 (1979).14. R.E. Marshak and R.N. Mohapatra, Proc. of the Orbis ScientiaeConference, 1980. See also, L. Maiani, CERN preprint 2846-(1980).15. This point has been noted in the context of the S0(10) model byW. Maag and C. Wetterich, CERN Th. preprint and the SU (2) |_ x SU(2hs tn W.8 (1) model of Ref. 10 by R.N. Mohapatra and G. Senjanovic,Fermilab preprint (1980). See also R. Barbiere, CERN Th. preprint (1980).16. R.N. Mohapatra and R.E. Marshak, Ref. 9- For a review of the question of Majorana neutrinos and n-h oscillations and their impacton the future of particle physics, see R.N. Mohapatra, Proc. of theFirst Workshop on Grand Unification, Durham, New Hampshire (1980).17- It has been argued in the second paper of Ref. 10 that it may be desirable to split the fermion and W boson masses, scaled by intro­ducing two sets of Higgs multiplets <j>i and <(>2 (1/2, 1/2, 0)-type, each with a different transformation property under a discrete symmetry. All the remarks of this paper remain unchanged after the inclusion of this additional Higgs multiplet. For details see Ref. 10, second paper.18. H. Fritzsch and P. Minkowski, Ann. of Phys. (N.Y.), 93., 193 (1975);H. Georgi in Particles and Fields - 1974, ed. C.E. Carlson (AIP, New York, 1975) p. 575.18a. Our order of symmetry breaking - from S0(10) to SU (2) |_ x SU(2)r t  SU(4') - has the additional advantage that it avoids the "monopole catastrophe" [see G. 1t Hooft, Nucl. Phys. B79, 276 (1974); A.M. Polyakov, JETP Lett. 20, 194 (1974)].19. T. Goldman and D. Ross; S. Rajpoot; Q. Shafi, P. Sonderman and C. Wetterich; R.N. Mohapatra and G. Senjanovic, [reported in the talk of G. Senjanovic at the VPI Workshop, (Proceedings 1980)].20. V. Elias, J.C. Pati and A. Salam, Phys. Rev. Lett. 40, 920 (1978);V. Elias and S. Rajpoot, Phys. Rev. D20, 2445 (1979TT21. A. Halprin, P. Minkowski, H. Primakoff and S.P. Rosen, Phys. Rev.D 13, 2567 (1976); cf. also S.P. Rosen, Purdue Univ. preprint (1980).21a. Very preliminary results from a recent experiment searching for the double g decay of 82Se [M.K. Moe and D.D. Levinthal, Univ. of California at Irvine preprint (1980) ] give values like Ti/2^ 80) ^3 x 1021 yr and n ^ 1.2 x 10_1+ but the authors point out that the systematic errors are not completely understood.21b. Nuclear structure computations by the Los Alamos group [W.C. Haxton,G.J. Stephenson and D. Strottman, Los Alamos preprint (1980)] for 82Se, 128Te and 130Te appear to predict lower half-lives (by factors of 10 or more) for double 3 decay (with neutrinos) than are observed by geochemical means.21c. In comparing our numbers with those in Ref. 21, it is important to29keep in mind that our heavy (Majorana) neutrino will be smaller by the factor [see Eq. (30)].22. G. Feinberg, M. Goldhaber and B. Steigman, Phys. Rev. 18P, 1602 (1979). 22a. See E. Fiorino (CERN preprint, 1978) and H.H. Chen and P.J. Doe (UCIInternal Report, 1980).22b. The best present limit on the y e + y branching ratio is 1.9 x 10“ 10 J.D. Bowman et al. , Phys. Rev. Lett. 42^ 556 (1979).23. See, for example, S.M. Bilenky and B. Pontecorvo, Phys. Reports 41C,276 (1978).24. R.N. Mohapatra and R.E. Marshak, Ref. 9 and Ref. 22.25. See M. Goldhaber, Proc. of Neutrino '80 (Erice, 1980).26. S.L. Glashow, Future of Elementary Particle Physics, HUTP-79/A029 andHUTP-79/A059.27. L.N. Chang and N.P. Chang, Phys. Lett. 92B, 103 (1980).28. See, for example, P. Frampton, S. Nandi and J. Scanio, Phys. Lett.85B, H. Georgi and C. Jarlskog, HUTP-79/A026 (1979), R.N. Mohapatra and D. Wyler, Phys. Lett. 89B, 181 (1980), S. Nandi and K. Tanaka,Phys. Lett. 921, 107 (198oT7"29. S. Weinberg, Phys. Rev. Lett. _43, 1566 (1979);F. Wilczek and A. Zee, Phys. Rev. Lett. 43, 1571 (1979).30. It has been pointed out to us by L.N. Chang and N.P. Chang [Phys.Lett. 92B, 103 (1980)] that if one includes in the model an additional{45}_dimensiona1 Higgs multiplet (besides (15)), there exists a dia­gram that can lead to a reasonably small tn-h (~107sec).31. G. Senjanovic, private communication.32. R.N. Mohapatra and J.C. Pati, to be published; one undesirable pre­diction of [SU(4)]1+ is sin20w ~  0.30 which is somewhat high. The closely related group [SU(6)]Lf allows for neutron oscillations com­petitive with proton decay and predicts the more plausible value sin20 w —  0.23 (J.C. Pati, private communication).33. This formula was first derived by V.A. Kuz'min, Pis'ma JETP J_3, 335 (1970) and then rederived by S.L. Glashow, Ref. 26. Recently, R.N. Mohapatra and R.E. Marshak [Phys. Rev. Lett. 94B, 183 (1980)] have obtained a lower value of tn->-n (by a factor of 10) by taking into account the overlap (~10-2) between the Nj and N2 wave functions.Also Kuz'min et al. (K.G. Chetyrkin, M.A. Kazarnovsky, V.A. Kuz'min and M.E. Shaposhnikov, preprint of Institute for Nuclear Research, U.S.S.R. 1980) have reconsidered the problem and obtained a higher value of tn->n (by a factor of 10) by effectively choosing a smaller value of M (~100 MeV). A reasonable compromise is Glashow's value (see text).34. R.N. Mohapatra and R.E. Marshak, Ref. 33; see also M. Ba1do-Ceolin, preprint from Univ. of Padua (1980) and V.A. Kuz'min et al. , Ref. 33-35. R. Wilson, Oak Ridge Research Reactor (Harvard Proposal 1980);M. Ba1do-Ceolin, ILL Reactor at Grenoble (Ref. 34); T. Pinelli, S.P. Ratti and P. Trower, 1NFN Reactor at Pavia (private communication).30SUMMARY OF DISCUSSIONMEASDAY asked if neutron oscillations could be observed by searching for annihilation of the antineutron in a nucleus or if the nuclear magnetic field would suppress it.MARSHAK stated that the magnetic energy of the neutron is small compared to the difference between the effective potentials of the neutron and the antineutron which is of the order of several hundred MeV in the nucleus. He referred to Ref. 33 and equations (52) and following, (6m 5s; vfM) , relati ng the experimental limits on nuclear stability to the characteristic oscil­lation time for n->rT in free neutrons, namely that the limit of >103° years for T-1 gives an oscillation time for free neutrons (tn-HT £*+f/6m) of~0.1 yr, which should be possible to observe with reactors. This reflects the large nuclear potential difference and not the nuclear magnetic field.YAMAZAKI asked if experiments were being planned to look for n -> n with cold neutrons. MARSHAK pointed out that while the probability of getting an antineutron increases with increasing time of flight (as for cold neutrons) this advantage would be lost because the intensity of cold neutrons is much lower (e.g. at Grenoble). At this point, therefore, thermal neutrons have the advantage because of intensity but experi­ments are nevertheless being planned with cold neutrons.In response to a later question by DEUTSCH on the same point, MARSHAK elaborated: The number of ant i neutrons, Nf^ , is~ Nn(“ z )  >\tn->n /where =  is the time of flight, t^pp the oscillation time and Nn the number of neutrons. The time of flight of thermal neutrons over say 20 m is ~10-2 sec; for cold neutrons it may be 10-1 sec. To use this formula the condition2y Bt << 1, where y is the neutron magnetic moment and B the effective field, must hold. This gives the requirement for degaussing and illustrates that a factor of ten gain in time of flight would be off­set by a factor of 10-2 loss in intensity.In comparing the Grenoble cold neutron experiment with the Oak Ridge thermal neutron experiment (Ref. 35), the Grenoble effort may be able to detect a mixing time of 106-7 sec, whereas the Oak Ridge effort will be a few orders of magnitude more sensitive.S1RLIN commented that left-right symmetric models involve really two parameters of phenomenological interest at low energies: mW|_/mWR and £ where £ is the mixing angle between W|_ and 0 s in forming the mass eigen­states. Phenomenologica11y £ is small, and he asked if this can be explained in a natural way. He remarked that left-right symmetric models which do predict £=0 may be interesting.MARSHAK stated his model has no way of determining £ at this point, but that if £ is taken to be zero that gives the usual standard model relationmW = mZ cos20w-31STROVINK asked if it would be desirable to improve the experimental limit on the mass of the right-hand W, for example in the measurement of the ? parameter in muon decay.MARSHAK thought that would be difficult but it should be done. He re­marked that "if the desert of grand unified theories is going to bloom after 102 GeV, the right-handed W will have to be lC^-lO4 GeV". However, to force the limit up to 1 TeV (a factor of three) would be useful. In the context of the workshop he pointed out the importance of improving limits on y-* ey, y->-e because there is now a theoretical basis for branch­ing ratios of the order of 10-12 to 10“ 13; the standard model did not predict anything above 10-21+ so we are now in an interesting region. But it should be stated that other than the n -> "FT measurement, it will be difficult to push m ^  above 103 GeV.WAPSTRA asked if neutron oscillations mean that (B-L) is not conserved; if so, is any other number of fundamental particles (e.g., rishons) con­served in the left-right symmetric model.MARSHAK answered that the question of (B-L) conservation is central to the distinction between what he considered were the two basic theoretical opt ions:In the electroweak model with SU(2)|_ x U(l), which is extended to grand unification via the group SU(5), global (B-L) is conserved; this model predicts proton decay and no neutron oscillation in the basic version of Weinberg, Salam and Glashow. Some extensions, for example by Glashow using extra Higgs and by the Changs using other complicated structures, apparently can get some neutron oscillation in the SU(5) model.On the other hand, the left-right symmetric model, taken up to the partial unification level, i.e. SUl(2) t  SUr(2) t  SU(4), predicts the neutron oscillation 6-quark diagram (AB=2) as dominant rather than proton decay, and then global (B-L) is not conserved because of spontaneous breaking of local (B-L) symmetry.Basically the question will be what is the competition between proton decay and neutron oscillation. The left-right symmetric model does not say that SU|_(2) x  U (1) is wrong but it is rather a limiting case; as one goes into the TeV region one should start to get the left-right symmetric model and all kinds of new effects, with right-handed currents operative.In response to WAPSTRA's question about other conservation laws, MARSHAK remarked that electric charge is always conserved. On the question of rishons, Harari's theory deduces two conserved quantum numbers: Q and (B-L). But by interpreting (B-L) as the weak hypercharge, it gauged and so it can be broken spontaneously. This is just the phyiscs of the left- right symmetric model— the breaking of the local (B-L) symmetry gets connected with parity breaking. This leads automatically to two Majorana neutrinos in each generation— a light one of several eV and a heavy one around 100 GeV. The same Higgs boson with nonzero expectation value in the right-hand sector gives these Majorana neutrinos and gives the32neutron oscillations. One does not have to introduce a lot of extra parameters, but once one accepts (B-L) as the weak hypercharge, then the physics carries one along to all these predictions.For example, having two Majorana neutrinos, the light one dominantly left- handed, the heavy dominantly right-handed, predicts neutrinoless double g-decay at a level which is experimentally accessible (— 1023 yr half-life or so). It also predicts numbers for the rare muon decays yey, y -* 3e, y" -* e~, y" ■> e+ . The theory may be right or wrong, but it is accessible to experimental tests.DEUTSCH asked about the mechanism of confinement in the neutron experi­ments. One has Bragg diffraction at the walls which takes time and this direct mechanism may be different for the neutron than for the very small amplitude from the component of the antineutron.MARSHAK said that the experiment should be done in free space; indeed the different behaviour of the n and IT in reflections would be a serious problem unless one is very ingenious to compensate for the different phase shifts.33MUON 1C MOLECULES AND ATOMS Hubert SchneuwlyInstitut de Physique de 1 ' Universite, CH-1700 Fribourg, SwitzerlandABSTRACTA brief and perforce incomplete survey is given from muonic atoms as a tool in nuclear physics, through the chemistry involved in the atomic capture mechanism, to a speculation about energy production using muonic molecules.INTRODUCTIONWhen a few tens of years ago the muon was discovered, it was an unex­pected particle. For many physicists this muon became an undesirable or at least an unwanted particle. However, it has turned out during the last twenty years that it was good luck that nature has given to us an electron which can be distinguished from others. Because the muon is so similar to an electron it can be used as a probe in nuclear physics. By considering the positive muon as a light proton it became a tool in solid state physics. With a little bit of optimism it may well happen that the neg­ative muon in the future will even give us interesting information about molecular structures.MUONIC ATOMS AND ELECTROMAGNETIC STRUCTURE OF NUCLEIBecause of its mass 200 times heavier than the electron, the muon in low atomic orbits moves near to the nucleus and is therefore an ideal probe of its electromagnetic structure. This has been recognized very early, and high energy resolution detectors used today allow a precise determination of the charge size, isotope shifts, isomer shifts, electric quadrupole moments, etc.1Nuclear rad i iFrom precise experimental energies of muonic transitions to atomic levels sensitive to the nuclear extension, one can determine the electric size of nuclei assuming the validity of quantum electrodynamics, electron screening and nuclear polarization correction calculations. The informa­tion one extracts from the data is not immediately model independent. Through the potential it generates in a given atomic level, the muon is sensitive to a generalized moment of the nuclear charge distribution. For a transition i -> f the generated transition potential can be approximated, e.g., following Barrett2 , byVy(r) - vj(r) = A + Brke"ar.Vy (r) is calculated from the components f(r) and g(r) of a numerical solution of the Dirac equation using, for example, a two-parameter Fermi charge distribution3^p(r) = p0 |j + exp (kin 3 —  )where c is the half-density radius and t the skin thickness.Figure 1 shows the best fit c values with parameter t = 2.3 fm keptfixed for all measured "spherical" nuclei. It is a smooth curve andpeculiarities of magic proton numbers do not appear in the scale chosen here.The generalized moment of the nuclear charge distribution p(r) which i s measured i s then/CO p (r)e-arrkr2dr .oWith this moment an "equivalent radius" is usually defined:3Rk3 f RW e-ctrrkr2dr = < e “arrk > .ois the radius of a sphere with constant density with the same general­ized moment as the charge distribution p(r). R^ can be taken as a model- independent radius parameter corresponding to a specific muonic transition. As an example, Fig. 2 illustrates measured R^ values for 150Sm (Ref. 3)- The parameter a can be kept constant for a given nucleus and k is then the only parameter dependent on the specific muonic transi­tion in this nuc1eus.There is still interesting activity in this field especially because muonic atoms and elastic electron scattering give complementary information concerning the radial shape of the nuclear charge distribu­tion.3-6Fig. 1. Best fit c values of the Fermi charge distribution (t = 2.3 fm) from muonic transitions for all measured "spherical" nuclei.1’3-5Fig. 2. Example illustrating the dependence of parameter k of the generalized charge distribu­tion moment on the specific muonic transition.335Isotope shiftsEnergy shifts can be measured with very high pre­cision when muonic X rays of different isotopes are measured simultaneously.7-11 Figure 3 shows the muonic 2p-ls X-rays of such a measurement of the three 95>97>98Mo isotopes.The decrease in energy corres­ponds to an increase in charge size. The experimental un­certainty which can be achieved in AR(< values is of the order of 2 x 10-l+ fm.The main uncertainty remains the nuclear polarization cor­rect i o n .Whereas in the c values the magic proton numbers do not clearly appear (Fig. 1), the neutron shel1 and sub- shel1 structure is clearly visible in the isotope shifts. Starting from a closed neutron shell the AR^ diminishes linearly when adding neutron pairs and jumps when a new subshell is populated.11Quadrupole momentsFor low atomic levels the quadrupole splitting, e.g., of a muonic 2 p 3 / 2  state in a heavy atom, is large but depends on the spatial distri­bution of the nuclear charge rather than on an integral property like the quadrupole moment. For states the orbits of which are large compared to nuclear dimensions the quadrupole splittings are basically described, asin ordinary atoms, by a quadrupole coupling constant which is the pro­duct of the nuclear spectroscopic quadrupole moment and the field gradient of the muon at the nucleus. In contrast to the electronic case, thisfield gradient can easily be calculated because the muonic atom is a hy­drogen-like system. The measurement of the quadrupole splittings of such states allows for a precise determination of spectroscopic quadrupole moments.12-15In light elements the quadrupole splittings are too small to be measured with the conventional Ge-detector technique. Because of their very high energy resolution bent-crystal spectrometers are able to measure such splittings. In a recent experiment performed at SIN with the bent-crystal spectrometer installed there,16 using the continuous angular scanning mode,17 one has measured the spectroscopic quadrupole moment of aluminium.18 Figure 4 shows the measured 3d5/2 " 2p3/2 X-ray line in 27A£ at 66 keV. The hyperfine structure is clearly visible with an experimental resolution of 20 e V .Fig. 3. Muonic 2p-ls transitions in 95,97,98^q _ isotope shifts are of7 keV and 5 keV respectively at 2.7 MeV.1136Fig. k. Quadrupole splitting of the Z^s/ 2  ~ transition in 27A£measured with a bent-crystal spectrometer.18Nuclear excitationThe muonic cascade can excite the nucleus so that dynamic moments can also be determined.15 The nuclear excitation is interesting by it­self. The resonance excitation is not limited to E2 resonance in strongly deformed nuclei, but other multipoles are possible provided the resonance condition19 is fulfilled and spin and parity of the corresponding mixed levels are the same. Up to now several types of resonances characterized by the Ml, E2 and also El and E3 multipole moments could be identified.The purpose of a very recent experiment20 performed at SIN was the search for an EO-resonance effect which was predicted to exist in the 68Zn isotope.21 The effect of the nuclear EO-excitation causes a split­ting of the 2s level into two levels with an energy separation of about 20 keV. Only the lower of these two levels is expected to be sufficiently populated to observe the subsequent 2s - 2p transitions. This 2s level should be shifted by about 200 eV compared to the unperturbed level towards higher binding energy. One should then observe a shift of both the 2s - 2p transitions to lower energies. The experimental result differs both in magnitude and sign from the theoretical predictions, but is still compatible with a zero shift at the level of three standard deviations.Other nuclear physics problemsThe muonic isomer shift is related to the difference between the monopole charge distribution of an excited state and that of the ground state and can therefore provide a sensitive test of the understanding of the nuclear structure.22-25The virtual excitation of the nucleus of a muonic atom is a signif­icant part of the nuclear polarization. Its effect is an increase in the binding energy of the muon, specially in low states. The Is binding energy is increased in carbon by about 2 el/ or 3 x 10-3%, and in lead by37about 5 keV or 5 x 10-2%. However, the calculation of these nuclear pol­arization corrections is well known to be difficult and imprecise.Typical quoted uncertainties range from 10 to 50% of the total nuclear polarization effect. These uncertainties limit the accuracy of the nuclear charge parameters determined from muonic atoms.26In the heaviest nuclei, the muonic cascade can eject neutrons27 by radiation 1 ess transitions and also lead to fission. 