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UBC Theses and Dissertations

Part I : a simple model of the hypertriton. Part II : muon capture by 3He Congleton, J.G. 1992

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PART I: A SIMPLE MODEL OF THE HYPERTRITON.PART II: MUON CAPTURE BY 3He.ByJ.G.CongletonB. A. (Physics) Oxford University, 1987.M. A. (Physics) Oxford University, 1991.A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF PHYSICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAMarch 1992© J.G.Congleton, 1992Signature(s) removed to protect privacyIn presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)________________Department of PR Y5/CSThe University of British ColumbiaVancouver, CanadaDate HIhCCH ?DE-6 (2/88)Signature(s) removed to protect privacyAbstractThe thesis is in two parts. Part I is covered in chapters 1 and 2 and concerns a simplemodel of the hypertriton developed by the author. The model is based on the fact thatthe lambda particle is loosely bound and so a lambda-inert core approach should bereasonable. The core is taken to be exactly like the free deuteron and a separable ANpotential is used to construct the l)inding potential for the A particle. The model istested in chapter 2 by calculating the ratio of two body to all pionic decay rates of thehypertriton and the result is found to agree well with experiment.Chapters 3 to 7 concern muon capture by 3He. Using the elementary particle model itis shown that the spin observables for quasi-elastic muon capture by 3He are much moresensitive to the nuclear pseudoscalar form factor ( and hence the nucleon pseudoscalarform factor ) than is the rate. Reliable and sophisticated wavefunctions for 3He andare then used to find the muon capture Hamiltonian in the impulse approximation.The result differs from that found in the elementary particle model in that the magnetic( and dominant ) part of the Hamiltonian lacks strength. In chapter 5 new theory isdeveloped for the muon wavefunction overlap reduction factor leading to the result C =0.979. Chapter 6 details a calculation of muon capture by 3He leading to the deuteronneutron break-up final state in the plane wave impulse approximation. Finally, theprocesses leading to muonic atom formation are considered in chapter 7 with particularreference to final hyperfine population densities and their dependencies on target andbeam polarization. It is shown that if only the intra-atomic processes are included, theresults for the hyperfine population densities are unreliable.11Table of ContentsAbstractList of Tables viiList of Figures ixAcknowledgements xi1 A Simple Model of the Hypertriton1.1 Introduction1.2 Model1.3 Formalism1.4 Lambda Nucleon Potential1.5 Lambda Deuteron Potential1.5.1 Spin Average1.5.2 Momentum Average1.5.3 Evaluation of the Ad Potential1.5.4 A Separable Fit to the Ad Potential . .1.6 Solution of the Schrödinger EquationApplication of the Hypertriton ModelIntroductionLambda Particle Decay Amplitude . .Two Body Decay Rate112681111141618232 An2.12.22.3272728311112.4 Total Decay Rate 352.5 Results 383 Muon Capture by 3He: Rate and Spin Observables in the ElementaryParticle Model and Their Sensitivity to the Pseudoscalar Form Factor. 413.1 Introduction 413.2 Definition of Elementary Particle Model and Kinematics 423.3 The Hadronic Current 433.4 Rate and Spin Observables 483.5 Results 524 Muon Capture by 3He in the Impulse Approximation 574.1 Introduction 574.2 The Trinucleon Bound States 584.3 The Kamimura Wavefunctions 604.4 The Effective Harniltonian 644.5 Matrix Elements 694.6 Summary and Conclusions 725 Muon Wavefunction Overlap Reduction Factor 735.1 Introduction 735.2 Method 785.3 Corrections 805.3.1 Perturbation of the Muon Wavefunction 805.3.2 Relativistic Effects 835.:3.3 Numerical Solution of the Dirac Equation for an Extended ChargeDistribution 87ivo.3.4 Neutrino Wavefunction 895.4 Summary 906 Muon Induced Break-up: the Deuteron Channel 936.1 Introduction 936.2 Kinematics 946.3 PWIA 976.4 Two Body Break Up Momentum Distribution 1016.5 Muon Wavefunction Overlap Reduction Factor for Two Body Break Up . 