28-30 The measurement of muonic lifetimes in actinides31 has to take into account that the muon may be captured by a fission fragment.32If the muon would not simply be a heavy electron, then it should not only be an electromagnetic and weak interacting particle. Through a pre­cise measurement of 2p - Is transitions in muonic 7Li, 12C and 13C using the bent-crystal spectrometer at SIN one hopes to clarify the situation concerning an anomalous muon-nucleon interaction.33 One hopes that the uncertainty in the nuclear polarization correction will not be limitative.FORMATION MECHANISM OF MUONIC ATOMSThe description of the formation mechanism of muonic atoms is rather complicated because we have to deal with a many body system and it is not easy to know which approximations are allowed in a calculation. Presently one may distinguish three problems in the capture mechanism which are, however, interconnected: On what does the capture probability depend in an atom? Which is the first bound state of a muon? How does the muon deexci te?First bound stateUp to now it is not clear whether the first bound state of a muon can be a molecular state. One generally assumes that the muon is captured in a very high atomic state and there is experimental evidence34 that the probability for direct capture in states n < 20 is low. The intensity structure of the muonic X-rays strongly depends on the initial angular momentum distribution. Because of the complexity of the problem, one can make several hypotheses about the reasons for different initial angular momentum distributions.A recent experiment on muonic sodium K-series intensities35 sheds some light on this problem. Within the experimental uncertainties no differences have been found between the sodium Lyman series in the com­pounds NaN02, NaN03, Na2S03, Na2S01+, Na2Se03, Na2Se01+, Na2S, Na2Se, NaC£, NaBr and Nametal- ln investigated targets the sodium can be consid­ered as an ion. From these experimental results one may deduce that neither the conductivity of the target materials nor the interatomic dis­tances affect the initial angular distribution and the muonic cascade.If the electron density distributions, which are the same for all considered sodium ions, are responsible for the muonic initial angular momentum distribution,36 one can expect to observe changes in the muonic cascade intensities between allotropic modifications of a given element. This is indeed the case as it has been shown in phosphorus37 (Fig. 5) and very recently in carbon38 (diamond-graphite). It is not evident that these changes can be simply correlated to what one calls "ionic radii" even if in a number of cases such a correlation has been shown.3938Fig. 5- Relative muonic X-ray intensity ratios of phosphorus in three allotropic modifications.37Muonic cascadeFrom the initial capture level down to the Is state the muon cascade proceeds mainly through Auger electron ejection and radiative transitions, in some cases through nuclear excitation and in the heaviest atoms also through nuclear Auger effeet.40>41 Presently one measures only the radi­ation emitted, but plans exist to measure also the Auger el ectrons.1+2 For practical reasons, cascade calculations start from initial levels between n = 14 and n = 20. The cascade mechanism is rather well under­stood4 3>44 and a recently written cascade program45 including higher multipoles, higher electron shells as well as treatment of the penetration part of the Auger matrix element is now in general use.The muonic cascade is strongly influenced by the electron refilling.Spectacular density depen­dence effects have been ob­served in gases.34’46’47 The refilling rates of the electron shells is one of the main problems of the muonic cascade. They may be deduced from a comparison of measured muonic X-ray intensities with cascade calculations or from accu­rate muonic X-ray energy measurements and deduced electron screening.The electrons are per­turbed by the presence of a muon in the atom and, depending on the level in which the muon stays, the screening of the nuclear charge by the muon is incom­plete leading to a shift (Fig. 6) of the electronicFig. 6. Electronic K X-rays emitted in presence of a muon inlr (prompt) and after weak capture (delayed).39X-ray energies compared to the (Z - 1) atom.1*8 ’1*9 In heavy muonic atoms it has been found that the refilling times of the K-, L- and M-shells are not much longer than in corresponding normal atoms and that the inner electron shells are essentially instantaneously refilled compared to muon i c t i mes.50 *51Other problems of the muon capture process and the muonic cascade are the depolarization of the muon52-55 and the anisotropy of muonic X-rays with respect to the beam direction54 and with respect to the crys­tal axis.56Atomic capture rate of muonsTwo experimental methods are commonly used to determine relative atomic capture rates, the one is the summing up of the K-series intensi­ties, the other the lifetime method, still employed.57 There are also two different approaches to describe the atomic capture. The first one assumes that chemical effects are perturbations to the main mechanism of the capture process58-64 or that they can be taken into account through ionic radii.65 The other one takes the chemical bond into account from the beginning.66"68 That the shell and subshell structure of the atoms must play a decisive role is supported by calculations59*69 showing that muons should be captured at kinetic energies of a few tens of eV. Then, no reason forbids capture below 10 eV and, thus, the valence electron structure should also become important. That the chemical bond cannot be considered only as a perturbation, at least not in light elements, has been confirmed by experiment.70 Even if the recent semiempirical formula­tion for the prediction of capture ratios68 fits the experimental results of a large class of compounds quite well, there is only fair agreement for the capture ratios in noble gas mixtures.71In the semiempirica1 formulation,68 the atomic capture rate depends on the valence electron distribution expressed by a single number, the ionicity of the bond. A recent measurement has shown72 that the relative capture N/0 is not the same in a mixture N2 + O2 and in NO. The predic­tion is in excellent agreement for the mixture. Assuming the validity of this model in its simpler version,68 one has extracted from the experi­mental capture ratio N/0 in NO the ionicity a of the N-0 bond. This is a little bit bold. However, the important result is that the NO molecule should have an -N0+ configuration rather than the expected +N0“ config­uration, i.e., the bond should have a negative ionicity (a =*-0.1). In the same paper72 the capture ratio C/0 for carbon monoxide is reported.The analysis in terms of the model gives for the C-0 bond also a negative ionicity indicating that the configuration should be "C0+ , which is con­firmed by the measure of the value and the sign of the dipole moment of the molecule.73In order to understand the capture mechanism, capture ratios and muonic cascades on well-chosen molecules and solids should be measured and also further systematical investigations of capture ratios should be performed.74-76AOMUONS IN HYDROGEN-CONTAINING SUBSTANCESIn hydrogen compounds the initial (n,Z)-distribution of muons in an element Z of the compound is not only influenced by the chemical bond, but also by the transfer. One expects very light elements to be especially sensitive to such effects.In a recent experiment muonic X-ray intensities in low Z elements and their hydrides have been investigated.77 In beryllium, one observes the greatest chemical effect on muonic X-ray intensities ever seen.Transfer plays a role too. However, because one is not yet able to mea­sure the relative capture rate in hydrogen in compounds, one does not know which part of the effect is due to transfer.There is a renewed interest in the transfer mechanism78’79 and recent theoretical papers have shown that it could be used to study surface struc­tures.80 82 The effect of hydrogen isotopes on muonic X-ray structures has also been investigated in water.83,8kRecently the hydrogen isotope effect has been measured in the trans­fer to aluminium.85 Muons are transferred to the host atom Z into low angular momentum orbits so that higher members of the Lyman series are strongly enhanced compared to normal capture in atom Z. This enhancement is less marked when the yA£ system is formed via muon transfer from muonic deuterium (Fig. 7)• This indicates that the initial angular momentum distribution of the yA& generated through transfer from deuterium is weighted to higher angular momentum states than from protonium.The only case where it is clear that a muon can have a molecular orbit is in hydrogen isotope mixtures. During the recent years groups, especially Russian groups at the JINR in Dubna, have studied experimen­tally86’87 and theoretically88’89 the resonant formation mechanism of hydrogen mesic molecules. The predicted temperature dependence and the absolute value for the ddy formation rate have been confirmed experimen­tally (Fig. 8). The maximum formation rate for the ddy molecules was predicted to be 0.8 x 106s- 1 . For the dtp molecule formation theFig. 7- Muonic aluminium X-rays Fig. 8. Resonance formation ratesobserved after muon transfer of the ddy molecules: comparison ofa) from free yd atoms experimental temperature dependenceb) from free yp atoms.36 with the theoretical predictions.89---- 1____ I_350 450- RAY ENERGY (KeV)41resonance rate is predicted to be about a hundred times higher. Experi­mentally a lower limit of 108s“ 1 has been found, but no temperature depen­dence has been observed.87 The fusion rate is predicted to l O ^ s -1 and the probability for the muon to stick to the fusion product 9He is only 10”2. This means that every muon in the deuteriurn-tritiurn mixture should be able to produce about 100 fusion reactions and release an energy which is about 20 times the rest mass of the muon. However, the sticking pro­bability to the 4He is of crucial importance for the cyclic process and should be checked experimentally.The possibility of using muon catalysed dt fusion for energy produc­tion has already been considered several times. With the new experimental results87 and the agreement with the theoretical predictions, the situa­tion has evolved. A new speculation has been published,90 where "it is shown that the use of the muon catalysed dt fusion combined with the fissile nuclides blanket can provide a positive energy gain."CONCLUSIONElectromagnetic properties of a great number of nuclei still remain to be studied with the use of muonic atoms. Many earlier experimental results can easily be improved using the high fluxes of present meson factories. Individual isotopes can today be investigated because only little target material is needed.To understand the formation mechanism of muonic atoms further exper­imental data are needed, namely systematic studies of capture ratios and cascade intensities as well as measurements on particular molecules or solids and mixtures of atoms and molecules. Taking into account the different masses, the gross formation mechanism of other exotic atoms +V =T K, E ,  p, ft) should not be different from that of muonic atoms. Data from muonic atoms could therefore be used as an input, if needed, in the study of strong interacting exotic atoms. The great strides made during the recent years in the understanding of the formation mechanism gives new hope that muonic data might provide in the future information on molecular or solid state structures complementary to ySR.Muonic hydrogen will remain a very special and very interesting field of investigation (see also Ref. 91). The formation of muonic atoms through transfer from hydrogen should be easier to treat theoretically. There is renewed interest too.I would like to thank John A. Macdonald for a critical reading and Miss Ruth Hayoz for typing of the manuscript.REFERENCES1. R. Engfer, H. Schneuwly, J.L. Vui11eumier, H.K. Walter and A. Zehnder,ADNDT _l_f4, 509 (1974) .2. R.C. Barrett, Phys. 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Stolupin andV.V. Fil'chenko, Phys. Lett. 94B, 476 (1980).88. S.I. Vinitsky, L.l. Ponomarev, I.V. Puzynin, T.P. Puzynina, L.N. Somov and M.P. Faifman, Sov. Phys. JETP 47, 444 (1978).89. L.l. Ponomarev, Proc. 6th Int. Conf. on Atomic Physics, ed.R. Damburg, (Plenum, New York, 1978), p.182.90. Yu.V. Petrov, Nature 285, 466 (1980).91. E. Zavattini, these proceedings, p.62.45SUMMARY OF DISCUSSIONYAMAZAKI asked what is known about the initial Jt-d i s t r i but i on deduced from K series intensities and its chemical dependence.SCHNEUWLY stated that the initial £-distribution is inferred from long and complicated cascade calculations. Generally in heavy nuclei the K series intensities are consistent with a statistical distribution of the muon in a level between N ~  14 and N ~ 2 0 .  For lighter nuclei around Fe the ini­tial distribution is flat in i. In light nuclei electron refilling becomes important and, for example, in Na one sees a population of Z = 0 and 1, low population of intermediate £, and again high population of high ft-levels. The effect of the chemical structure on the initial &-distribu- tion in, for example, the Lyman series intensities for an element and its compounds or its allotropic forms would require extremely elaborate calcu­lations. There is not much information on this.ECKHAUSE asked if there is any evidence of difference in electron refill­ing time between insulating and conducting species of a particular element. He remarked that any such effects might perhaps be used to enhance popula­tion of the (2s) level.SCHNEUWLY pointed out there are very few data but that in the case of sodium one sees no difference between the metal (a good conductor) and several of its compounds (insulators). He remarked also, however, thatelectron refilling times in light elements in the presence of a muon andwithout the presence of a muon can be quite different.DEUTSCH asked about current knowledge on (2s) population, especially in light p-mesic atoms.SCHNEUWLY responded that not much is known, but that if one wanted to populate (2s) levels it could most probably be done by transfer from hydro­gen where one would have a population of low H-states. In general the probability of populating (2s) states in light nuclei is higher than in heavy.DEUTSCH also asked in the Zn resonance experiment (Ref. 21) if a nuclear de-excitation gamma-ray from the 0+ state was observed. He pointed out the similarity to the case of some deformed nuclei in which due to quadrupole interactions the first 2+ state is excited, and that in the Zn case from the splitting one should expect some excitation of the 0+ level in thetransition from the (2s) atomic level.SCHNEUWLY stated that such an effect was not reported; very roughly the population of the 0+ level was about 4% (overall).46It was noted in comments by DEUTSCH and MEASDAY that the economics of muon catalysed fusion are currently highly unfavourable; that at TRIUMF a few megawatts of power are consumed to produce a few watts of muons.MEASDAY then asked whether the information about relative capture inhydrogenous compounds one obtains in pionic atom experiments, where one can easily verify that the pion has ended up in hydrogen, can be trans­ferred directly to muonic atoms.SCHNEUWLY thought it might be possible. However, the problem is that in muonic atoms it is mainly the Lyman series which is measured; but inpionic atoms one cannot generally measure the Lyman series and so one can­not check the comparison. If one tries to make comparisons in higher transitions, even the muonic atoms often have not been measured with great precision so the comparison remains difficult.YAMAZAKI pointed out, however, that even the Lyman series measurements have been informative in deducing angular momentum population.The field of muonium chemistry was not covered by Schneuwly and LAGANA made a comment regarding the importance in theoretical chemistry and gas phase scattering of experiments in which a y+ acts as a nucleus (proton) in the formation of muonium; that it has helped to elucidate isotopic effects in reaction mechanisms including such features as tunnelling, the details of the potential energy surface, and the effect of atomic vibra­tional states. [This field was covered in some considerable detail in the Vancouver ySR2 conference held the following week, the proceedings of which are to be published separately. - Ed.]47MUON SPIN POLARIZATION PHENOMENAToshimitsu Yamazaki Department of Physics and Meson Science Laboratory, University of Tokyo 7-3"1 Hongo, Bunkyo-ku, Tokyo 113, JapanAbstractProspects for future ySR are reviewed (y+ SR method­ology, new pulsed muon facility, y“SR physics, etc.) Cir­cular polarization of muonic x-rays and y-rays after muon capture is also discussed. Advantages of using a polarized target in muonic atom spectroscopy and in muon capture experiments are emphasized.§1. y+SR METHODOLOGYy+ SR spectroscopy is by now widely used for studies of solids, chemistry, etc., yielding an enormous amount of new information and thus constituting a new frontier of interdisciplinary science [1]. As the 2nd International Conference will be held here next week, I will not present specific examples of applications and their consequences, but rather concentrate on the methodological aspects which should be devel­oped in the near future so that present-day difficulties may be over­come. In this connection I will report in §2 on the new pulsed muon facility which has just been completed in Japan.There are three methods of ySR, as illustrated in Fig. 1. The first one, y Spin Rotation, is most popular. This method employs a transverse magnetic field (external and/or internal) in which one ob­serves a spin rotation pattern in a y + e decay time spectrum,N(0,t) = N 0exp(-t/x )[1 + A*G (t)*cos(9-u) t)] ,y x yfrom which one can determine the precession frequency and also the trans­verse relaxation function Gx (t). The frequency tells us about the in­ternal field of this hydrogen-1ike local impurity, which depends drasti­cally on the magnetic property of the host. The Gx (t) provides also important information on the fluctuation of the local field. Many people are studying diffusion properties of positive muons through Gx (t).However, Gx (t) sometimes gives unclear information, because static de- phasing and dynamic process cannot be discriminated.The phase memory is essential in the ySRotation method. Namely, if the local field changes from the initial phase (Phase I, local field H^')) to next phase (Phase II, H^)), as time 9Qe? on, then the first precession signal will fade out in a time (Yy[Hy^ " H^2)])"l. Therefore, in ySRotation, one cannot distinguish between tne two totally different processes: signal fading due to phase change and spin relaxation. A typical example is damping of the muonium signal in a chemical reaction (Mu -*■ y+ ) . Here, the precession frequency changes by a factor of 100: thus there is no possibility to observe the second precession. The opposite case (y+ -* Mu), which may occur in He gas with a very smallFig. 1 Three ways of \iSR.RWK Tdito rs:Jo.VFig. 2 Time gated paramagnetic shifts of y+ in MnO, observed by Uemura et al. [2 ].49admixture of Xe, is also tedious to study by the ySRotation technique.A similar situation takes place in a ferromagnetic material, where trapping of y+ immediately results in loss of precession. The method to overcome this difficulty is the phase-insensitive magnetic resonance method, as will be discussed later.Time-Gated Frequency Measurements should be useful in a paramagne­tic case, where the change of frequency should be small compared to the time scale. Here, such a signal fading does not occur and the precession signal persists. However, if we look into carefully, 0)^  may show a time dependence. An interesting example is the paramagnetic shift of y+ in MnO, studied by Uemura et al. [2], who observed a drastic change of fy, as shown in Fig. 2. This is direct evidence for the change of location of y+ (in this case from an interstitial site to a point vacan­cy). One can obtain two internal fields, one for each phase. In the near future such a method will be extended to many diamagnetic metals, where the Knight shift is of order of 50 ppm, but supposed to change from one s i te to another.The second method is \iSEelaxation. Here we apply a longitudinal field along the initial muon spin and observe the forward/backward asym­metry as a function of time. It provides a longitudinal relaxation function Gz (t). The theoretical framework was formulated by Kubo and Toyabe in 19&7 [3], and the basic properties of Gz (t) were studied both theoretically and experimentally by the University of Tokyo group [4-6].The unique feature of this function at zero external field is its sensitivity to the correlation time in a wide range, especially in aslow modulation scheme. If gR j G  1 (A = Yy6H, 6H being the random fieldcomponent), Gz (t) shows a recovery to 1/3 following the initial damping. This part will decay with a decay constant of yUOR c . The Gx (t), on the other hand, has no such recovery, insensitive to Tc as far as gR j > 1 (see Fig. 3).Now, this method is being applied to spin-glass systems. Here, the field distribution is Lorentzian (see Kubo [7]) and a stochastic theory is formulated by Uemura [8]. The first experimental results are reported by Uemura et al. [9]. Typical relaxation functions observed are shown in Fig. 4.Let me summarize the situation in Fig. 5, which shows the range R j to be determined by the transverse-field method (shaded) and the extend­ed range of Tc to be determined by the zero-field method. In this sense,the zero-field ySR is unique in studying spin dynamics. In addition,application of suitable longitudinal field will tell us about the correlation time [4],ySResonanceThe resonance method gives u>u as well as Gz (t) and line shape.The precision is limited by the natural width and thus we don't gain over the ySRotation method except for very high frequency, where con­ventional ySR cannot be applied. However, there are some unique features. The first depends on the fact that phase memory (t = 0) is not required. One can observe a magnetic resonance pattern in an arbitrary time region. Therefore, we can determine the frequency and its amplitude even in50Ot RO CFig. 3 Zero-field relaxation functions Gz (t) and transverse-field relax­ation functions Gx (t) for randomly oriented internal field of Gaussian distribution. Taken from Hayano et al. [6].jJL in CuMn (l.l at.%) T g = l & 8 KHext = 0-i---- 1---“ I----1— T r P i"2 3 4 5 6 7TIME (//sec)0. 0.2 0.4 06 08 1.0TIME (^zsec)fjf in AuFe (I at.%) Tg = 9.IK Hext= 0)Fig. 4 Observed zero-field relaxation function Gs (t) for y+ in Au Fe and Cu Mn at various temperatures. They are fitted to the theoreti­cal functions. Taken from Uemura et al. [9].51Phase II. A simple example is slow Mu formation from the initial y+ phase.The second feature is that the longitudinal field applied can de­couple unwanted random field such as nuclear dipolar field. Suppose, muonium is surrounded by nuclei. The nuclear dipolar field of 2 Oe gives a transverse damping time as short as 50 nsec, and thus it is difficult to observe spin rotation. On the other hand, an external field of 10 Oe, which corresponds to the resonance frequency of 14 MHz, is sufficient to decouple the nuclear dipolar field. If we apply r.f. amplitude of 2 Oe, comparable to the line width, and larger than the natural width, we expect a full amplitude in the time-integrated resonance pattern.Correlation time (sec)10'6 I0"9 10"'2-I------- 1------- 1------- 1------- 1------- 1 I I II / A  / *S  Rotation>///////////////////////////////////////Ze ro - f i e ld  RelaxationNeutron scattering Crit ical  regionFig. 5 Range of correlation time of random spins in spin glass system to be determined by various methods.52§2. NEW PULSED MUON FACILITY "BOOM"As one of the parasitic uses of the KEK 500 MeV boostersynchrotron, a new meson facility, so called BOOM, has been built. This project has been conducted by Meson Science Laboratory of the University of Tokyo.A superconducting muon channel, with the efforts of K. Nagamine, H.Nakayama and J. Imazato, started a successful operation on July 16, 1980, and immediately high quality backward muons emerged through the muon channel. The layout of BOOM is shown in Fig. 6. See Nagamine's report [10].The design concept of the BOOM muon channel is similar to that of SIN, but has different features, too. The superconducting solenoid (field strength 5T, length 6m, effective inner diameter 12cm) has a warm iron cylinder yoke in contrast to the cold one at SIN in order to reduce the cold weight considerably and thus the cooling down from room temperature to 4.5 K takes only 35 hours. The coils are cooled by supercritical helium flow which is produced through a heat exchanger on the side of the refrigerator cold box to simplify the solenoid cryostat structure. This superconducting solenoid has proved to be operating stably and safely.The extraction of backward muons from the superconducting sole­noid employs the principle of achromatic transport, thus assuring a small beam spot without any collimation. Actually, we have found a 5 cm x 5 cm spot in a preliminary tuning.The KEK boostersynchrotron has a unique time structure: 50 nsec width (due to single-turn extraction) and 50 msec interval (20 Hz repi- tition); accordingly the BOOM meson beam is extraordinarily sharplypulsed. Its duty factor is 10~6! The beam intensity designed is1 x 105 y+ per burst, or 2 x 106 y+ per sec. Those for y- are factors of 4 less. This intensity is limited by the primary proton intensity of 6 x 1011 ppp, but already comparable to those used at meson facili­ties. Furthermore, the surprisingly high instantaneous intensity is beyond comparison.The pulsed character of BOOM is extremely suitable to studies of delayed phenomena in a wide time scale because of virtually no back­ground. At usual meson facilities it is rather difficult to measure ay-e time spectrum for more than 10 ysec because of unavoidable constantbackground and limited time gate due to second y and various origins of beam associated electrons. The importance of slow relaxation of muon spins has already been emphasized in §1. The BOOM facility is expected to be ideal for such purposes. A typical ySR spectrum by this pulsed beam is shown in Fig. J. In spite of an untuned beam line and incomplete shielding, it shows promise for the future. Furthermore, due to the extremely small duty factor BOOM offers unique opportunities to apply an extreme external condition such as radiofrequency field and laser.How can we measure y-e time spectrum in such a pulsed beam condition? For 105 y+ arrival per burst we should obtain 101* y-e events in a flash. For this purpose we have developed two methods, as illust­ra ted in Fig. 8.53Fig. 6 Layout of the pulsed muon facility BOOM.Fig. 7 A typical \iSR spectrum observed with the BOOM pulsed muon beam.]) Digital Counting (microscopic)A conventional counter telescope to observe decay electrons one by one. The logic signal is analyzed by a logic analyzer with time bin of say, 20 nsec or 50 nsec. Since the logic analyzer answers yes or no, the number of events m per time bin per burst should be maintained considerably lower than 1 to avoid distortion. The observed number of event n is related to m in the following wayn = 1 - e-m or m = 1 n (1 - n)With 50 nsec time bin, a reasonable time-integral- rate per burst may be 20. Therefore, we need many counters, say, 16 telescopes. Then, 320 events per burst, 6,000 events per sec.2) Analog Counting (macroscopic)This is based on entirely different concept. In view of a huge number of events per burst, it may become impractical to handle each event one by one. Instead, it is far easier and more economical to handle the bunch of y-e decay events macroscopica11y . We use a single, but 1 arge-solid-angle Cerenkov counter in which all the y-e events are piled up. We expect a single pulse showing y-e time spectrum. This pulse is analyzed by a fast transient digitizer.In BOOM, a single Cerenkov counter already shows a nice time spectrum with precession. Therefore, this method looks promising. Here, we have no limit on the counting burst ±  50 nsec Clock pulse Li i i i i inii inium i i i i im ini i i i  mn     i h i i i i m i i i i  110 1 2  3 4 5 6fi sec Digital outAnalog outFig. 8 Two ways of observing y-e time spectrum in a pulsed muon beam.- g )/g55§3. SOME ASPECTS OF y SRShorter lifetimes, smaller electron yield and smaller residual polarization in the lsj/2 state of muonic atoms makes y"SR unfavourable. Nevertheless, y~SR has very unique features [11,12].3.1 Bound muon magnetic momentThe magnetic moment of negative muon at its 1s ]/2 state bound by a nucleus is subject to change compared to its free value. The most important effect is so called Breit effect [13]- It is given byA a  . - ■ (aZ)2for a point nucleus. Ford et al . [14] calculated the correction fora finite-size nucleus. The second effect, much less in magnitude, is the nuclear polarization effect. This process can be compared with the core polarization effect in nuclear magnetic moments [15]. The magnetic core polarization in the latter, induced by the interaction of an extra nucleon, is understood by the first-order configuration mixing effect of Arima and Horie. However, some difficulties remain. First of all, the bare Ml operator for a nucleon is not known a priori because of meson exchange effects.Fig. 9 Deviation of the g factor of bound negative muon, as observed by Hutchinson et al. [16] and Yamazaki et al. [17].Atomic number (Z)56Secondly, the nucleon destroys the core to a greater extent than assumed in the first-order theory. Therefore, the extra particle is not a good spectator to observe the core polarization (see [12] for detailed discussions). On the other hand, the bound y- interacts with the nucleus only through the electromagnetic interaction. It is too weak to destroy the nucleus to any extent, but can excite the nucleus to giant Ml states with a small amplitude (hyperfine coupling energy / Ml energy 'v 2 keV/10 MeV 'v 5 x 10" *0 . This effect causes a small but important change in the bound-muon magnetic moment.The present status of experiments is rather poor. Old measurements ([16],[17]) are presented in Fig. 9- In the near future we hope to im­prove accuracy so that we can argue about the magnetic polarization.3.2 Giant Hyperfine AnomalyBecause of the finite extension of the magnetization of the bound muon outside the nuclear sphere, the negative muon is expected to probe the hyperfine field from electrons in somewhat different ways. The difference between the hyperfine field felt by y“Z and that probed by a corresponding nucleus of charge (Z-l)e is a new type of hyperfine anomaly, called the giant hyperfine anomaly [18,19].The muonic hyperfine anomaly with respect to a fictitious point probe is expressed in terms of the normalized probe magnetization distribution m(r) and the electron spin density p(r) as follows,ZRo r^o / \ r2 , 2ZRo r00 / \ r-Ro , ZRo r°° / \ .e = - — - J m(r) F -r dx - — j m(r) -=-2- dx - — JL m(r) dx (1)y a o o Ro 9o Ro Ro so RoThis is an extended type of Bohr-Weisskopf effect. In the case of nuclear hyperfine anomaly only the integral inside the nuclear sphere is present, while in the muonic atom the integral outside plays a decisive role.Even in the small Z limit, £y is non-vanishing; a rough estimate in the non-magnetic case givesEp = “3 ~  = “ 1.bS% (2)Illyirrespective of Z, which was introduced by Yamazaki et al. [18], who repor­ted the first experimental observation. This is a unique way of probing spatial distributions of electron spin densities to examine the core polar­ization scheme of Freeman and Watson [20].Freeman et al. [21] calculated the muon-electron-nucleus system in a unified way. The present status of experiments is summarized in Ref. [19]. Here, it is emphasized that the y_He atom hyperfine coupling constant determined by Arnold et al. [22] provides the most basic giant hyperfine anomaly, proving the estimate (2).§4. CIRCULAR POLARIZATION OF MUONIC X-RAYS AND MUON-CAPTURE Y-RAYSA muonic atom is polarized with respect to the initial polarization of muons. Therefore, each muonic x-ray is circularly polarized with respect to the incident beam, as illustrated in Fig. 10. The circular polarization Px of Kctj x-ray (2p3/2 Is 1/2) ‘s as much as 50%, and that of Ka2 x-ray (2p1/2 -* ls1//2) is "33%- Yamazaki et al. [23], placing a transmission type Compton analyzer plus a Ge detector down­stream of a target where muons stopped, found Px (Kcti) ” Px (Ka2) =0.80 ± 0.30, a maximal circular polarization, in Cd and Pd metals.The primary aim of this experiment was to see whether or not the circular57polarization is attenuated in a Pd metal, where the spin precession signal is anomalously small. Abela et al. [24] reported attenuated circular polarization in Se, but the reason is not clear.There are some interesting points in circular polarization measurements.i) As far as the cascade depolarization is well understood, the circular polarization determines uniquely the muon longitudinal polarization and thus the muon neutrino helicity without using any assumption on the weak interaction. This can be compared with the famous experiment of Goldhaber et al. [25], who determined the electron neutrino helicity by measuring the circular polarization of gamma rays, whose neutrino recoil was indicated by resonant scattering.ii) The circular polarization, if the maximal neutrino helicity is assumed, is a good signature of cascade depolarization. This is more sensitive than the ySR asymmetry.iii) Each nuclear excited state after muon capture is also polarized, depending upon the capture prpcess involving the nuclear structure. The polarization can be measured only through the circular polarization of the deexciting x-rays. Such a y-ray is Doppler broadened due to the neutrino recoil RED  I  10/A Z), if the lifetime is shorter than 10-13 sec. This is the case with the y-rays in 12C + y_ -> 12B + Vy. The Doppler shift tells us the neutrino momentum. The circular polarization leadsto the polarization of the nuclear sj^ate populated by muon capture. So, we have a combination of ay , pv and I. This is an extended version of the average polarization and the longitudinal polarization experiments on the B B decay by the Leuven -ETH group [26]. There has been no such experiment on y-rays, but in the near future it is highly possible.I nnni ilrrfinn non-statisticolstatisticalIFig. 10 Muon spin trajectory in amuonic atom with a spinless nucleus.1 stretch /  I stretch/non-stretch ITion-stret t tj-l=J-l + l/2 j— I kHDLi LiUy58§5. REPOLARIZATION OF NEGATIVE MUONThe negative muon at the ground state has a very small polarization ^ 1/6. This polarization is being used for y"SR experiments and also for radiative capture and neutron asymmetry experiments. It would be excit­ing if one could repolarize the y~ spin. In 1975 Nagamine and myself proposed two ways [27]; one is thermal equilibrium repolarization, another is repolarization by hyperfine coupling with polarized nuclei. Let us suppose that the polarized nuclear spin I is coupled with an unpolarized muon spin, forming a hyperfine doublet F = I ± 1/2, as shown in Fig. 11.In heavy nuclei, the upper state decays immediately to the lower state.Then we expect the following residual polarization: [12]P (F<)yPy (F>)41(1+1)(21-1) (21+1)34l (21+3)(21+1)3 + 100% for I -+ <for yj > 0for yi < 0 .respecti vely. For 2 0 9 Bi we expect - 80%Both approach polari zat ion.Such a high polarization of the ground state muon permits precise determination of the spin dependent quantities. For instance, in the radiative muon capture process the y-ray shows asymmetryW (0) = 1 + a-P *cos0ywhere a  tells us the pseudoscalar coupling constant [28]. In nuclear matter the gD is expected to be reduced considerably compared with itsPCAC value nucleus. promi s i ng(9p ^ 7g/\) • Such a problem should be examined only in a heavy Therefore, the repolarized muon in polarized 209Bi seems to beToJWB-d8 1a8dlWedKWaiFig. 11 Hyperfine coupling of unpolarized muon spin and polarized nuclear spin makes the muon spin of the ground state highly polarized.59§6. CONCLUDING REMARKSIn view of various interesting subjects we need a fine muon beam of h i gh i ntens i ty , h i gh polari zat i on and low contamination. Such a beam can be obtained by extracting backward muons from a long decay section using superconducting solenoid.AcknowledgementThe author would like to thank his collaborators, especially Prof. K. Nagamine for daily stimulating conversations on muon physics.References[1] See, for instance, Muon Spin Rotation (North-Hol1 and Pub. Co., 1979); J.H. Brewer and K.M. Crowe, Ann. Rev. Nucl. Sci. 28^(1978) 239.[2] Y.J. Uemura,R.S. Hayano, J. Imazato, N. Nishida, T. Yamazaki andH. Takigawa, to be published.[3] R. Kubo and T. Toyabe, in Magnetic Resonance and Relaxation, ed.R. Blinc (North Holland, Amsterdam, 1967) P. 810[4] T. Yamazaki, Hyp. Int. 6^  (1979) 115.[5] Y.J. Uemura, R.S. Hayano, J. Imazato, N. Nishida and T. Yamazaki,Solid State Comm. y\_ (1979) 731.[6] R.S. Hayano, Y.J. Uemura, J. Imazato, N. Nishida, T. Yamazaki and R . Kubo, Phys. Rev. B20 (1979) 850.[7] R. Kubo, invited talk at ySR2 (Vancouver, August 1980), to be publi shed.[8] Y.J. Uemura, Solid State Comm, to be published.[9] Y.J. Uemura, T. Yamazaki, R.S. Hayano, R. Nakai and C.Y. Huang,Phys. Rev. Letters 45 (1980) 584.[10] K. Nagamine, Invited talk at ySR2 (Vancouver, August 1980), to be publi shed.[11] T. Yamazaki, K. Nagamine, S. Nagamiya, 0. Hashimoto, K. Sugimoto,K. Nakai and S. Kobayashi, Physica Scripta _l_l_ (1975) 133;K. Nagamine, Hyp. Int. 6^  (1979) 347.[12] T. Yamazaki, Nucl. Phys. A335 (1980) 537.[13] G. Breit, Nature 122 (1928) 649.[14] K.W. Ford, V.W. Hughes and J.G. Wills, Phys. Rev. JI29 (1963) 194.[15] A. Arima and H. Horie, Prog. Theor. Phys. _[2^  (1954) 623.[16] D.P. Hutchinson, J. Menes, G. Shapiro and A.M. Patlach, Phys. Rev.131 (1963) 1362.[17] T. Yamazaki, S. Nagamiya, 0. Hashimoto, K. Nagamine, K. Nakai,K. Sugimoto and K.M. Crowe, Phys. Lett. 53B (1974) 117.[18] T. Yamazaki, R.S. Hayano, Y. Kuno, J. Imazato, K. Nagamine, S.E. Kohn and C.Y. Huang, Phys. Rev. Letters 42^  (1979) 1241.[19] T. Yamazaki, invited talk at ySR2 (Vancouver, August 1980), to bepub 1i shed.[20] A.J. Freeman and R.E. Watson, in "Magnetism" ed. G.T. Rado and H.Suhl (Academic Press, New York, 1965) Vol. IIA, ch.4.[21] A.J. Freeman, M. Weinert, J.P. Desclaux and J.V. Mallow, to bepubli shed.60[22] K.P. Arnold, P.O. Egan, M. Gladish, W. Jacobs, H. Orth, G. zu Putlitz, W. Wahl, M. Wigand, V.W. Hughes and J. Vetter, contribution to the 8th International Conference on High Energy Physics and Nuclear Struc­ture (Vancouver, 1979) .[23] T. Yamazaki, R.S. Hayano, J. Imazato, K. Nagamine, C.Y. Huang andF. Boehm, Phys. Rev. Letters 39_ (1977) 1462.[24] R. Abela, G. Backenstoss, I. Schwanner, P. Blum, D. Gotta, L.M. Simons and P. Zsoldos, Phys. Letters 71B (1977) 290.