1056.6 Trinucleon Structure Functions 1096.6.1 General Considerations 1096.6.2 On-shell Impulse Approximation 1116.6.3 Off-shell Impulse Approximation 1126.7 Matrix Element Squared 1196.8 Results 1197 Muon Depolarization and Hyperfine Populations 1257.1 Overview 1257.2 Introduction 1257.3 Cascade Calculation 1287.3.1 In the Absence of Hyperfine Coupling 1287.3.2 With Hyperfine Coupling 1387.4 Attempt to Include Effects of Exterilal Collisions 1427.5 Summary and Conclusions 144Bibliography 146vA Natural Units 154B Model Calculation of VAd(q’,q) 156C Derivation of Two Body H Pionic Decay Amplitude 160D Derivation of Inclusive H Pionic Decay Amplitude 165E Three Body Basis and the Permutation OperatorsE.1 Three Body BasisE.2 The Permutation OperatorsF Expansion of a Function in the Gaussian Basis 176G Matrix ElementsG.l The Norm0.2 The Non Spin-Flip Matrix Element [i]°0.3 The Spin-Flip Matrix Element [6}010.4 The Spin-Flip Matrix Element []21G.5 The Form of (p,q)0.5.1 Momentum Space Representation0.5.2 An Expression for (p,q) .H Deuteron Spectrum in the PWIA 200I The Nucleonic Weak Current and1.1 General liltroduction1.2 The Vector Form Factors1.3 The Axial Form FactorsNucleon Form Factors 205205207211169169172177178187192195198198199viList of Tables1.1 AN Separable Potential Parameters 111.2 Strength of the Ad Potential 201.:3 Ad Potential Range 212.1 Measurements of the Decay Branching Ratio R 282.2 Theoretical Calculations of R 282.3 Parameters for the Expansion in Gaussian Basis 342.4 Summary of H —* 3He Form Factor Results 372.5 Bubble Chamber Results for T(H) 403.1 Sensitivity of Observables 0 to F 554.1 Properties of the Wavefunctions. 624.2 Channel Specifications for the 8 Channel Wavefunction. . 634.3 Channel Specifications for the 22 Channel Wavefunction. 634.4 The Reduced Matrix Elements 714.5 The Effective Form Factors 715.1 Previous Calculations of the Reduction Factor C 755.2 Summary of Corrections to Coarse Result 917.1 Transition Probabilities for the 3He-t Atom 1317.2 Muon Depolarization with 1 = 1 Populated Only 1367.3 Muon Polarization During the Cascade 137vii7.4 Residual Muon Polarization Pb, for Spin Zero Nuclei 397.5 Residual Muou Polarization per Unit Target Polarization for Various Values of ni1, 1427.6 Values of the A,B,C,D Parameters with External Depolarization 144C.1 Hypertriton Channel Specifications 162E.l Group Product Table for the Group G 172viiiList of Figures1.1 Contributions to the AN interaction in the Hypertriton due to ANcoupling 41.2 The ratio VAd(q’,q)/VAd(q’,O) for various values of q’. The solid line is thefull calculation and the dashed line is the fit for the soft-core potential andBonn C deuteron 221.3 Lambda part of the hypertriton wavefunction in momentum representation 26:3.1 Feynman diagram for muon captllre 423.2 Sensitivity of observables to F 543.3 Unpolarized rate versus F 546.1 The four momentum transfer squared q2 as a function of deuteron momentum 966.2 Figurative diagram of the PWIA 966.3 Diagrammatic expansion of the matrix element 1006.4 Two body break-up momentum distribution for 3He. Data are taken fromtwo experiments performed at Saclay [123,124] in 1982 and 1987. Thestraight line is a fit to the data 1036.5 d-3He overlap functions E>’. The solid line is the 22-channel and theclash-dotted the 8-channel 3He wavefunction 1076.6 The muon wavefunction reduction factor for two body break-up as a function of the deuteron momentum 1086.7 The off-shell parameter 6 114ix6.8 The matrix element squared 1206.9 Deuteron spectra for various ranges of d. The solid line is the off-shellimpulse approximation. The dashed line is the on-shell impulse approximation. The data are taken from Cummings et al [121] 1216.10 The neutron/deuterou momentum in the nd CM as a function of the LABdeuteron momentum 1226.11 The neutrino energy Q as a function of LAB deuteron momentum 1247.1 Term diagram 1287.2 Effective cascade route 1367.3 Rose and Mann 1 distribution 138B.1 The sinhc function 1581.1 Comparison of realistic continuation with simple