[25] M. Goldhaber, L. Grodzins and S.W. Sunyar, Phys. Rev. 109 (1958)1015.[26] See, L. Grenacs, in Exot? c Atoms (ed. K.M. Crowe, J. Duclos,G. Fiorentini and G. Torelli, Ettore Majorana International Science Series, 1979) p. 45[27] K. Nagamine and T. Yamazaki, TRIUMF Proposal E73 in Col 1ected Papers on Muon Spin Research through International Collaboration by the University of Tokyo Group (ed. T. Yamazaki, Japan Society for the Promotion of Science, 1979).[28] For instance, R.D. Hart, C.R. Cox, G.W. Dodson, M. Eckhause, J.R.Kane, M.S. Pandey, A.M. Rushton, R.T. Siegel and R.E. Welsh, Phys.Rev. Letters 39 (1977) 399.SUMMARY OF DISCUSSIONDEUTSCH asked about the momentum bite of the new KEK muon channel.NAGAMINE reported the range width for the muons is approx 1 g/cm2 (see also Ref. 10).In referring to the circular polarization experiments DEUTSCH remarked that these measurements were of greater value in the determination of gp in Be, Si and C than in the helicity determination. He stated that the Goldhaber analogy has some difficulty because of the spins involved and there is not a clear one-to-one correspondence for the helicity. Unfortu­nately, the gp experiment is about an order of magnitude more difficult than the X-ray circular polarization because the rate is lower and because there is no fast timing. With the polarized target, if one can measure the capture rates to the giant resonance levels from the two hyperfine states separately, this would be a very nice test of gp and on the renor­malization in 209Bi. In the gp determination in C, for example, where one has very prominent levels, one does not expect very strong renormalization. But in the giant resonance, because it is a collective state, it can be looked at as a big mixture of channel less states; this means capture on deep-lying nucleons so one should expect a much bigger renormalization of gp. The problem is how to measure it and the hyperfine effect would be one way.YAMAZAKI pointed out that this would only apply to light nuclei since in a heavy nucleus the upper hyperfine states have already decayed to the ground state (by Auger effect). DEUTSCH wondered whether with the polarized (Bi) target one could keep the upper level populated using a strong magnetic field. YAMAZAKI said no, that in 209Bi there is already612 keV separation and even in very light elements the upper state would decay and that in heavy nuclei this would certainly be true. It was men­tioned (by ZEHNDER) that the decay is in about 300 psec— less than 1 nsec. DEUTSCH observed then that it would remain a radiative capture process.In referring to the use of longitudinal fields in ySR to avoid problems that arise in the transverse field method, SCHENK pointed out one still has nuclear dipolar fields and asked how can one avoid dipole-dipole relaxation.YAMAZAKI stated that the nuclear dipolar fields are a few oersted and so with a modest external field of 10 or 20 Oe one can easily decouple them. Then one observes true resonance but because of this factor the resonance is broadened; the line broadening might be 2 or 3 G, even so, if we apply r.f. intensity of a few Oe, we still can observe very long longi­tudinal relaxation time and deduce the width that way. SCHENK agreed.BREWER asked, in view of the y+ surface beam facility at TRIUMF, the pulsed y± beam at KEK and the high flux y± facilities at SIN, what is needed most in the world of muon facilities.YAMAZAKI pointed out that like SIN, TRIUMF has a 100% duty cycle. To fully utilize this feature a dedicated muon channel providing high inten­sity, high polarization negative muon beams is necessary to study spin- dependent phenomena. A backward decay muon channel is a unique way to produce a high intensity, high polarization, low contamination muon beam.62MUON PROPERTIES, TESTS OF QED, "-ONIUMS"E. Zavattini CERN, CH-1211 Geneva 23, SwitzerlandLet me start by showing a table with some recent experimental results on the properties of the muons, from which I wish to start my discussion.Table I. Some properties of muons.Experimental cond i t i onsLifetime (Ref.1) t~ nsecDisappearance rate (g-2)y Factor (Ref.2)A-*- =  1 / t ± sec-1________________ ap-5-u at reststopped in u” liquid hydrogen (2.7 ppm deuterium)2197■1^8 ± 0.066219*1.903 ± 0.0661*55135.5 ± 13.7**55601.0 ± 13-7(1165911 ± 11)10“9(1 165937 ± 12)10-9The updated value of the y+ lifetime at rest, x+ and the accurate result on the negative muon total disappearance rate A- (for y“ stopped in liquid hydrogen contaminated with 2.7 ppm of deuterium) are the results recently published by the Saclay-CERN-Bologna1 collaboration group (SCB group). The difference between the disappearance rates A+ and A- can easily be understood: a negative muon, stopped in liquid hydrogen can dis­appear via the two elementary weak processesy“ e" + ve + (1)andy- + P N + Vy, (2)whereas the positive muon can disappear only via process (l). This diff­erence is due to the fact that a negative muon, once stopped, will form stable muonic systems of dimension ay ~  (ti2/mye2 ), where the overlapping between the muon and proton wave functions is big enough to make the weak process (2) observable. From the values of the rates A +  and A - , and taking into account minor corrections for the small presence of deuter­ium in the liquid hydrogen3 and a small relativistic correction of order a 2 to the ty- lifetime, due to the fact that the negative muon is in a bound state,4 the SCB collaboration group has deduced a quite accurate capture rate value A ^ s  f°r process (2), muons stopped in pure liquid hy­drogen. The value of Al^s appears, for comparison, in Table II together# with other accurate determinations of the rate of process (2), in which the process was identified by detecting the 5.2 MeV outgoing neutron.Table II. Measurements of muon captures rates in hydrogen.Cond i t i ons Group Capture: rate sec"• 1 StateParticledetectiongas 8 atm CERN-Bologna (Ref.5) XS,C = 652 + 57) (pp) neutronf X9 = 661 ± 37gas 42 atm Dubna (Ref.6) As,D = 686 ± 88 ) s i ng1et neutronliquid hydrogen Columbia (Ref.7) XL,C = 465 ± 42 ) (pup) neutron1 XL = 1*61 ± 181 iquid hydrogen SCB (Ref.l) AL(S = **60 ± 20 ) singlet electron612 keV separation and even in very light elements the upper state would decay and that in heavy nuclei this would certainly be true. It was men­tioned (by ZEHNDER) that the decay is in about 300 psec— less than 1 nsec. DEUTSCH observed then that it would remain a radiative capture process.In referring to the use of longitudinal fields in ySR to avoid problems that arise in the transverse field method, SCHENK pointed out one still has nuclear dipolar fields and asked how can one avoid dipole-dipole relaxation.YAMAZAKI stated that the nuclear dipolar fields are a few oersted and so with a modest external field of 10 or 20 Oe one can easily decouple them. Then one observes true resonance but because of this factor the resonance is broadened; the line broadening might be 2 or 3 G, even so, if we apply r.f. intensity of a few Oe, we still can observe very long longi­tudinal relaxation time and deduce the width that way. SCHENK agreed.BREWER asked, in view of the y+ surface beam facility at TRIUMF, the pulsed y± beam at KEK and the high flux y± facilities at SIN, what is needed most in the world of muon facilities.YAMAZAKI pointed out that like SIN, TRIUMF has a 100% duty cycle. To fully utilize this feature a dedicated muon channel providing high inten­sity, high polarization negative muon beams is necessary to study spin- dependent phenomena. A backward decay muon channel is a unique way to produce a high intensity, high polarization, low contamination muon beam.62MUON PROPERTIES, TESTS OF Q.ED, "-ONIUMS"E. Zavattini CERN, CH-1211 Geneva 23, SwitzerlandLet me start by showing a table with some recent experimental results on the properties of the muons, from which I wish to start my discussion.Table I. Some properties of muons.Experimental condi t ionsLifetime (Ref. R z nsecDisappearance rate (g-2)p Factor (Ref.2) X* = eUR ! sec'1 dii1u at reststopped in y' liquid hydrogen (2.7 ppm deuterium)2197-148 ± 0.066219L.903 ± 0.066*•55135.5 ± 13.7*•55601.0 ± 13.7(1165911 ± 11)10--(1165937 ± 12) 10“ 9The updated value of the y+ lifetime at rest, RE and the accurate result on the negative muon total disappearance rate A- (for y- stopped in liquid hydrogen contaminated with 2.7 ppm of deuterium) are the results recently published by the Saclay-CERN-Bologna1 collaboration group (SCB group). The difference between the disappearance rates A+ and A” can easily be understood: a negative muon, stopped in liquid hydrogen can dis­appear via the two elementary weak processesy" -* e" + ve + vy ( 0andy" + P N + vy , (2)whereas the positive muon can disappear only via process (l). This diff­erence is due to the fact that a negative muon, once stopped, will form stable muonic systems of dimension ay ~  (fi2/mye2 ), where the overlapping between the muon and proton wave functions is big enough to make the weak process (2) observable. From the values of the rates \+ and A- , and taking into account minor corrections for the small presence of deuter­ium in the liquid hydrogen3 and a small relativistic correction of order a2 to the T y  lifetime, due to the fact that the negative muon is in a bound state,^ the SCB collaboration group has deduced a quite accurate capture rate value gl -o for process (2), muons stopped in pure liquid hy­drogen. The value of A ^ s  appears, for comparison, in Table II togetherwith other accurate determinations of the rate of process (2), in which the process was identified by detecting the 5.2 MeV outgoing neutron.Table II. Measurements of muon captures rates in hydrogen.Cond i tions Group Capture! rate sec”■ 1 StateParticle detect iongas 8 atm CERN-Bologna (Ref.5) XS,C = 652 + 40 V (pp) neut ron( Xg - 661 ± 3 7gas *t2 atm Dubna (Ref.6) XS,D = bFb + FF k s i nglet neutronliquid hydrogen Columbia (Ref.7) XL,C = *465 + ncu ) (ppp) neutronv XL = *461 ± 18liquid hydrogen SCB (Ref.1) XL,S = *460 + 20 ) s i nglet electron63It is clear that A ^ s ,  obtained essentially from the difference between the values of A” and A+ , relies on the assumption that the CPT theorem8 is valid in weak interactions, at least up to a level of 10-5. On the other hand it has been possible to deduce a rather accurate value for A|_ s in liquid hydrogen without the necessity of knowing the absolute detection efficiency for the neutron in process (2). By this novel method, one could today probably reach an even higher accuracy on A|_. Unfortu­nately, it is difficult to see how one can reliably, by this same technique, obtain an accurate value for the capture in gas Ag. The reason for this will appear clear later. I wish at this point from the experimental results of Tables I and II to deduce some general properties concerning the muons.From the measured value of the total disappearance rate A" and the directly measured captured rate A(_ c by the Columbia group one can get, for the negative muon the best present experimental determination of its lifetime, ty-, at rest:ty- = 2197-102 ± 0.22 nsec. (3)This value represents a substantial improvement over the previous direct determination.9 Combining the value (3) with the value of x+ from Table I one can form the quantityDw = X+~T/~- = (0.021 ± 1). (b)TSuch a quantity, expected to be zero because of the CPT theorem, repre­sents quantitatively a level at which this theorem has been experimentally verified directly in weak interactions; for the previous limit see Ref. 10.On the other hand, assuming the CPT theorem one can combine the valuet+ of Table I with value (3) to obtain the best determination of the"muon" lifetimeTCPT = 2197.\bb ± O.O63 nsec. (5)At this point it would be nice to be able to deduce from value (5) a moreprecise value for the Fermi coupling constant gp. However, it seems thatthis cannot be done until one knows how the higher order QED corrections affect the theoretical value for process (l). From the quantities given in Table I, one can also form= aM ~ ay = (0.22 ± 0.1A)10-1*, (6)«ywhich is a quantity, also expected to be zero, that represents quanti­tatively a level at which the CPT theorem has been experimentally verified in QED.Finally let me point out another consequence which one can deduce from value (3) of t y - .  In Table III are presented the results of the measurements of the negative muon lifetime done by the (g-2) group at CERN.2*10 The interesting features are that these measurements were done: 1) using high energy negative muons (as well as positive muons) of 3 GeV/c momentum; 2) confining the muons in a circular orbit. The rela- tivistic factor for these muons was yy = 29-327 ± 0 .00b.Table III. Comparison of the negative muons lifetimes directly measured.Authors Lifetime in flight Lifetime at rest U . ° rS_______________________ i t ______________________________ 1MZ_____________J. Bailey et at. ,, ,,0 ^ nYu - 29.327 ± 0.004 64.368 _ 0.029 useepresent discussion 2.197102 ± 0.00022 useevalue (3)From the values of Table III let us form the quantity, expected to be zero,64L _ ti T A i  - tp-) _R : .(1±0.5) x io“ 3. (7)Its value is of the same order of the corresponding limit set on L+ for positive muons in Ref. 10;L+ = (0.2 ± 0.9)xl0"3. (8)Limits (7) and (8) are the most accurate test of the relativistic time dilatation using elementary particles and antiparticles.The additional particular interest of limits (7) and (8) is that thecomparison is made with (positive and) negative muons subjected to a constant transverse acceleration (the muons being confined by a magnet in a circular orbit) of 1021 cm sec-2 or 1018 g.As remarked by J. Bailey et at. in Ref. 10, the particles and anti­particles in the ring perform a round trip and so when compared with thecorresponding particles and antiparticles decaying at rest in the labora­tory simulate closely the so-called twin and antitwin paradox.One can improve the accuracy of the negative muon lifetime at rest, from which the quantities (4), (5) and (7) have been deduced by also using the experimental result for Ag in Table II. To do so, however, one must know in what different muonic systems the negative muon is bound, before decaying, when stopped in hydrogen at different densities.Such a knowledge is also necessary when one wants to extract fromthe experimental quantities of Table II any information on the weak coupling constants determining the rate of process (2), in particular on gp. In fact this is one of the most important motivations for accurately measuring Ag or A|_. However, because of the extreme spin dependence of the V-A interaction, one has to know, in order to compare the experimental results among themselves and with the theoretical predictions, the rel­ative orientations of the proton and of the muon spins, as well as the overlapping of their wave functions at the instant of the absorption process (2) for different experimental situations. This necessity brings us right into muon-molecular physics;3 let us discuss it a little.In Fig. 1 is shown the tree of molecular processes happening when negative muons are stopped in pure hydrogen; A0 is the rate of the decay channel (1). The symbols (pyp)0 ar|d (pyp)p represent the ortho and para muon-molecule states, respectively, with rotational angular momentum L=1 and L=0; the ortho state is about 148 eV higher than the ground para state.65Fig. 1. Muon molecular reac­tions in liquid hydrogen. At any time the muon can decay (rate A0 ) or be captured in various atomic or molecular systems. The dashed line represents the transfer of the muon from the (up) atom to a possible impurity pre­sent within the liquid hydro­gen target. Ap m ,A0m and A$,At are capture rates (seeApp are ratesTable I V ) ; App and molecular formation (see text) and A0p is the transition rate from ortho to para muon-molecular ion.The molecular formationomimpurity where presentrate App is proportional to the hydrogen den-106 sec-l atsity and it has been experimentally found to be around 2 xthe liquid hydrogen density.3 It has also been shown by S. Cohen et al. 11by very detailed calculations that A^p cs 3 * 10-5 A ^ p . Therefore, in what follows we will assume with good confidence A^p = 0. The rates Aom and Apm are the capture rates of process (2) respectively from an ortho and para molecular state. For the initial neutral atomic system (yp) it hasbeen experimentally shown12’13 that it becomes a Is singlet state quitesoon after the negative muon has stopped.In Table IV are presented the results of the most recent calcula­tions1^ of the different capture rates of interest, assuming the standard values for the coupling constants involved. In the table the rates A$ and Ay refer to process (2) taking place from a (py)ls atomic system in a singlet and triplet state, respectively.Table IV. Expected theoretical capture rates calculated by Primakoff (using "standard" values for the coupling constants).Rates State Ca1cu1ated value sec-1s i nglet 66^ ± 20*T triplet 11.9 ± 0.7Aom = 2y0 (3AA$ + e g g H h 2y0 = 1.01 ± 0.01 ortho (pyp)i_=i 506 ± 20Apm = 2yp (3AAj + u ( ( l ) 2Yp = 1.15 ± 0.01 para (pyp)|_=0 201 ± 8The quantities Yo ancl yp 15’16’17 are the ratios of the muon densities at one of the protons respectively in an ortho or para (pup) molecular ion (averaged over the separation between the protons) to the muon density at the proton in a (yp)is neutral atom. The errors given on the quantitiesYo and Yp are of computational origin.Given the value of App it is clear that practically all muons stoppedin liquid hydrogen after a few microseconds will be bound in the molecularionic system (pyp). This is rather important since transfer of a muon to a possible impurity contained in the hydrogen target can only happen from a neutral muonic system (yp).Therefore, in this case the quick formation of the ortho molecular66state prevents muons from being transferred at a later time and then cap­tured by impurities. The situation is quite different when one stops neg­ative muons in a hydrogen gas target where, due to the lower density of the hydrogen, the muon mostly remains bound in a neutral atomic system (yp)is . In this case the probability of a later transfer and successive nuclear capture of the muon by an impurity element in the target may be significant even at very low impurity concentration, given the extremely high rate of transfer of a negative muon from the (yp)is atom to the im­purity . 3 This is the reason why the lifetime technique introduced by the SCB group to measure the rate of process (2) in hydrogen is suitable to measure X(_ but not sufficiently reliable to measure Xg .To measure this latter quantity it appears necessary to detect the5.2 MeV neutron released in process (2). Let us now go back to Fig. 1.S. Weinberg18 has shown that due to magnetic moment interactions within the (pyp) 0 excited molecule an Auger electric dipole transition is poss­ible leading to an ortho-para transition. In Fig. 1 the rate of such an Auger transition is labelled Xo p . Moreover with a qualitative procedure Weinberg findsXop < 1.6 x lo*4 sec-1. (9)Because of this limit one has always assumed in the past that X0p = 0 and X[_ in Table II has to be interpreted as Xom. In this hypothesis a compar­ison of X|_ with the theoretical value X0m calculated by Primakoff, and given in Table IV, can be done and one obtainsXL - x££ = A5 ± 27 sec- 1 . (10)The agreement is moderately good. If one also includes the value of Xomdeduced from Xg (Table Ii), Xom = *499 ± 28 sec- 1 , and combines these threevalues (for which x2 —  0 ?= k~12 ± 15 sec-1and thenXL - Xom =3*4 + 25 sec-1which again is moderately good. Notice that now the experimental value is more precise than the theoretical.Another way of showing the comparison in a more general way is pre­sented in Fig. 2. There one sees how the experimental values X|_ 5 andXc,s determine Xom even for the case where Xop ^ 0. In the same figure isshown the theoretical value for Xom, as calculated by Primakoff and the value of XQm deduced from the experimental value of Xg given in Table II (assuming for the small capture Xy the theoretical value of Table IV).To fully exploit the new accurate value of X|_ 5 with maximum confi­dence, it is necessary to clarify the situation regarding the transition rate Xo p . To do this, the SCB group surrounded the target, used for the measurement of X- , with neutron counters in order to measure the time distribution of the neutrons emitted from the absorption process (2) of the negative muon. As first pointed out by Bleser et al . 19 a comparison between the time distribution of the decay electrons (which is practically an exponential shape) dNe (t)/dt and the time distribution of the neutrons from process (2) taking place from the (pyp) 0 molecule dNn(t)/dt, and assuming a V-A type of interaction, gives the possibility of measuring67Fig. 2. Allowed region of Aom and A0p values: defined by the value A 5 |_ ob­tained by the SCB collaboration from Table II (thin line); defined by the value Aq^l obtained by the Columbia group from Table II (thin dash); defined by the value of Aom expected from the value of Ag given in Table II (dash-dot); and by the value of A calculated by Primakoff and given in Table IV (thick line).omX&p(l04sec'1)the ortho to para Auger conversion rate A0p of the (pyp)0 orthomolecule formed after the negative muons are stopped in liquid hydrogen.Referring to Fig. 1 it can be shown that for time t >> 1/App one has for the neutron time distribution dNn (t)/dt an expression of the type3- - J —  “  C ( * o m  -  A p m ) e - A o p t  +  A p m ] e - A o t  ( n )in which one expects Aop << A0 . In an interval of time dt so thatdtAop << 1 the distribution (11) will, in first approximation, appear asgiven by(P . e"[A0 + (Aom - Apm ) Aop/AomJ t ^ ^ 2)dtThat is, it appears to be an exponential but with the rate bigger than the corresponding rate for the electron decay distribution by an amount approximately equal to (Aom - Apm )A0p/Aom-This method was applied at Columbia since 196219 and a limitA0p < 5 x 1014 sec- 1 (13)was obtained.Very preliminary results20 were presented here in Vancouver a year ago obtained from the first runs of this modified set-up; other runs have lately been completed, and the data are now being analyzed. So far the situation is that the group confirms the presence of a positive effect with A0p around the value (13)- If such a result is confirmed, then the problem of ortho-para conversion deserves a closer look theoretically in order both to explain such an apparently big rate for A0p and to calcu­late it with a precision of a few per cent.The precise knowledge of such a high rate for A0p could offer an in­teresting possibility, especially now that rather intense fluxes of68stopping muons, with very small spread in range, can be available at the pion factories. Let me do this exercise with some details. Formula (11) after inserting the different expressions for Aom and Apm can be « e-A0t [i + 2(A-^Bx)e-A°Pt] (14)d twhere x = (Aj/Ag) = 1.18^ (see Table IV) and A and B are two expressions of Y0 and Yp which take a value near one. On the basis of formula (14) it is therefore tempting to try to collect a sufficiently high number of neutron events from process (2) in order to extract, by a proper fit, both the quantities A0p and x = (Aj/As). It is c 1ear, therefore,that a reliable calculation that could give A0p with sufficient accuracy would help quite a lot (see Ref. 21).In order to study the sensitivity of such a method I have done a Monte Carlo calculation in which events were generated with their time distribution according to expression (14), x = (Aj/As) was put equal to the theoretically expected value, and A0p put equal to the value (13).This distribution was then fitted with an expression similar to (14) where x and A0p were left as fitting parameters (but with the additional condi­tion that A0p was constrained to be within ±7% of a large theoretical value equal to value (13). No background was added in the fit and the quantities A and B were assumed to be perfectly known. I found that to determine x with an error of 50%, at least 2 x 107 neutron events, counted after 3 microseconds from the stopping time, were needed. (The calcula­tions originally were adapted to the Saclay beam.) Of course, in the fit only statistical fluctuations were considered.Clearly the experiment looks impossible. Moreover, this is not the only difficulty; another problem arises in trying to extract x from the coefficient 2(A-4Bx) of expression (14) even if this coefficient could be determined experimentally with sufficient precision. It is easy to show that, because of the uncertainty on Yo ar|d Yp (see Table IV), the coeffi­cient A has an uncertainty of a few %; this is highly disturbing since x is only 1.18%. At this point to help the situation a bit an improvement in the calculations of the quantities 2y0 and 2yp is needed.Nevertheless, such a difficult experiment to stop negative muons in a pure hydrogen target and measure the time distribution very carefully of >> 107 neutron events, in order to deduce either A0p or both x and A0p with good precision, is to my mind highly welcome at one of the pion factories.To complete the exercise let me suggest a little improvement. One of the worst (but not the only) backgrounds in identifying process (2) via the detection of the outgoing neutron is given by the photoneutrons.The decay electrons of process (l) make lots of photons, via brem- sstrahlung on the target material and these photons will produce lots of photoneutrons. A way to limit this background is to introduce an anti- coincidence within the hydrogen target itself where the muons are stopped (this idea was already used by A. Alberigi et al.5). This can be done, for the case of muons stopping in a liquid hydrogen target, by putting in anticoincidence the signal of a photomultiplier which detects the Cerenkov light emitted by the decay-electrons of process (l). One therefore needs an aperture in the target from which such light can be extracted.I have shown you this exercise because one of the fundamental69problems in the physics of muon capture is the experimental determination of the capture rate Ay [or x = (Ay/As)] and I see it also as one of the hardest challenges to pion-factory experimentalists.I hope that the high value for A0p given by (13) can be confirmed in the near future. The way I have outlined here, although difficult, looks to me sufficiently promising that it is worth trying. This concludes my discussion of muon properties connected with the weak processes (l) and (2 ).Let me now turn to the simple muon-atomic systems like (yp) and (y-1+He)+ in order to see what one can deduce about electromagnetic inter­actions, in particular about quantum electrodynamics.By stopping negative muons (the same is true for all other negative particles) in a target element of atomic number Z one can get bound systems where the bohr radius ay iswhere Ay is the reduced Compton wavelength of the muon and a is the fine structure constant. One expects that the energy levels of these systems will depend on all interactions taking place between the muon and the nucleus.One of the most fascinating, and deve1 oping, experimenta1 subjects is the branch of muon spectroscopy to determine experimentally, as accurately as possible, those energy levels in order to compare them with the theor­etical expectations. Through this comparison one checks, to a high degree of precision, the extent to which the muon (or eventually the pion or kaon) obeys the relative QED corrections, properties like the form factor, the polarizabi1ity of the central nucleus, the existence of new types of interactions, y-e universality, and possibly in the future, properties of weak interactions. 21The experimental methods with which one has proceeded so far are:i) Direct measurement: In this case a negative muon is stopped in a chosenmaterial and during its process of de-excitation X-rays of different ener­gies are emitted. Some of these are searched and carefully measured by means of either a solid state detector or a crystal spectrometer.ii) Study of the muonic atom levels using the double resonance method.22,23 (Perhaps this method can also, in the future, be extended to the study of pionic or other atoms.)I will not discuss the first method; achievements and progress lately obtained with this technique can be found in Refs. 21*, 25 and 26. The second method, so far, has been successfully employed to clarify the n=2 (n principal quantum number) levels spectroscopy of the muonic system (y_t+He) + .In what follows I will first illustrate this last method and mentionthe significance of the experimental results obtained with it and thengive some information on foreseen future applications in some laboratories.In Fig. 3 are shown schematically the lower levels of the muonic system (y_1+He)+. tr is the lifetime for spontaneous radiative decay of the level27; internal Auger transitions are not considered since the system is an ion. Only for pressure larger than 50 atm does the external Auger de-excitation (due to electrons of the surrounding helium atoms in the target) become important, and this only for the metastable level70Fig. 3. Schematic of the lower levels of the (ti_1+He )+ system (not to scale) t r  is the lifetime for spontaneous radiative decay, and xj the lifetime of the level.(u-l+He)Is -28 As shown in the figure the only stable level is the Is; the 2s is metastable since the radiative decay to the Is level via a dipole electric transition is forbidden. The most probable transition is a two- photon transition for which the rate is expected to be28:gy3 = 105(|)6 = 105 sec-1.The value (16) has been experimentally verified to about 20%.28Table V. Decay channels for the 2S p-l+He state.(16)Process Symbo1 Rate (sec-1)muon decay 4 .54 x 105muon capture *c 45 ± 5two quantum decay A2y 1.06 X 105spontaneous Ml transition AMl 0.53total disappearance rate X 5 . 6 x 105at zero dens i ty Jexternal Auger de-excitation *a(p) (2.4 ± 1.4) x 10 3P*Stark collision de-excitation As (p) < 300 P*P=pressure in atmospheres71Table V gives the rate of the different channels through which the metastable level can disappear (for negative muons stopped in a helium gas target at a pressure less than 50 atm). The table has been taken from Ref. 28 and for information about the rates see references therein.The experimental method consists of irradiating with tunable electro­magnetic radiation the muonic system initially prepared in the state i (for the case of the (y-ttHe) + system i = 2s) so that at a certain value of the radiation wavelength the transition i f i s i n resonance (in the case of Fig. 2, f S 2p3/2). The occurrence of the transition i -> f is signalled by the successive decay of the f level into the level labelled g (in the case of Fig. 2, g = Is). The levels i, f and g are chosen so that a suitable tunable electromagnetic source can be found and so that the X-ray emitted in the spontaneous f -> g transition can (easily if possible) be detected. (For the case of Fig. 2, the f->g transition gives an X-ray of 8.226 keV.) In this way the difference in energy between the two levels i and f can be measured rather accurately.In order to pursue this method various requirements must be met.Two of these are:a) the level i must have a non-zero populationb) the transition line f -* g must be distinguishable from other f -> g transitions emitted during the cascade by the muons when they are brought at rest in the target. Almost all muons will go to theIs ground state.The initial (non-zero) population, for the case when i = 2s, can be quite generally measured using a simple theorem relating this population (e2s) with the ratio Ka/Kj28 between the number of Ka rays and the sum of all X-rays belonging to the Lyman series (Kj = Yu + Kg + ... + Kn) emitted per muon stopped in the target. The relation is:(|7)1 + F - Ka/Krwith F = 7-2 ± 0.4.Formula (17), proved in Ref. 28, is valid when muons are stopped in a gas target so that Auger de-excitation to the 2s level is negligible (in general up to many atmospheres). For the particular case of the (y-ltHe)+ system one gets e2s «  5% for helium pressure in the range of 10 to 50 atm. Formula (17) will be very useful to understand what is going on for negative muons stopping in hydrogen gas.28In Table VI the most recent calculations of the fine structure dif­ferences in the n=2 levels of the (p-1+He)+ ion are presented30; in the last line are given the experimental results obtained at CERN.22’23 In the calculations the helium rms charge radius obtained by I. Sick31< r2>l/2 „ k 6 7 4 + 0 _oi2 fm (18)from the analysis of electron-helium scattering experiments has been used.Note that the experimental errors on the measured values of S3 and S2 are much smaller than the errors of the calculated values which are lim­ited by the uncertainty in the helium rms charge radius (18). From Table VI one sees that the Dirac contribution with a Coulumb potential to Si is zero; clearly this prediction is grossly inconsistent with experiment and form factor contributions or contributions due to fluctuations of the electromagnetic field (vertex contributions) cannot explain the big72Table VI. Fine structure differences for n=2 (p-1+He) + .Contri but ionsTransition energy (MeV)) ykyQOUyFy) iUy s l=2Pl/2_2s1/2Dirac contribution with Coulomb 1 h^ 70 0potential and pointlike charges ./uNuclear polarizabi1ity 3.1 ± 0.6 3.1 ± 0.6Fi ni te s i ze -288.9 ± 4 .1 -288.9 ± 4.1Electronic vacuum polarization:Uhling corrections first iteration 1664.44 1664.17higher iteration 1.70 1.70Kallen-Sabry term a2Za 11.55 11.55a(Za)n>3 -0.02 -0.02a2 (Za)2 0.02 0.02Muon vacuum polarization 0.33 0.33p-e vacuum polarization 0.02 0.02Hadron vacuum polarization 0.15 0.15Vertex corrections and (g-2):a (Za) -10.52 -10.85a(Za)n ,n > 1 -0. 16 -0.16a2Za -0.03 -0.03Recoil terms:Brei t 0.28 0.282 photons -0.44 -0.44Sum:theory MeV 1527.2 ± 4.2 1380.9 ± 4.2experiment MeV 1527.5 ± 0.3 1381.3 ± 0.5discrepancy. Here we see directly, and without applying any QED formalism, that the Coulomb law of interaction between two charges is not valid when the distance between the charges is of the order of ay. This unambiguous and direct finding is essentially one of the main contributions of muon spectroscopy to the clarification of the electrostatic interaction between two charges when these are separated by a very small distance. This fact has long been predicted by QED; in fact it was one of the first typical QED predictions soon after field theory was introduced in physics. In 1935 E.A. Uhling32 showed that, to first order in a, the potential between two charges V(r) had to be modified, due to the polarization of the vacuum (e1ectron-positron virtual pair formations), byV(r) = Vc (r)[1 + A(r)a + ...] with Vc (r) the Coulomb expression for the potential andI , R  s  6  s2 l 3 r3 y u9lVo3 uMlsn  r8YVwhere X. is the reduced Compton wavelength of the electron.Analyzing the comparison of the theoretical expectations with the experimental results of Table VI one sees that the contributions due to the polarization of the vacuum are tested at a level of 0.25%.22,2373Table VII. Experimental tests of QED vacuum polarization corrections.Exper i ment Preci sion (ppm)Contr i but i on vpTest of vp contri but ionMomentumtransferLamb-Shift in hydrogen (Ref. 30)20 0.025 0.3% (Ref. 29)amec9e-2(Ref. 3 00.2 0.0001 0.25% mecfs spli tting in n=2 levels of (y_4He)+ (Theory) (Refs. 22,23)2000 1.2 0.25% arriyCy28Si and y24Mg (Ref. 32)10 0.0035 0.24% 7 Otffly cyPb and yTl (Ref. 33,34)10 0.005 0.2% 20amyC9y-2 (Ref. 2)8 0.005 0.25% my ca = fine structure constant me = electron mass my = muon mass c = 1i ght ve1ocityThis result is shown in Table VII together with the strongest present experimental evidence of the QED vacuum polarization corrections. One sees that these corrections can be considered to have been experimentally checked to a level of 0.2-0.3% and this for a large range of values for the momentum transfer between the charge and the electromagnetic field.33 Substantial improvements, with respect to the results given in column 4 of Table VII, can in principle be reached with the method of double resonance.We have seen that for the case of the fine structure splitting in (y_4He)+ it is not the experimental accuracy nor the theoretical QED cal­culations which limits the precision of the comparison between experiment and theory but rather the uncertainty in the helium rms charge radius.Experimental plans, conceived in order to improve the situation, are being made at SIN (Villi gen), by a group of the E.T.H. (Zurich)34 and atB.N.L. (New York)21 by a Columbia University group. The experimental direction that the group of E.T.H.34 wants to develop consists in perform­ing accurate measurements of the 2s-2p energy level differences both in muonic hydrogen (yp) and in muonic deuterium (yd). From these measure­ments, and with the accurate knowledge that can experimentally be obtained of the ratio R of the rms charge radius of the deuteron over therms charge radius of the proton <rp>1//2m .D <rp>(20 )they will be able to deduce with very high accuracy the rms charge radius of the proton and to check the vacuum polarization QED corrections.For this case applying the relation (17) one can easily get an idea of the number of (yp)2s initially formed (for the (yd) system one expects the same), for the case of negative muons stopped in hydrogen gas at dif­ferent pressures; this is shown in Table VIII. In this p 1 an, however, there is an additional complication, when one compares it with the one on the fine structure splitting of the (y-1*He)2s system.74Table VIII. Experimental values of the ratio Ka/Kj for stopping negative muons in low density hydrogen as function of pressure.Pressure (room temperature) hydrogen gas target0.25 torr 150 torr 600 torr 4 atm(Ref. 34) (Ref. 35) (Ref. 35) (Ref. 36)Ka/KT 0.88 0.53 ± 0.07 0.54 ± 0.08 0.42 ± 0.1e2S 2 % (6.6 ± 1)% (6.5 ± \)%CN+1OOThis is due to the fact that the yp system, being neutral, can often have very close collisions with the neighbouring hydrogen molecules. In this case, because of the Stark 2s-2p de-excitation, the 2s level has an extremely short lifetime (the two photon decay as shown by formula (16) in this case is negligible). Therefore to distinguish the few 2p -*■ Is X-rays emitted following a 2s -*■ 2p transition of a short-lived (yp)2s atom, from the large amount of prompt 2p -*■ Is X-rays emitted by the major­ity of the muons stopped in the hydrogen gas target is problematical. As one sees, unless the pressure of the hydrogen gas in the stopping target is very low (according to the E.T.H. group less than a few torricelli at room temperature31*) condition b) mentioned earlier cannot easily be met.For this reason the group has developed a very original set-up in which 40 MeV/c negative pions (produced at the SIN machine) are sent along the axis of a magnetic bottle which is capable of confining most of the neg­ative muons coming from the decay of the 40 MeV/c negative pions within the bottle.35In such a device the group succeeded in having quite a high number of (yp) systems formed per second even at a pressure as low as a fraction of a torricelli. Moreover, a special development of the internal part of the target make them able to know the time of formation of the (yp) atom to within about 100 nsec.31*’35 Concluding from the proposal it appears that the group hopes to determine the proton rms charge radius with an accuracy better than 1% and to check the QED vacuum polarization corrections to about an order of magnitude better than the one shown in Table VII. In Table IX, as an indication taken from Ref. 33, are shown the expected 2s -*■ 2p energy differences in the (yp)2 system: the proton rms charge radius has been assumed to be 0.87 ± 0.02 fm.37 The necessary tunable source is being developed by the group itself.35In the proposal made by the Columbia group27 the proposed set-up is quite different from the one proposed at SIN31* or the one realized at CERN.22’23 In this case the central idea is to avoid studying transitions involving s states so that form factor corrections and nuclear polariz- ability contributions, which give the biggest uncertainty in the theore­tical calculations, are negligible. Therefore in testing the QED contri­butions here one can fully exploit the excellent experimental accuracy.75Table XX. Contribution to the 2s -*■ 2p splitting for the pp muonic atom. The states are in­dicated as 2F + Lj, where F = j + nuclear spin X. Energies are given in units of a2Ry = 0.13^6! eV. The disappearance rate of the 2p state is 1.2 x 1011 sec"1 : in the given units the width of the 2p level is 0.0006 a 2Ry (which for A causes a linewidth T of 20 A).A rather important characteristic for the target is to maintain the helium pressure where the muonic ions (y1+He)+ are formed as low as ^1 atm in order to increase the number of muons passing during their cascade through the 3d states and to decrease (by reducing the Stark effect and the Auger transitions) the number of prompt Kg lines emitted when one in general stops muons in a helium gas target.Calculations based on the measured value of Kq/Kj, made by A. Placci et aZ.28*88 are shown in Table XI. One sees that the fraction of stopped muons emitting a Kg radiation, in general, is about 5% for muons stoppedCont r i but i onsS  + 3Pjj Isj, -v 3p3* -5Trans i t i ons3sp. -*■ !Pj, 3Sj, ->• 3Pj, 2^. 