linear extrapolation ofform factors 2101.2 The nucleon form factors 214xAcknowledgementsFirst, I thank Dr.Harold Fearing for his supervision of this work and also his encouragement and advice.In physics we move forward by building on what we know already. The foundationsare thus of great importance and I thank Mr.C.Moss aild especially Mr.B.J.Garbutt fortheir sound introduction to the subject which I received at Rutherford ComprehensiveSchool. At Wadham College I was tutored by Dr.D.T.Edmonds, Dr.G.A.Brooker andDr.G.C.Ross who taught me to think, challenged me relentlessly and inspired me often.To them i owe much and I thank Dr.Ross for encouraging me to pursue graduate studies.On the subject of tuition I also thank Dr.G.Jones, Dr.J.Iqbal and Dr.B.Jennings for manygood lectures on nuclear physics at the graduate level.I thank the Canadian Rhodes Foundation for their financial support during my firsttwo years.On the personal side there are many I should mention but first I thank my wife and‘schat’, .Joke for her love and support during my Ph.D. I also thank my father and motherwho, without ever pushing, have encouraged me to pursue high goals throughout my life.I thank my father for proofreading much of the thesis.During my stay in Canada, I have been fortunate in choice of ‘landlady’. I thank Ronand Collette MacFarlane, Max and Sally Haugen and David and Jane Armstrong for thehelp and friendship. It is not easy to live in a foreign country and much adjustment hasto be made. I thank my ‘native’ friends Andrew, Reena and Don for their friendship andhelp. I have learnt much besides physics during my stay and I thank the following fortheir sporting contributions. Steve (skiing), Jeff (curling), Russ and Richard (ice-hockey)xiand Kelvin (belaying). Also thanks to Dave (how to throw a party), Simon (how to winusing the Neapolitan Club), Craig (see you in Australia) and IJBC cricket club for lotsof fun. A quick mention for Jack and Sylvia who organize the excellent TRIUMF socialevents and I should be able to get away with finishing here, having left out only a fewjeop1e who I should have mentioned.xiiChapter 1A Simple Model of the Hypertriton1.1 IntroductionThe hypertriton piays a very important role in hypernuclear physics due to the fact thatit is the lightest hypernucleus. The A has isospin zero and so the AN interaction has noone-pion-exchange tail. The AN interaction thus does not support a deuteron like boundstate [1].There have been many studies of the hypertriton using model potentials [2]-[9], the resonating group method [10], variational methods [11,12] hyperspherical harmonics [13,14]and also using potentials based on boson-exchange [15,16]. It is now possible to obtainhighly accurate solutions of the Schrödinger equation for a three particle system usingeither the Faddeev formalism [171 or sophisticated variational techniques [18] using realistic potentials. One could therefore hope to make exact calculations of the hypertritonwavefunction and binding energy but, because of lack of knowledge of the fundamentalYN interaction, ( Y=hyperon, N=nucleon ), these solutions would be of limited quantitative significance. The receit YN interaction analysis of Maessen, Rijken and De Swart[19] was based on only 35 cross section data points with hyperoll laboratory momentaranging from 110 to 300 MeV.The solutions listed above [2]-[16] show what effects are likely to be important in anaccurate wavefunction for the hypertriton, where such a wavefunction would be basedon a YN interaction known to a much higher degree of precision than it is today. It1Chapter 1. A Simple Model of the Hypertriton 2was thought that a simple model would complement the existing descriptions of thehypertriton for two reasons.