2 2 3S j,3 S j , 3 s h  - 5p^Electronic a(aZ) vacuum1.5225 1.5225 1.5225 1.5225 1.5225 1.5225polarization a2 (aZ) 0.0112 0.01 12 0.0112 0.0112 0.0112 0.01 12Fine structure - 0.0625 - - 0.0625 0.0625Vertex correction a(aZ) -O.OOA9 -0.001(9 -0.001(9 -0.001(9 -0.001(9 -0.001(9Hyperfine structure 0.11(17 0.1135 -0.081(5 -0.0280 -0.0563 -0.0337Finite-size correction -0.0292 -0.0292 -0.0292 -0.0292 -0.0292 -0.0292=-0.03859 <r2> ± 0.0013 ± 0.0013 ± 0.0013 ± 0.0013 ± 0.0013 ± 0.0013Total (inoa2Ry) 1.6A13 1.6756 1.1(151 1.1(716 1.5058 1.528i(Total in A 56116 51(967 65086 62587 61166 60261± 51 ± 51 ± 51 ± 51 ± 51 ± 51For practical reasons (mainly connected with the availability of the necessary laser) they have chosen to measure the 3d-3p fine structure splittings in the (y_1*He)+ system. In Table X are given the expected energy differences of the 3d~3p states of interest.38Table X. Contributions to the 3d-3penergy level differences in (y1+He)+ given in MeV.Trans i t ionsContr i but i ons3d5/2"3P3/2 3d 3 / 2”3p1/2 3d3/2”3p3/2Fine structure 14.388 46.165 0Vacuum polarization:aZa f i rst i terat ion 110.514 110.594 110.511ala higher iteration 0.049 0.049 0.049a 2 Za 0.927 0.928 0.927a ( Z a ) 3 -0.004 -0.004 -0.004a 2 ( Z a ) 2 0.002 0.002 0.002Vertex contributions:self energy and anom­ -0.036 +0.029 -0.069alous magnetic momenttwo photon recoi1 0.005 0.005 0.005Total in MeV 125.84 154.77 111.42Total in A 98525 A* 80109 A* 111276 A*’■natural width r = 360 Ain a helium gas at a few atmospheres. Moreover, it can be shown that the fraction of muons passing in their cascading down to the lower levels, through the d levels, is about 60%. Another characteristic of this pro­posal is that one chooses to work with highly unstable levels.76Table XI. Expected values for the Kg/Ky ratio as function of the helium target density (room temperature).Approximate helium gas pressure (Ref. 39) (room temperature) Kg/KT — Y R0.1 atm 0.036 0.9061 0.05 0.85510 0.075 0.750100 0.080 0.560Experimentally found Yu UYH = 0.62 ± 0.08 at 7 atmAs shown in Fig. 2 the lifetime of the d and p levels are respec­tively Td = 5 x 10-12 sec and xp = 2 x 10“ 12 sec. This causes the lines of Table X to have a natural width T of aboutT = 360 A. (21)The experiment is therefore done by looking at a substantial increase (at least by a factor of two) in the Kg intensity (3p -*■ Is transition) when the laser wavelength passes through a value of one of the 3d-3p reso­nances. Clearly for this, good energy resolution in detecting the X-rays must be achieved (to distinguish the Kg from the other K lines) and one must have available quite a powerful laser. Moreover, due to the rela­tively large natural width T of the 3d~3p transitions it is not neces­sary to have a continously tunable laser. In fact, C02 lasers (to be used) emit in bands of lines about 100 A apart; it is of course necessary to choose a band that can cover one of the 3d-3p lines.Given the extremely short lifetime of the 3d level (see Fig. 2) one needs to fill the target cavity with the laser light, by the time the neg­ative muons arrive in the target. The experiment is planned to be perform­ed at the A.G.S.(B.N.L.) where intense muon beams as short as 50 nsec are available. The idea is to pulse the laser in phase with the pulsed beam from the A.G.S. Measurements on the low energy muon beam intensity per­formed by a Columbia group1*0 that can be obtained at the A.G.S. have shown that one can indeed have a sufficient number of negative muons per pulse stopped in a gas helium target at 1 atm, to perform the measurements descr i bed.With this project the authors hope to measure the energy difference of at least one of the 3d-3p lines to a precision around 10 - 15 A, and therefore to check the QED vacuum polarization contributions (see Table X) to at least a 0.1% level.I wish to make a last remark on this proposal. Since the initial state here is not on an s level, one can imagine doing this type of exper­iment also using pions as stopping particles (or even kaons). Of course a study on the proper levels corresponding to existing powerful radiation sources must be done in order to determine the feasibility. To give an77example measuring the 3d”3p energy levels difference in a (ir-1+He)+ ion one can check, with rather good accuracy, the formula for the energy levels given by the Klein-Gordon equation since the pion is a boson particle.Of course, in this case also (as for the (y_1+He)+ system) the differ­ence will be dominated by the vacuum polarization contribution; however, the Klein-Gordon formula contribution is about 30 MeV which is not so small. To have an idea of the sensitivity needed one sees that, using the old Sommerfeld formula, one obtains for the 3d-3p energy difference in (Tr-itHe)+ about 20 MeV.For muons the fact that they obey the Dirac equation (at least to 10 ppm) is experimentally proved by the (g-2) experiment. For pions one does not have this possibility. But measuring, with very great precision, the energy levels of pionic atoms can become equivalently interesting as a check of the validity of the Klein-Gordon equation.REFERENCES1. G. Bardin et at., CERN preprint EP/8-121 (1980) to be published.2. J. Bailey et al., Nucl. Phys. B150, 1 (1979).3. See for instance E. Zavattini, Muon Physics vol. II (Academic, New York, 1975) p.219 and reference therein.A. H. Uberal 1, Phys. Rev. 365 (i960).5. A. Alberigi Quaranta et at., Phys. Rev. 177, 2118 (1969).6. V.M. Bystristky et at., Dubna preprint DI 7300 (1973).7. J.E. Rothberg et at., Phys. Rev. 132, 2664 (1963).8. G. Luders and B. Zumino, Phys. Rev. 106, 385 (1957)-9. S.L. Meyer et at., Phys. Rev. 132, 2693 (1963).10. J. Bailey et at., Nature 268, 301 (1977).U . S .  Cohen et at., Phys. Rev. _M_9, 397 (i960).12. A. Alberigi Quaranta et al. , Nuovo Cimento 47B, 72 (1967).13. A. Bertin et al. , Phys. Lett. 88B, 185 (1979)•14. H. Primakoff in Nuclear Particle Physics at Intermediate Energy (Plenum, New York, 1975) p.l.15- W.R. Wessel et al., Phys. Rev. Lett. 13, 23 (1964).16. A. Halpern, Phys. Rev. Lett. 13, 660 "0964).17. P.K. Kabir, Z. Phys. 191, 447~Tl966).18. S. Weinberg, Phys. Rev. Lett. J_3, 660 (1964).19. E.J. Bleser et al. , Phys. Rev. Lett. _8, 288 (1962).20. A. Magnon et al. , in Abstracts of Contributed Papers, 8th Int. Conf. on High Energy Physics and Nuclear Structure, Vancouver, August 1979,P.137.21. For a discussion on some of these points see E. Zavattini, Lectures at Int. School of Physics of Exotic Atoms (Erice, April 1977) ed.G. Fiorentini and G. Torelli.22. G. Carboni et al. , Nucl. Phys. A278, 381 (1977).23. G. Carboni et al. , Phys. Lett. 73B, 229 (1978).24. T. Dubler et al. , Nucl. Phys. A295, 397 (1978).25. M.S. Dixit et al. , Phys. Rev. Lett. 35, 1633 (1975)-26. H.J. Leisi, Nucl. Phys. A335, 3 (198oT-27. A. Sachs et al. , Columbia University proposal for an experiment at the A.G.S., B.N.L. (New York) 1979-28. A. Bertin et al. , Nuovo Cimento 26B, 433 (1975).29. A. Bertin et al. , Lett. Nuovo Cimento 18, 277 (1977).7830. E. Borie and G.A. Rinker, Phys. Rev. Al8 , 324 (1978).31. I. Sick et al. , Phys. Lett 64B, 33 (1976).32. E.A. Uhling, Phys. Rev. 48, 55 (1935).33- See E. Zavattini, 7th Int. Conf. on High Energy Physics and Nuclear Structure, ed. M. Locher (Birkhauser, Basel, 1978) p.49.34. H. Anderhub et al. , ETH Zurich, proposal at SIN (Villigen) R— 78-15.1.35. H. Anderhub et al. , Phys. Lett. 71B, 443 (1977)-36. A. Placci et al. , Phys. Lett. 32B, 413 (1970).37- F. Borkowski et al. , Z. Phys. A275, 29 (1975)•38. E. Borie and G.A. Rinker, Karlsruhe preprint TKP-80-6 March 1980.39- A. Placci et al. , Nuovo Cimento J_A, 445 (1971).40. J. Fox et al. , A measurement of the production of low energy pions at large angles from the A.G.S. proton beam, Nevis (Columbia) internal report (unpublished).SUMMARY OF DISCUSSIONOn the subject of testing the Klein-Gordon equation with pionic atoms DEUTSCH asked about dealing with the effect of the strong interaction on the levels. ZAVATTINI said that one can choose higher levels (say d-states or above) where the strong interaction would be small. It was pointed out by ECKHAUSE that the K-G equation had been tested in pionic atoms by Cal Tech experiments. ZAVATTINI agreed, saying these measure­ments were essentially X-ray spectroscopy but that the resonance experiment he was discussing should be a more precise (about 0.1%) test.In response to a question by DEUTSCH regarding measurements to determine Xj by detecting the neutrons, ZAVATTINI pointed out the difficulty in com­paring results at different pressures because of the variation of neutron detection efficiency and emphasized the value in this regard of the tech­nique used by the SCB group in which the time distribution of the neutrons is measured.In the tests of CPT using the compilation of muon lifetimes, DEUTSCH noted that the phase space is different for a bound muon; ZAVATTINI said this had been taken into account.SIRLIN asked whether the hadronic corrections in vacuum polarization are calculated with dispersion relations. ZAVATTINI did not know off hand but cited the reference (?) and remarked that the corrections were small.SIRLIN also asked about the experimental situation regarding bound states of the pion-muon system (from K° decay). ZAVATTINI stated there has not been an effect observed, but referred to a current experiment of Schwartz (which will look for n-p) .WARREN asked about the reliability of current cascade calculation for pre­dicting the population of the 2s state in y"H and y~He systems and whether any pressure dependence is indicated. Referring to Schneuwly's discussion of initial muonic atomic states, ZAVATTINI said there is a question of &-distribut ion in light nuclei, but that in the cascade from a high enough79N (~20), because of C 1 ebsch-Gordan coefficients the muon tends to be pushed to circular orbits and such calculations give about the results seen experimentally for helium. There is the difficulty of incorporating the Stark effect at higher pressures. The situation for hydrogen he did not know.OLIN asked whether these laser pumping experiments did not require a very low duty cycle beam to match the duty cycle of the laser. ZAVATTINI pointed out that lasers can be pulsed at intervals of a few seconds (con­trollable at a few tens of nanoseconds) with pulses of a few nanoseconds wide; if the target cavity is well made one can have a large number of re­flections (more than 50) and therefore the light may last in the cavity for a time sufficiently long to certainly catch a spike of a pulsed (fast) beam. He stated that such an experiment is being planned at the AGS (at a type of low duty cycle neutrino beam) as shown in Ref. 27.NAGAMINE asked about the present status of the Columbia (y-He) Lamb shift measurement (Ref. 27) and the expected precision compared to the SIN experiment (Ref. 3*0 •ZAVATTINI reported that the experiment has been approved and funded but is awaiting the installation of the low energy beam line at the AGS. The experiment should start by the end of 1981 .Both experiments aim at a factor of 10 improvement. He stated that the SIN experiment is similar to the older experiment (Refs. 22, 23) and is rather specific for the 2s-2p measurement. The Columbia technique he claimed is more general, being able to handle very short-lived leading levels, and could be used with tr, y or K provided suitable lasers and detectors exist.YAMAZAKI asked how one detects individual X-rays in such a very short burst. ZAVATTINI said one uses many small detectors. There is a back­ground problem, however, for example Ka 's from other processes.BREWER asked for an estimate of the residual polarization of a y" (from a 100% polarized beam) when it first reaches the Is atomic orbital of y”p.He was interested in the possibility of taking advantage of the asymmetry properties of the triplet state to determine the rate (Ay).ZAVATTINI stated that at 1 atm the system is immediately in an F=0 state and therefore isotropic. It was suggested, however, that at a pressure of 0.1 atm the triplet state (F=l) may survive long enough to be measured. ZAVATTINI also pointed out that since the muon is stopped with an energy 10 x the splitting of the singlet and triplet levels one can have some re-excitation to the triplet; the problem in a gas is that this energy is quickly (10-30 nsec) lost by collisions and an atom remains in the singlet state once it decays. The question of depolarization is very difficult to predict; however, one should try to measure the electron asymmetry and "rather than having a problem, you have an experiment" (see Ref. 13).ECKHAUSE asked about the pressure dependence of the 2s population in He.It was about (5 ± 2)% from 7 to 50 atm— approximately independent. In80hydrogen there appears to be some variation from atm to 60 Torr but the errors are rather large.There was a question about looking at the 2s state in other than He such as Li or B. ZAVATTINI said yes, one could do such experiments in the gas phase.DEUTSCH asked about a Russian gedanken experiment using lasers to equal­ize the two hyperfine levels. One would then measure the neutron yield as a function of laser energy. It was proposed for H or 3He. ZAVATTINI thought such an experiment would work in principle, provided the pressure was low enough so that the laser pumping to the triplet state was not defeated by depopulation due to collisions. Moreover, in such an experi­ment, at a fixed pressure, the efficiency of neutron detection would be constant.81MUON DECAYA. Si rlinDept, of Physics, New York University, New York, NY 10003ABSTRACTA number of subjects concerning muon decay are briefly discussed. These include: the canonical analysis, predictions of the two-component and V-A theories, T-violating amplitudes, radiative corrections, relation at the one-loop level between the fundamental parameters of the SU (2)l 2  U(l) theory and Gy, possible effects of lepton mixing in y decay, obser­vations about the high and low energy parts of the spectrum, and con­straints imposed on theories with manifest left-right symmetry.CANONICAL ANALYSIS 1We consider y“ ->■ e" + vj + v2 , y+ -*■ e+ + Vj + v2 with massless neu­trinos. We leave open the possibility that Vj and v2 may be distinct par­ticles, or identical particles, or particles related by particle-anti- particle conjugation. We further assume that the final state is unique.If vj = v2 or vi = v2 the uniqueness of the final state follows if we assume the additive law of lepton conservation. If ^ v2 , the same conclusion follows if we assume the additive laws of lepton and muon number conservation.In the general four-fermion theory the energy-angle distribution of e“ (e+ ) from the decay of a fully polarized y- (y+ ) at rest is given bydN(x,e) = ^  a|s(1-x) + v ( j x  - I - 3^ )  ++ 6 „  T K  cos  9 f 2 ( l - x )  +  1 .6 (3 *  -  I -  3 ^ 3 . )  j ,  (1 )where the upper and lower signs refer to y- and y+ , 0 is the angle betweenthe e+ momentum and the spin direction of the y + ; E0 = (m2 + m|)/2mM) is the maximum e+ energy, p and E are the momentum and energy of e+ and 3 = p/E.The parameters A, p, n, B, and 6 are ratios of bilinear functions ofthe ten coupling constants g;, gf (i = S, V, T, A, P) of the charge reten­tion order [(ey)(vi v2)]. Alternatively, they can be expressed in terms of the ten constants g;, gf of the charge exchange order [(ev2)(vjy)].The latter are related to gj, gf by Fierz transformations. The parameter A is simply related to the muon lifetime. Another important piece of ex­perimental information is the longitudinal polarization of e+, namely P = (N+ - N-)/(N+ + N-) where N+ and N- are the total number of e+ with helicities h=l and h=-l in the rest frame of the y+, integrated over all energies and directions of e+.Most of the experimental work in the past has addressed itself to the task of determining the parameters t, p, n, E, 6 and P. Recent values of p, E, <$ and n are282Table I.Parameter Exper i menta1 Va 1 ue V-A ValueP 0.752 ± 0.003 0.755 0.972 ± 0.013 16 0.755 ± 0.009 0.75n -0.12 ± 0.21 0IM 1.00 ± 0.13 1More recently attempts have been made to measure the T-odd correlation <CTe* Pe x CTp> » ar|d preliminary results put a 10% limit on the T-viola- t i ng ampli tudes.3In this connection, it is worth while to recall that for the case in which the e + polarization is detected, there exist more general expres­sions for the decay probability involving the direction of the e+ spin, the energy and direction of e+ and the spin of y +, as well as general expressions for the components of the e+ polarization in arbitrary direc­tions.4 For instance, to gain a more complete knowledge, one could attempt to measure the component of the electron polarization perpendicu­lar to pe but parallel to the plane defined by <Oy> and pe , i.e. the correlation <ae • Pe x Pe x ®y>.The general discussion shows that if the neutrinos are not observed one can at most determine ten parameters while the four-Fermi Hamiltonian involves 19 coupling constants (10 complex couplings minus one arbitrary phase). The decay probabilities for y+ and y“ are further related by the TCP theorem.4TWO-COMPONENT THEORYIf the two neutrinos in y decay are two-component spinors, from the fact that the helicity of positrons near the end-point energy is h «  1, it is seen that and v2 have opposite helicities and thus Vj ^ v2 . The two-component theory implies9s = 9 s' = 9p = 9 p ' = 9j = 9j' = (2a)and furthermore gf = g; (i = V, A in charge retention order) if the hel­icity of vi and \i2 is h = -1 and g; = -g; if h * I. As the helicity of theneutrinos from y decay has not been detected, it it impossible to say whether gf = g; or gf = -gj . It is currently assumed on theoreticalgrounds that v2 is the same particle that occurs in 8 decay:n -*■ p + e” + ve which is known to have positive helicity. Thusgf = g;(i = V, A). (2b)Comparison with the general expressions of the four-component theory leads to83P = 6 = (3a)* r i9Ai2 - x P y n " 2 L|9aI2 + |sv12]• (3b)r = p + = -p . = - (gygfi + 9A9p) (3c)5 Pe+ Pe |gv12 + |gA 12 *If terms of 0(me/Ee ) are neglected, then it is easy to see that the longitudinal polarization of e+ from the decay of unpolarized muons is independent of energy. Thus, in the decay of unpolarized muons, Eq. (3c) is valid for the differential as well as the integrated e + longitudinal polari zation.It is useful for applications to