• It is easily applied to calculations of processes involving the hypertriton such as itsproduction via strangeness exchange or associated production and also the mesonicand non-mesonic decays of the hypertriton. Hence,• it is a reference point against which more sophisticated descriptions can be compared.1.2 ModelThe hypertriton was modelled as a deuteron and a lambda particle in an effective Adpotential. The deuteron was taken ( as a first approximation ) to be the free deuteroni.e. it was assumed that the NN configuration in the hypertriton is exactly that of thecleuteron with 3S1 and 3D1 partial waves unperturbed by the presence of the lambdaparticle. This model was inspired by comments made by Gibson [20] in which the hypertriton is described as a system in which “the A clings tenuously to the deuteron inalmost a molecular type state”. The justification for such a model is that the A is veryloosely bound.BA EB(H)— EB(d) 0.13 + 0.05 MeV [21] (1.1)This lambda separation energy is only 2% of the separation energy of a neutron in thetriton ( 8.48 — 2.22 6.26 MeV) although the more pertinent comparison is to the totalbinding energy of the system. The ratio of the lambda separation energy to the totalbinding energy is 5%.An important feature of the model is that it inherently includes the NN 3S1 tensorinteraction. Gibson and Lehman [5] have shown that “proper attention to the tensorChapter 1. A Simple Model of the Hypertriton 3nature of the up triplet interaction is necessary ... “. This is true because the np interaction is responsible for the bulk of the binding in the hypertriton and the binding in adeuteron like NN state is sensitive to the tensor coupling between the 3S1 and 3D1 partialwaves.The experimental data on the hypertriton is limited. The lambda separation energyis known, equation (1.1), and also the total angular momentum is known to be [22][23] [24]. The measurement of the ratio R of two-body decays to all decays (in which a7T is produced ) of the hypertriton is,= F(H‘. + 3He)= 0.35 + 0.04 reference [25] (1.2)F(H —> ir + all)and the implication that follows because if the spin were then only the p-wavepart of the amplitude could contribute and this would give a value of R far too lowirrespective of the exact form of the wavefunction. Evaluation of R will provide a test ofthe wavefunction and is presented in chapter 2.The only other inputs for this calculation are the scattering lengths and effectiveranges for the AN and 3S1 partial waves and four different deuteron wavefunctionsfrom the Bonn potentials A,B and C [26] and the Paris Potential [27]. The AN values aretaken from the Nijmegen analyses [19,28] of YN scattering data which use a one bosonexchange potential approach and SU(3) flavour symmetry for the coupling constants.The scattering lengths and effective ranges should provide reliable constraints on the lowenergy behaviour of the AN — AN T-matrix.The model being proposed includes no coupling of AN to the N channel. Althoughthis is rather unrealistic as far as describing the full AN interaction goes, it may be areasonable approach for use in a system where the NN components of the wavefunctionare small. This statement is motivated by considering a variational approach to theChapter 1. A Simple Model of the Hypertriton 4Figure 1.1: Contributions to the AN interaction in the Hypertriton due to AN —*coupling.problem. In that case, matrix elements of the Hamiltonian would be minimized in orderto find the ground state wavefunction and these matrix elements would involve only ANAN parts of the AN interaction. That is not to say that intermediate N stateswould not make a contribution, but rather that the intial and final state would haveto be AN. Thus, provided the single channel potential reasonably approximates the fullAN — AN T-matrix and there are negligible N components in the final wavefunction,then this single channel approximation will be adequate for the purpose of finding thewavefunction. Dabrowski and Fedoryñska [9] found a probability of 0.36% for the NNcomponent in the hypertriton using the phenomenological YN coupled channel potentialof Wycech [29].Of course, this ignores the dispersive effect of the spectator particle, ( figure 1.1 )which tends to weaken the coupled contributions to the AN — AN T-matrix and thatwill have to be accepted as an approximation of the model. The three body force arisingfrom the coupling to ZN , shown on the right hand side of figure 1.icouple, will also beneglected as will the tensor coupling in the AN 3S1 channel. The latter has been shownnot to make an appreciable contribution [15].The wavefunction derived herein was forced to have the correct binding energy byadjusting the parameters of the Ad potential. The amount of tuning which needs to beperformed will give an indication of the credibility of the model.Chapter 1. A Simple Model of the Hypertriton 5An outline of the method followed is given below.• A separable fit was made to the AN s-wave potential.• The potential was spin averaged for the AN configurations found in the hypertriton.• The AN 1)otential was summed over the two nucleons and averaged over the deuteronwavefunction.• The resulting Ad potential was fit to a separable form and only the s-wave partwas used. The other partial waves were taken to be zero.• The Schrödinger equation for the lambda l)art of the wavefunction was solved andthe binding energy was forced to the experimental value.Without tuning, it would not be expected that the model would provide a good valuefor the binding energy of the hypertriton. As is known for the trinucleon system it is verydifficult to find agreement with experiment in the value of EB and this is even more truefor the hypertriton where the balance of kinetic and potential energies is so precariousthat it has even been speculated that the nucleus is only bound because of the presence ofa ANN three body force [3]. However, for a good description of the hypertriton, it is highlydesirable for the wavefunction to possess the correct binding energy since this determinesits asymptotic behaviour, ( the asymptotic limit is large distance or correspondinglysmall momentum ). This has been demonstrated in the trinucleon system where scalingis found for all the low energy observables i.e. no matter what potential is used theproperties of the wavefunction such as r.m.s. radius, D-state probability, etc. all showsimple power law dependence on the value of the binding energy which the input potentialyields [30]. The argument for scaling is that the low-energy observables depend stronglyon the asymptotic part of the wavefunction and in this region where the potential is veryChapter 1. A Simple Model of the Hypertriton 6weak the wavefunction depends only on the binding energy. One would thus expect thesame scaling behaviour for the hypertriton. The value of the ratio R ( equation (1.2) )has been shown to depend strongly on BA [ii] in a calculation which employed variouspotentials, all consistent with low energy Ap scattering and binding energies of is andip shell hypernuclei.1.3 FormalismFollowing the general definition of Jacobi coordinates for a three body system, a lineartransformation (k1,k2,k3) — (K, ) was made in order to separate out the centre ofmass motion. In the following equations (abc) take the values (123), (231) or (312), ka isthe momentum of particle a and mc, is the mass of particle a.A— ka+kb+kc(m— mbk)/(mb + m)= [(me + mc)a — ma(b + )] /(ma + mb + m) (1.3)If the lambda particle is labelled particle #1 then writing as (),= kA+k2+k3== [2MN — MA(k2 +k3)]/(2MN + MA)In the centre of mass frame (CM) this simplifies to, (i.4)I = 0=(2—i3)IT = ‘CAThe internal momentum coordinate of the deuteron is the ‘pair’ momentum and thelambda momentum is the ‘spectator’ momentum IChapter 1. A Simple Model of the Hypertriton 7The Hamiltonian governing the system is H which in the CM is,-*2 -*2= —+ up++ Ad2m 2u= Hdeuteroii + Hianibda (1.5)The Hamiltoniaii can be split into two parts because a static approximation has beenmade for the lambda-deuteron potential VAd: i.e. the influence of the lambda particleon the nucleons is neglected. The Ad potential will thus only depend on the variable( after spin dependence has been taken care of ). The reduced mass of the two nucleonsis rn = 2.379fm1 and the reduced mass of the lambda-2N system is ji = 3.547fm1The wavefunction can now be written using the fact that the hypertriton has spinone half and using the model for the nucleon part.