express Eqs. (3b) and (3c) in terms of the coupling constants of the charge exchange order:= l Re gyS's " " 2 (I§VI 2 + 1/MSsl2) ’( Mwhere« = Pe* ' -Pe- - ] % ' ■ I'.b)I §V I + l/^l3sI2 9s = -9p = - (gv + 9a)» (5a)gy = = (gy ” 9C h " G  (-*b)gf = 9i (i = S, V, A, P), (5c)gj = 9T* = °* (58)V-A THEORYThe V-A theory in the case of y decay implies the validity of Eqs. (2a) and (2b) and furthermoregA = -gy. (6a)Alternati vely,§S = 9S ' = 3p = 9p' - 9R k oR ­ = °» (6b)9V ’ ‘SA * V  * ' V  -V  <6C>and we arrive at the famous phenomenological interaction^ e f f  = [veYy (l-Y5)'i've] ['f'vyYyd-Ys)^] + h -c -» (7)where Gy = /2gy and have assumed v2 = ve , Vj = vy.The predictions are:p = 6 = |, (8a)84To what extent can accurate experiments determine the interaction? There is a large class of interactions very different from the V-A theory that also leads to Eq. (8a). If, however, experiments should also show with precision that £ = 1 and P^+ = +1, then one would conclude that the interaction in the charge retention order is of the form- ^ e f f  =  | j > e Y y  0 " Y 5 H y ]  [ $ v i Y y  ( 9 V ~ 9 V ' Y 5H V 2]  +  h - c - >  ( 9 )which corresponds to g^' = -gy, g/\ = -g\i'. In order to determine the interaction completely, it is necessary to measure gyVgy. This has not been achieved so far, as the relevant experiments require the detection of the neutrinos from y decay. Jarlskog5 has suggested the study of the e-v angular correlation as the most realistic in this class of possible experiments.Equation (9) leaves quite open the nature of and v2 . If further experiments should show that g y ' = gy, we will learn that the neutrinos involved are two-component fermions and have opposite helicities and, therefore, ^ \>2* We could not decide from this knowledge whether V} and V2 are distinct or identical particles. Experimental determination of the "flavor" of vj and v2 > i*e - whether the usual identification V2 = ve and vj = Vy is correct, requires the study of reactions induced by the neutrinos from y decay.RADIATIVE CORRECTIONSQED or photonic corrections have played a very important role in the analysis of y decay. In the case of the two-component theory of the v or the V-A theory, in which only the V, A, V' and A' interactions survive in the charge retention order, the radiative corrections are finite in the local four-Fermi theory. This is true to all orders in a but first order in Gy. For S, P, T, S', ?' and T' interactions of the charge retention order the corrections are divergent, which is a reflection of the non­renormal i zab i 1 i ty of the local theory. In comparing experiments with the general four-component theory, it has become customary to describe the radiative corrections by means of the finite expressions obtained for the V, A, V' and A' interactions. The justification for this procedure is that the experimental information is consistent with pure V, A, V' and A' interactions and, therefore, terms of order a/2iT times g?,gf2 0  = S, P, T) are regarded as being of second order in the small quantities. From a theorist's point of view, it is more satisfactory to restrict oneself to the two-component theory (in which case the corrections are finite), attempt to determine the parameters E, and n, and otherwise verify the quali ty of the fit.In particular, with the inclusion of the QED corrections, the total decay probability in the V-A theory becomes85. PR OC  E 51 _ GylTlyN G S TUU MFrom the experimental value t^+ = (2197-1^*8 ± 0.066) nsec6 one obtains:GM = (1.16631 ± 0.00002) x 10-5 GeV-2. (10b)It has been estimated that the effect of the radiative corrections on the spectrum induces a shift in the measured value of p somewhat larger than 5%. On the other hand the corrections to t amount to only 4.2 x 10-3.This can be traced to the fact that the terms of 0(a) containing electron mass singularities cancel in the integrated spectrum but not, in general, in the differential distributions. An observable for which the corrections of order a become very large (several tens of per cent) is the asymmetry of low-energy e+. It may be interesting to study this in detail, to see whether the present theoretical description is satisfactory or whether it is necessary to include corrections of 0 (a2).The only effect of 0(a2) calculated so far is that part of the cor­rections to 1/xy which contains electron mass singularities. The result can be readily obtained using the theorems on cancellation of mass singu­larities and is given by7:P  R O CE DI N GSP  INTC I UM TFHY H/EA LThe third term in Eq. (10c) contributes -3-5 x 10-5. Its inclusion would raise the value of Gy given in Eq. (10b) by ~ 2  x 10-5.SU(2)L x U(l) THEORYIf all the neutrinos are massless or if there are global conserva­tion laws of e, y and x numbers, the sequential SU(2)l x U(l) theory leads naturally to the successful V-A theory of y decay. There are small corrections of 0(my/Mw) 2 to the various parameters, but for values Mw «  80.3 GeV they are negligible. For example the p value should be pIB = 3/i, + l/3(my/Mw) 2 = 0.75 + 5.8 x 10" 7 and £ = 1 + 3/5(my/Mw) 2 =1 + 1.0 x 10-6. It should be emphasized that the present experimental value for the parameter £ is less than unity by slightly more than two standard deviations. It is important to improve the accuracy in these measurements.A new and potentially significant theoretical development of this year has been the detailed analysis of the relation between the basic parameters of the SU(2)[_ x U (1) theory and Gy, at the one-loop level. 8 At the tree level we have the relations:Gy _ 92 (Ha)ft 8MW 2 ’e = g sin By, (1 lb)cos 0w = (Mw/Mz) • (He)86Equation (11c) is valid, for instance, if the Higgs transform as I = 1/2 multiplets under weak isospin. The question is: what are the corrections to these relations at the one-loop level? The answer, of course, depends on the precise definition of g and 0w. In the renormalization framework used in Ref. 8, Eqs. (lib) and (lie) are satisfied exactly with the under­standing that e is the conventionally defined charge of e+ and and M2 are the physical masses of W and Z. If you wish, Eq. (lie) may be viewed as the precise definition of cos 0w> and then Eq. (fib) is the precise definition of g. From the study of the weak corrections of 0(a) to y decay (i.e. all the corrections of O(Gpa) with the exception of the traditional photonic corrections), one finds=  1__8Mw2 Jl (1 + Ar) ’ UZaJwhere Gy is conventionally defined according to Eqs. (10a) and (10b). The correction Ar turns out to « 6 . 6 %  plus a bosonic contribution which ranges from -0.2% to + 1.2% in the interval 0 ^ (Mh2/Mz2) - 100 where Mh is the mass of the physical Higgs scalar. From Eqs. (lib) and (12a) it follows that[Trg | ]/iGyJMlI , Z6 1/2 (1 + Ar/2) 37.281 GeV (1 + Ar/2)' s in  0W "  -------------sTrTiw---------------‘ ° 2b)If we knew sin 0w accurately, we could predict Mw with high precision.For example if sin20w = 0.23 and M^ «  Mz we get M^ = 80.A GeV, instead of the 77-7 GeV predicted without radiative corrections. Radiatively cor­rected predictions for Mw were first given by Marciano who did not do a complete calculation but rather obtained the leading log contributions by a simple use of renormalization group arguments.9 His numerical results are close to those of Ref. 8. Other authors, Veltman10 and Antonelli, Consoli and Corbo11 also found substantial mass shifts for M^ and Mz, but their calculations differ somewhat from Refs. 8 and 9 at least partly because a different definition of sin20w was employed. Clearly a precise measurement of sin20w is of utmost importance, as it also has important implications for grand unified models and the proton instability. To help in this question the corrections of 0 TVqjPh to v-induced neutral current phenomena have been studied in a recent paper.12 These lengthy calculations of radiative corrections could be carried out in a reasonable amount of time because of the simplicity of the renormalization method adopted in Ref. 8 and the use of current algebra techniques in evaluating non-photonic vertex diagrams.POSSIBLE EFFECTS OF LEPTON MIXING IN y DECAYSuppose that the neutrinos have mass and there are no global conser­vation laws of e, y  and U  numbers. Then, with three lepton families:(13a)87where v{, v|> V 3 are the combinations coupled to y, e, t in the charged current, vj, V2, V 3 are the eigenstates of the mass matrix and U is in general a 3 * 3 unitary matrix. The virtual exchange (v{ y) -*■ W (e v£) leads to an effective interaction2 3^ e f f  = " 8 ^ 2  . 2 ^  u2ju lk [eYy (l-Y5)vj][vkYy(l-Y5)liJ. (13b)If the masses are negligible and we do not observe the reactions induced by the neutrinos,nothing especial happens: we get nine amplitudes each of the V-A type which contribute incoherently to y decay. Because of the properties of the unitary matrices the overall normalization in the prob­ability is3(g2/8M^2)2 £  Iu lkI 2 IU2jI 2 = (g2/8MW2)2 ,j,k=lthe same as in the canonical case. Suppose now that one of the masses,M 3 = Mx , is significant. We may distinguish two cases: M 3 > my - me , in which case vx cannot participate in the decay, or M3 < my - me . In the first case the effective coupling in y decay becomes(g2/8Mw2) -^ 1 - | U j 3 |2 -y/l - | U23 I 2 < (g2/8Mw2).This may affect the discussion of Cabibbo universality and the constraints placed on the Cabibbo angles. For example, if we call V the unitary ma­trix analogous to U which describes the mixing of u, c, t in the quark sector, the ratio of 3“ and y-decay couplings which is usually called cos 0C becomes cos 0C = V n ( | U i i |2 + | U1212)~ 1/2 while the ratio of Ke3 and y-decay couplings becomes V2l(|^lll2 + I 12 12) - 1/2 (Ref. 13)- Using the radiative corrections to the 3 and muon lifetimes one knows from ex­periment that (|V1112 + |V2112) (|Uij | 2 + 1U12 I 2) _ 1 's quite close to 1.In the usual analysis one assumes no mixing, i.e. U13 = 0, and therefore |U1112 + |U!2 |2 = 1. From the property of unitary matrices,£  iviji2 = 1, j=iand the fact that | V u |-2 + | V21 12 one gets |V31|2 = 0.003 ± 0.00A. Defining V3j = sin 0C sin 03 one obtains an upper bound sin ©3 ~ 0.36.1I+ When U 13 0 the analysis becomes much less stringent. For example if|Ui3 |2 is large so could be |V31|2 . If M3 < my - me, vx can participate in y decay but because it is massive its contribution to the decay prob­ability is not exactly of the canonical type. To simplify the discussion let us assume that U23 = 0 so that vx does not interact with e but U13 ^ 0 so that vx couples to y. The transition probability becomes:dN (x, 0) = (1 - | U 13 | 2)dN (°> (x,0) + | Ux 3 | 2dN < 1 > (x, 0), (l*ta)where dN^°^(x,0) is the usual distribution of the V-A theory with massless neutrinos while dN( 1 )(x,0) is the distribution obtained if we assume that88the v coupled to the muon has mass MVx. It was calculated many years ago by Bachall and Curtis15 and, neglecting me , may be written asdN(^(x,0) = const 8 (xmax-x) x2dxdfi[- AllII,3- 2 x + AL^>- ++ cos e[] - 2x - — (14b)2 2where x = 2E/my, r = Mvx/my, xmax = 1 - r. Using these expressions one finds for the integrated asymmetry- j  + 2 | U ! 3 | 2 r [2 + 5 r - ^  r2 - y -  - 2r(2r + 3)£n|]a = r= —3 pj . ( 1 4c)1 +  2 | Ux 3 | 2r| -^4 ( 1  - r2) -  —  ,  6r £ n y jFor small r this is in the direction of decreasing the magnitude of the asymmetry and, therefore, the magnitude of £. For instance, for r = 0.1 a = -(1/3)[1-|U13|20.578]/[1-|U13|20.516]. For |U13|2 = 0.1 this leads to a = -(1/3)0.993, a 0.7% effect. Much more interesting is the effect on non-integrated observables such as the spectrum shape. Approximating the normalized spectrumby Sj (x) = 12 (1 —x) + 8peff(4/3x - 1) one finds for r = 0.1 and |Ui3|2 = 0.1, Peff 0.70, the precise value depending on what region of the spectrum we examine. Clearly this is ruled out experimentally. Even for |U13|2 = 0.01 (and always r = 0.1) the effect is significant: peff = 0.743 to 0.746 in the range 0.3 - x ^ 0.95. This is more than 2 standard deviations away from the experimental value throughout a very significant range of the spectrum, and it is possible that careful comparison with present data may already lead to the conclusion that |U13|2 < 0.01 for r = 0.1.The origin of the effect is clear: the mixing to a massive neutrino pro­duces a shift from the high end to the lower parts of the spectrum.Recently Shrock has proposed sensitive correlated tests for lepton mixing and neutrino masses mainly based on TRR-Yh(y decays.16 During the course of this Workshop he has informed me that he has also independently studied the effect of neutrino mixing in y decay.17 It seems clear that, because of the well known helicity effect, the two-body tests are much more sensitive than y decay in the analysis of U23, i.e. the mixing para­meter between the e and t neutrinos. To decide whether this is the case for U 13, i.e. the mixing parameter between the y and t neutrinos, requires detailed study. Shrock's conclusion is that even in the case of U33 the two-body decays provide a more sensitive test. It should be remembered, however, that experiments currently under way may significantly decrease the error on p and thereby improve the sensitivity of y decay as a tool to investigate U 13.HIGH AND LOW ENERGY PARTS OF THE SPECTRUMThe deviations from the Michel-type spectrum given in Eq. (l^d) are most important in the range 1-x ~0(r). If r is small it may be interes­ting to study with great precision the slope of the spectrum near the end89point. There is another reason why the study of the high end of the spec­trum is interesting. The radiative corrections contain a contribution 1 + (2a/ir) [£n(mp/me) - l]£n(l-x) which diverges as x 1 . Theoretically the problem is solved by consideration of multiple soft-photon emission which is expected to lead to an exponentiation of the type'  + v [ l n ( i % ) '  ' ] l n ( | - x)  *  e xp j v [ l n © '  ' ] t a ( | - x ) j= O - x ^ a A O U n H / m e H ]which vanishes as x -> 1 rather than diverging. How close to x = 1 must we get in order to be sensitive to the terms of order a2? Expanding, we find that the terms of 0 (a2) in the above expression give rise to correc­tions of 0.43, 0.96 and 1 .7% at 1-x = 10-2, 10-3 and 10-1+, respectively, while the corresponding contributions from the terms of 0 (a) in the same exponential are -9*3, ~l^ and - 19%, respectively.On the other hand, a detailed study of the low energy part of the spectrum is also very important because it may improve the determination of the parameter r). A deviation of this parameter from zero would indi­cate the presence of non V,A couplings in the charge retention order.These could arise, for example, from the exchange of exotic charged Higgs mesons coupled to (ev2) and (v^y) with large coefficients not proportional to the lepton masses.THEORIES WITH MANIFEST LEFT-RIGHT SYMMETRYThese are theories in which parity conservation is broken spon­taneously in such a manner that left-handed and right-handed charged currents coupled to and W^, respectively, have identical transforma­tion properties in flavor space and one current is obtained from the other via Y 5 -* ~Y5. Particular realizations have been constructed in the frame­work of SU (2) |_ x SU(2)r x  U(l) and higher groups such as SU(A)l x SU(4)r 2  U(l). The relevant interaction involving the charged intermediate bosons is of the form:I = - ^ [ ( V - A ) pWP + (V+A)pWP + h.c.]. (16a)W|_ and 0s derive their masses via the Higgs mechanism. Under the spon­taneous breakdown of parity conservation, the right-handed field 0 s be­comes more massive than Wi_. In general there is mixing and the eigen­states of the mass matrix are orthogonal combinations of W|_ and 0 s dVli = Wl cos Q - Wr sin £, (16b)W2 = W|_ s i n x. + Wr cos (16c)The effective interaction at low energies is18eff = - 7 f [ VJvP + nAAAJAP + nAV T—:aP + APVP)1> (17a)9 0The V-A limit corresponds to ? 0 and (m2/mi) -»■ °°. Comparison with thefour-fermion theoryshows that gy = G//2, gA = -(G//2)pAA, §y' =■gA' = “ (G//2)nVA> 9i = 9i ' = 0 (I = S, T, P). The coupling constants in the charge retention order are obtained by Fierz transformations:9S = "9p = (G//2) TRPCg - 0 .  9V = "9A = (G//2)(l + : C C hUy - p; k 9- p — ­ k"9a f = “ (G//2)ny/\, gy = g y ' = g$ ' = gp' = 0. As far as y decay is con­cerned, these identifications lead to the relations:P = f [ 0  + hAA)2 + ^ h V A ^ / D  + hAA2 + 2r>VA2]» (l8a)6 = \  , (18b)E = -2pV A (l + nAA)/[l + nAA2 + 2nVA2], (18c)n = 0 , (18d)P(e+) = +E . (I8e)If C = 0, nAA = 1 and we get gy = -gA , g y ' = -gA'* In that case p = 3/4. This still is not the V-A theory because nvA / “ 1 ancl therefore gy = .?C  / § V f = ~9Ar• As a consequence E / 1 even when t; = 0.One determines -r)AV / HAA from the longitudinal polarization of electrons in Gamow-Teller transitions and then : C C from the p value. Allowing a spread of two standard deviations it was found that 0.813 -PAA - 1.23, 0.698 ^ /:g— - 1.11 which translates into 2.76 5 m2/m1 ^ °°,-0.060 ^ tan E - 0.054.18 As far as y decay is concerned, the analysis leads to the predictions of Eqs. (18b), (l8d), (l8e) and 0.959 - E - 1. This is one more interesting reason to measure E very accurately.REFERENCES AND FOOTNOTES1. For a more detailed discussion of the canonical analysis and experi­mental aspects of muon decay see, for example, A.M. Sachs andA. Sirlin, Muon Physics II, Weak Interactions, edited by V.W. Hughes and C.S. Wu (Academic Press, New York, 1975), p.^9, and references cited therein.2. Review of Particle Properties, Revs. Mod. Phys. ^2, No. 2, Part II (1980).3* F. Corriveau et al., paper contributed to the 8th Int. Conf. on High- Energy Physics and Nuclear Structure, Vancouver, Canada, August 1979*A. T. Kinoshita and A. Sirlin, Phys. Rev. 108, 8AA (1957).915. C. Jarlskog, Nucl. Phys. 75, 659 (1966).6. G. Bardin et al. , CERN-EP7S0-121 (1980) and report by E. Zavattini,these proceedings, p.62.7. M. Roos and A. Sirlin, Nucl. Phys. B29, 296 (1971)-8. A. Sirlin, Phys. Rev. D 22, 971 ( 1 9 W -9. W.J. Marciano, Phys. Rev. D20, 27*t (1979).10. M. Veltman, Phys. Lett. 91B , 95 (1980).11. F. Antonelli, M. Consoli and G. Corbo, Phys. Lett. 91B , 90 (1980).12. W.J. Marciano and A. Sirlin, Phys. Rev. D22, 2695 (1980).13. A similar discussion was given by B.W. Lee and R.E. Shrock, Phys. Rev. D16, 1A44 (1977).1h. A. Sirlin (unpublished). A detailed published analysis was given byR.E. Shrock and L.-L. Wang, Phys. Rev. Lett. 1*1, 1692 (1978).15. J. Bachall and R.B. Curtis, Nuovo Cimento 21,~~522 (1961).16. R.E. Shrock, Institute for Theoretical Physics at Stony Brook report ITP-SB-80-23.17- R.E. Shrock (manuscript in preparation).18. M.A. Beg, R.V. Budny, R. Mohapatra and A. Sirlin, Phys. Rev. Letters 38, 1252 (1977).SUMMARY OF DISCUSSIONSTR0VINK began with three comments. First, that it is at the high energy end of the y-decay spectrum that the sensitivity is highest, both to the parameter r (which is nonzero if one or more neutrinos is massive) and to the parameter £; therefore, it would be natural to combine the two measurements in a single experiment.Secondly, in the numerical integration of the muon decay rate using the prescription of Sirlin and Sachs for 0(a2) corrections at the end point, these corrections raised the decay rate by a factor of 1.00075. SIRLIN pointed out the difficulty in the formula since the exponentiation is supposed to be correct near the end point but not necessarily far away from the end point. Even this is debatable as the rule for exponentiat­ing is not clear.Thirdly, the lower limit derived (with B6g, Budny and Mohapatra) for the right-handed W mass used experimental input from polarized nuclear B-decay. That is, if the 0 s .0r/ mixing angle is zero, the B-polarization is 1-2A2 , where A is TJP/UJs hy a On the other hand, if one measures the 5 parameter in the decay of muons whose polarization is inferred from ir-decay kinematics, ^effective = 1-^A2 ; that is, it is twice as sensitive. Physically the reason is that the departure from (V-A) affects both the it- and y-decay processes. SIRLIN agreed that a £ measurement was a good i dea.MEASDAY asked if measurements of radiative muon decay y -*■ evvy were more directly sensitive to radiative corrections in muon decay. SIRLIN was not sure whether this was so. He pointed out that radiative decay depends92on the same combination of coupling constants according to a theorem of Pratt 20 years ago. The early interest in radiative decay was the possi­bility of measuring the time reversal amplitude. But this was ruled out by Pratt who showed that if the two neutrinos are massless, all the correlations that have time-reversal odd must be pseudoscalar. There have been some useful measurements of the electron-photon angular corre­lation to determine the ratio of the scalar amplitude to vector amplitude in muon decay.In going to very small values of 1-x in muon decay to study the nature of radiative corrections, DEUTSCH asked if there was a theoretical relationship between these corrections and the infra-red tail of radiative muon capture. He remarked that it would be easy experimentally to measure the soft photons of the infra-red tail in the radiative capture process.SIRLIN explained that the calculation of radiative corrections includes both real photons and virtual photons; that both real and virtual photons have infra-red divergences which mutually cancel. At the end of the decay spectrum the real photon cannot be emitted because it has no energy and so in a sense it cannot cancel the infra-red divergences from the virtual photons. This is the origin of the logarithmic singularity as x -+ 1. Thus, the detailed nature of this singularity is a consequence of a kinematic constraint imposed by the three-body muon decay. In the two- body capture process the effect of phase space may be different.WARREN asked if any additional information is gotten from an accurate measurement of the electron polarization as a function of energy and angle for 100% polarized muons.SIRLIN said yes, that the formulas of Ref. k describe the electron polar­izations in all directions; the dependence is contained in ten quadratic combinations of coupling constants, a result which is also explained in an old paper by Pratt on the radiative decay.DEUTSCH mentioned that such a measurement has been done at SIN by Egger and collaborators; it is in the process of final analysis and not yet publi shed.93FUTURE MUON FACILITIES AT TRIUMFJ. Doornbos, J.A. Macdonald and J.N. Ng TRIUMF, kOOk Wesbrook Mall, Vancouver, B.C., Canada V6T 2A3J.H. BrewerUniversity of British Columbia, Vancouver, B.C., Canada V6T 2A6The workshop was conceived as a forum in which to consider the muon facilities at TRIUMF in relation to the experiments likely to be forth­coming in the next decade. The programme was revised at the last moment, in order not to conflict with a memorial service for R.M. Pearce, by delet­ing the discussion of the question of facilities at TRIUMF. However, the workshop has subsequently initiated some thoughts and discussions by users. A preliminary version of this section was presented at the TRIUMF Users Annual General Meeting in November 1980 for discussion, and the two recommendations below were approved.The papers presented at the workshop and discussions of them illus­trate some of the promising areas of muon physics to be investigated in the next few years. While one cannot predict what new phenomena may be revealed, it is true that we are beyond the first generation of experi­ments in many areas; that future experiments will be of yet higher preci­sion, often requiring exotic or technically complex targets and apparatus and good background suppression. Correspondingly, the muon beams most suitable to the new generation of experiments will be those of not merely high intensity but of high luminosity, high stopping density and low con­tamination. Because the muon spin is an important or central feature of many studies, beams with high polarization are also desirable.Previous reports1’2 have described in general terms the capabilities of present facilities and the options for future facilities at TRIUMF.Table I summarizes some of the characteristics of surface y+ beams avail­able at LAMPF, SIN and TRIUMF. It shows that the TRIUMF surface beams are quite competitive, especially with the use of the dc separator to complete­ly remove the positron contamination. In addition the separator can be used at high fields to produce a transversely polarized beam which can be stopped in a beam-axial field at the target. Both LAMPF and SIN are currently developing electrostatic separators for their surface muon beams. For many experiments with y+ , surface beams with positron separation meet the criteria of high polarization and high luminosity.For experiments using negative muons (or requiring positive muons of higher momentum) either cloud or decay muons are possible. Only backward decay muons simultaneously meet the characteristics of high polarization and low contamination. High stopping luminosity is also attainable with careful beam line design.In Table II— several y" beams are compared— they are all backward decay beams except for TRIUMF-M9 and S e1.RR E 3 which are cloud. The table shows that cloud beams can be comparable in stopping luminosity to some decay beams, but the polarization is low (<50%) especially at momenta below 100 MeV/c.3 The table also shows that stopping luminosity of an order of magnitude greater is possible at the internal target used for the bent-crystal spectrometer at SIN. Decay channels can be tuned for y+ withTab le  I .  Comparison o f  s u r fa c e  y+ beams ( p o l a r i z a t i o n  100%).Faci1i ty Product i on ta rgetSpot area (cm2)Lumi nos i ty (y+sec“1pA-icm-2) (before separator)Maxi mum 1umi nos i ty (y+sec_1cm"2)Sepa rate e+?RRU2 precess ion?TRIUMF:M9 10 cm Be 0.5-10 (col 1imated)103 105 (yes)c (yes)cMl 3 1 cm C 6COOX 1.7 x 105 poss i ble dc separatornonew M20 10 cm Be 12 (FWQM) 2 x 103LDoXCSl yes yesSINa :coLUt= Mo? 15 4.7 x 103 4.7 x 105 by degrader (dc separator under development)noLAMPFb :SMC 10 cm C 1-100 (0.3-1-7)xl03 (0.05-1)xl06 by degrader (dc separator under development)noaTable from SIN report #5/78/2000, update August 1979-G. Sanders, private communication.CThis will not be possible with the RF separator to be installed in 1981.95rates a factor of k~5 higher; they can also be tuned to extract forward decay y+ or y” with an increase in intensity, but usually with large pion contamination. The polarization of decay muon beams is typically >80%.Table III shows a concise albeit oversimplified summary of some aspects of muon physics which are or could be studied at TRIUMF. It is not complete or absolute, but shows, as is evident already from Tables I and II, that experiments using negative polarized muons will be at a dis­advantage at TRIUMF, even with the new M20 upgrading.1*In the Appendix the capabilities of various possible decay muon channel configurations at TRIUMF are summarized and compared with SIN and the present M9 beams.The two general recommendations arising from users discussions andapproved at the 1980 Users Meeting are:1) TRIUMF has competitive surface muon beams now and should continue to support and further develop these.2) TRIUMF lacks, and needs, a channel capable of high stopping luminosi­ty and highly polarized negative muon beams.SURFACE MUON BEAMSTo address recommendation 1) several possibilities arise:a) Improve Ml 3:- Can M13 be improved with new magnets?- Can a separator be incorporated into M 13? This has been studied since the workshop . 5 A small separator to produce a clean surface beam is feasible.b) Improve M9:- Shorten Q1 and/or modify T2 shield to increase solid angle.- Other changes to improve optics— more magnets, slits, etc.?- Install a new improved dc separator with high transmission to replace the RF separator in order to restore the versatility of both cloud and surface beams.c) New or improved technologies:- Precision production targets optimized for surface muons.- Application of permanent magnets to surface muon (or other) channels.- Separator design (M9 and M 13)-- Low-energy y” production targets using magnetic bottles for PR= ad) Installation of new M20:The redesigned M20 improves the performance over the present M20 by about a factor of 6 and will produce an excellent surface beam with a dc separator for e+ rejection and for muon spin precession. In addition, the new M20 will produce a backward decay beam which will be within a factor of 2-3 of the lower rate beams available elsewhere (Table Ii). It will make some of the experiments envisaged with y" more feasible.Table XI. Comparison of p" beams (p+ generally A-5 times greater rates).[A] [B] [C] [D = C/AB]Fac i1i ty ProductiontargetTP(MeV)Tp (Me /c)Spot area (cm2)Ap/pc(«Range width (g/cm2 CH2)Flux (p~sec~ *pA- 1)Stopping(p'sec'1umi nosi ty "ipA* 3g-1)TRIUMFM9 (cloud,RF separator) 10 cm Be 500 77 7.5 x A 10 ~l .2 (±0 .2 ) 9 X 103 0.25 X I03new M20 (Ref. A) (bkwd decay) 87 it x A 7 ~l 1.5 x 103 0.1 X 103S 1 Na(bkwd decay)pEl 12 cm Be 580 85125 6 x A(12)H O1.5AA2x 10“ x )0 51.112.1XX103103PE2 90130 6 x 10W )(~6 )12.563X I03 x 10“0.10.2XXI03I03uE3 0-60 ~ 2 0  (internal) 30 X I03pEA 50905 x A.56 x 55 x A.5 6 x 5CO "cN  ^^0.350.091.80.A5A125X 103 X I03 X 10“ X 1030.510.370.A90.37XXX103I03103103uE3 (cloud) Mo? 60 3 x 5 0.65 1.6 X 10“ 1.6 X 103LAMPFbSMC(bkwd decay) 6 cm C 800 105 3.5 x 10.2 (12) 2.5 3 X 10“ 0.35 X 103j^Table from SIN report #5/78/2000, update August 1979.P.A. Thompson et al., Nucl. Instrum. & Methods 161, 39 (1979).Values in brackets calculated approximately from quoted range width.Table IIIMuon physics: Descriptions Muon facility requirements Importance (0- 10) of various features'Marks'forTRIUMFnow(0-10)General 'area (no order)1Specific 'programs' Comments FluxStoppi ng 1umi nos i tyBeam pur i tyH i gh polar i z .T ransverse po1 a r i z .p+SR Gas phase (mu formation, mu chemistry)No competitors yet. Surface p+ only5 9 6 10 8 10Liquid phase (mu chemistry, rad i ca1s)Competition at SIN 5 7 8 10 8 9Solid state (relaxation and spectroscopy)Strong competition at SIN 8 10 10 10 9 9p‘SR Solid state physics - espe­cially comparison of p~Z with (Z-l) impurities, sub- st i tut ionalTypical Tp~0.1 R! so need/ can take 10 times more rate than p+SR; cascade depolari- zat ion-<-need — 100 times as many events as p+SR for same preci s ion!10 10 10 109 if 29 MeV/cp-; 0otherwi se1p" atoms, p~ captureX-raysRadiative captureFew expts yet use polarized p, but could be exploited1081086105761Triplet p~p capture in gas H2 Must run in ~0.1 atm H2 8 10 10 10 10 3p"-induced F i ss ion 7 9 10 - - 7nuclear phys i cs Nuclear polarization effects See Louvain group and SIN8 9 10 10 7 1p+ rare processesp+ ■+■ e+y Requirement similar to those of p+SR except higher flux10 8 10 - - 8Excited muonium states Not many muonium formed per p+ incident -*■ higher flux10 10 7 10 8p” rare processesp"Z -+ e~Z 10 8 10 - - 7p decay parametersn,5Vp mass6 10 10 10 - 1098DECAY MUON BEAMSTo address recommendation 2) a superconducting solenoid decay channel is the facility most widely favoured to meet the need for highly polarized negative muons. The technology for such a device is now well established with two such channels operating at SIN (since 1975 and 1977, respective­ly) and one at KEK (since 1980s). The solenoid could be a near copy of one of these existing designs, while the injection and extraction could be designed at TRIUMF to meet our particular requirements. Such a channel can have good extracted muon beam optics— the KEK design has an M9_like doubly achromatic extraction section. In addition the internal muon beam could have ~50 times the stopping density of the extracted beam.The channel could be designed to be put on a new beam line BL2A. However, it could also be designed to replace M20, but would require a redesign of 1AT2 and the front ends of M9 and M8. A third plan would be to excavate part of the proton hall extension, to extend and upgrade beam line 1*A for high current operation and to install a production target and channel. This latter possibility could allow more efficient use of the unpolarized beam in the same way that BL1B does for the polarized beam and could be co-ordinated with an isotope separator facility. However, the serious impact this plan would have on the proton hall experimental pro­gramme has been pointed out by users.WHAT ABOUT HIGH ft CHANNELS?Lobb7 has proposed a type of channel using a very high acceptance (~1 sr) in order to achieve high flux. Because these channels use cloud muons and non-paraxial geometry, the muon polarization in these channels would be low. Nevertheless, these designs offer an efficient use of pro­duction target flux and could compensate for TRIUMF's ultimate beam intensity being substantially lower than that planned at SIN or LAMPF. However, current interest seems to stress the capability for polarized beams as well as intensity; this, coupled with the exotic nature of these designs, has resulted in, at least until now, a lack of enthusiasm forthis type of facility among users.The possibility of applying this design to a surface muon facility has not been examined in detail.APPENDIX: HIGH FLUX DECAY MUON CHANNELMonte Carlo calculations have been performed to obtain flux estimates for 80% polarized backward muons of 85 and 125 MeV/c. Backward muon beams are not contaminated by pions and electrons. The results are given in Table IV. Column 1 gives the measured results of the SIN superconducting solenoid decay muon channel pEl (Ref. 8) which takes off from the produc­tion target with an angle of 0° with respect to the 580 MeV proton beam.If this channel is reproduced exactly at TRIUMF, we can expect the resultsof column 2, obtained by multiplying column 1 with 0.5, the ratio of thepion production cross-sections at 500 MeV (TRIUMF) and 580 MeV (SIN).Column 3 is based on Monte Carlo calculations for a decay channel with an 8 m long SIN solenoid preceded by a front end consisting of a quadrupole doublet and a bending magnet taking off at an angle of 1*5° with respect to the proton beam (solid angle of 70 msr).99Column k gives the result obtained if the solenoid is replaced by an array of 20 permanent Rare Earth Cobalt quadrupoles, 15 cm long, 5 cm spaced apart, pole tip radius 9 cm and pole tip field of 8.5 kG, cost approximately $1 million. The flux is 2/3 of the flux for a solenoid channel. Column 5 is obtained if normal quadrupoles are used in the decay section. The flux is a factor 2.9 lower than for a solenoid channel. Itappears that a solenoid channel gives the best results. The relative performances of a solenoid channel, a permanent quadrupole decay channel and a normal quadrupole channel are similar to the relative performance of the solenoid channel, F0D0 compact channel and F0D0 normal channel studied by the IK0 group.9Column 6 gives the intensity of a beam 77 MeV/c unpolarized cloud muons, uncontaminated by pions and electrons at focus W3 of the M9 chan­nel, after the RF separator is installed. The solid angle of acceptance of this channel is 25 msr. An improvement of a factor 2 in intensity might be possible if the solid angle is increased to 50 msr by modifying the front end, see column 7-Table IVpu SIN TRIUMF Solenoid Permanent Norma] M9(MeV/c) pEI UEI 0° l*5‘1 quads 1*5° quads 1*5° present improvedU+ 125 6.7 » 107 (9.6 x I03)3.1* x 107 (4.9 x 103)1.6 x(2.3 x107I03)1.1 x (1.6 x107I03)5.6 x (0.8 xI06 1 o3)85 1.3 x I07 (5.1* x I03)6.5 x 106 (2.7 x I03)8 x(3.3 xI06103)5.3 x(2.2 xI061 o3)2.8 x (1.2 xI06I03)77 5 x 106(1.3 x 103)1 X !07 (2.5 x |03)IT 125 1.7 x 107(2.1* x 103)8.5 x I06(1.2 x 103)1*. 0 x (0.57 x106I03)2.7 x(0.39 xI06 1 o3)1.1* x (0.20 x106103)85 3.3 x 106 (l.l* x |03)1.7 x I06(0.71 x I03)2.0 x(0.83 x106 1 o3)1.3 x (0.51* xto6 1 o3)0.7 x(0.29 xto 6 1 o3)77 1 x 106(0.25 x 103)2 x |06 (0.5 x 103)Polarized 80% Unpolari zedFlux for: 100 pA protons on 10 cm Be in cm2 area at final spot with 8% Ap/pLuminosity in parentheses: p* sec-1 pA-1 g"1100REFERENCES1. J. Macdonald, TRIUMF design note TRI-DN-79“5 (1979).2. J. Macdonald, memorandum to D. Gurd "Report on new muon facilities to 1979 TRIUMF Users AGM";P.A. Thompson et al. , Nucl. Instrum. Meth. 161, 391 (1979) andG. Sanders, private communication;SIN report No. 5/78/2000 (1978) and C. Petitjean letter to J. Brewer, June 1979-3. N. Al-Qazzaz et al. , Nucl. Instrum. Meth. 1 ~Jk, 35 (1980).4. J. Doornbos, TRIUMF design note TRI-DN-80-4 (1980).5. J. Doornbos, TRIUMF design note TRI-DN-80-12 (1980) .6. K. Nagamine, in Proc. 2nd Int. Topical Meeting on Muon Spin Rotation,Vancouver (Hyperfine Interactions, in press).7. D.E. Lobb, University of Victoria report VPN-80-2 (1980) and references therein.8. C. Petitjean, private communication.9. P.F.A. Goudsmith et al. , The future IK0 muon channel, Interiko 77/1, Institut voor Kernphysisch Onderzoek, Amsterdam.101WORKSHOP PARTICIPANTSG. AZUELOS, TRIUMF, Vancouver, B.C., Canada M. BEAUDRY, TRIUMF, Vancouver, B.C., Canada E.W. BLACKMORE, TRIUMF, Vancouver, B.C., CanadaJ. BREWER, University of British Columbia, Vancouver, B.C., CanadaD.A. BRYMAN, TRIUMF, Vancouver, B.C., CanadaP. DEPOMMIER, University of Montreal, Montreal, Que., CanadaJ. DOORNBOS, TRIUMF, Vancouver, B.C., CanadaM. DOYAMA, University of Tokyo, Tokyo, JapanM. ECKHAUSE, College of William & Mary, Williamsburg, VA, USAG.T. EWAN, Queen's University, Kingston, Ont., CanadaH. FEARING, TRIUMF, Vancouver, B.C., Canada D.M. GARNER, TRIUMF, Vancouver, B.C., CanadaK. G0T0W, Virginia Polytechnic Institute, Blacksburg, VA, USAR.H. HEFFNER, Los Alamos Scientific Laboratory, Los Alamos, NM, USAW.-Y.P. HWANG, University of Washington, Seattle, WA, USAY. ITO, University of British Columbia, Vancouver, B.C., CanadaK.P. JACKSON, TRIUMF, Vancouver, B.C., CanadaH. KAJI, Tohoku University, Sendai, JapanS.N. KAPLAN, University of California, Berkeley, CA, USAH. KASPAR, Yale University, New Haven, CT, USAM. KRELL, University de Sherbrooke, Sherbrooke, Qu£., CanadaA. LAGANA, University of Perugia, Perugia, ItalyD.E. LOBB, University of Victoria, Victoria, B.C., Canada J.A. MACDONALD, TRIUMF, Vancouver, B.C., CanadaG. MARSHALL, University of British Columbia, Vancouver, B.C., CanadaH. MATIS, Los Alamos Scientific Laboratory, Los Alamos, NM, USAD. MEASDAY, University of British Columbia, Vancouver, B.C., Canada Y. MIYAKE, National Universities' Laboratory, Ibaraki, JapanK. NAGAMINE, University of Tokyo, Tokyo, Japan R. NAKAI, University of Tokyo, Tokyo, JapanB.W. NG, University of British Columbia, Vancouver, B.C., Canada J.N. NG, TRIUMF, Vancouver, B.C., CanadaT. NUMAO, University of Victoria, Victoria, B.C., CanadaB. OLANIYI, University of Victoria, Victoria, B.C., CanadaA. OLIN, University of Victoria, Victoria, B.C., CanadaC. ORAM, University of British Columbia, Vancouver, B.C., Canada J.-M. POUTISSOU, TRIUMF, Vancouver, B.C., CanadaR. POUT ISSOU, TRIUMF, Vancouver, B.C., CanadaB.C. ROBERTSON, Queen's University, Kingston, Ont., Canada L.P. ROBERTSON, University of Victoria, Victoria, B.C., CanadaE. RODUNER, Physikalisch Chemisches Institut, Zurich, Switzerland J.T. SAMPLE, TRIUMF, Vancouver, B.C., CanadaG. SANDERS, Los Alamos Scientific Laboratory, Los Alamos, NM, USA A. SCHENK, SIN, Villigen, SwitzerlandH. SCHILLING, University of British Columbia, Vancouver, B.C., Canada R.E. SHROCK, State University of New York, Stony Brook, NY, USAP. SOUDER, Yale University, New Haven, CT, USAW. SPERRY, Central Washington University, Ellensburg, WA, USAJ. SPULLER, TRIUMF, Vancouver, B.C., CanadaM. STROVINK, Lawrence Berkeley Laboratory, Berkeley, CA, USAJ. VINCENT, TRIUMF, Vancouver, B.C., Canada102A.H. WAPSTRA, IKO, Amsterdam, The Netherlands R. WOLOSHYN, TRIUMF, Vancouver, B.C., Canada R. YAMAMOTO, University of Tokyo, Tokyo, Japan A. ZEHNDER, SIN, Villigen, SwitzerlandK. ZIOCK, University of Virginia, Charlottesville, VA, USA


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