(7H;m) =bA(q) x x [YJ0()®x](1,S)=(O,)(2,) 2 3x [A(TJ. — j,fl//2] (1.6)In the above equation, /)() is the radial part of the deuteron wavefunction. Thespin part shows the coupling of the spin one deuteron to a spin one half lambda particleto give total spin or . The s-wave part of the deuteron requires S = and the d-wavepart requires S =X’Is = (SMs lmd; flA)Xi*tX1A (1.7)This spin vector is coupled to a spherical harmonic for the angular parts of thenucleons and lambda particle which has angular momentum 1 for the pair and 0 for thespectator A.YI(i3, ) = (irn l1m; 12rn2) }1rn1(l3)uirn2() (1.8)Chapter 1. A Simple Model of the Hypertriton 8The isospin part shows the trivial coupling of the isospin 0 A to the isospin 0 deuteron.A model calculation performed by Garcilazo [31] has shown that the isospin 1 Ann systemis not bound and thus in the limit that the An interaction is equal to the Ap interaction,the hypertriton must have isospin 0.Except for the spin part, the wavefunction is a simple product of a lambda wavefunction and a deuteron wavefunction.1.4 Lambda Nucleon PotentialThe ansatz for the lambda nucleon potential is a separable potential with dimensionlessform factors g(k). Only the s-wave were taken to be non-zero and the potential was takento have two free parameters, viz.,• The strength of the potential ).• The momentum space range of the potential AAN . This is often referred to as theinverse range since it corresponds to the inverse range of the potential in configuration space.These parameters were fixed by the scattering length and effective range for s-wave ANscattering and results were taken from the findings of the Nijmegen group [19,28]. Themore recent analysis [19] applies SU(3) flavour symmetry to extend the Nijmegen NNpotential to YN. The potential is based on exchange of ir, m, mj’, p, , , 6, € 8* particles andincludes .J 0 contributions from the tensor f, f’, A2 particles and Pomeron trajectory.It is characterized by Gaussian form factors for the couplings which give the potential softbehaviour near the origin. This is in contrast to the earlier analysis [28] which employedpotentials of the hard-core type. The results for the scattering length and effective rangewill be referred to as soft-core and hard-core for [19] and [28] respectively.Chapter 1. A Simple Model of the Hypertriton 9Taking the AN potential as separable means that the matrix element of the potentialVAN for a transition from initial state k to final state k’ is written as the product of afunction of k and a function of k’. Here, k aiid k’ are the iiiitial and final A momenta inthe CM.N(k’VAN) = -g(’)g() (1.9)MNkA—MAkN110— MN+MAChanging to the partial wave basis defined below arid writing the matrix element(k’00VANk00) as VAN(k’, k) we have for the s-wave part of the AN potential,VAN(k’,k) = _ANg(kI)g(k) (1.11)partial wave basis, (Iklm) = 2 7fl() (1.12)The T-matrix was found by solviirg the Lipmanri-Schwiriger equation ( equation(1.13) ) which is easily done for a separable potential. The Green’s function here isG0(E) [E — H0j where H0 is the free Harniltoiiiair.T(E) V + VG0(E)T(E) (1.13)Writing (k’OO T kOO) as t(Ic’, k; E) we have,t(k’,k;E)= (1.14)where,f dqq2[g(q)] = reduced mass of ANJE Jo [E —q2/2AN 2.5837fm’(1.15)Chapter 1. A Simple Mode] of the Hypertriton 10By taking the principal value of JE, equation (1.14) gives the r-matrix and the following identification with the s-wave phase shift can be made,ask—O 2kcot0 = —÷—-- + —rk (1.16)lr/IANr(k) a 2(the ON superscript indicates that E = k2/2j.i = k’2/2t so that the particles are “onshell” ).The scattering length a and effective range r reflect the low energy properties of theinteraction. A Gaussian form factor was used but results for the Yamaguchi type formfactor [32] are also shown for the sake of comparison to the nucleon-nucleon potential.g(k) exp[—(k/AAN)2] Gaussian1 . (1.17)g(k)= [1 + (k/AAN)2] YamaguchiThese results are shown in table 1.1 for hard-core (HG) and soft-core (SC) AN potentials.The AN potential has roughly the same range as the NN potential and about two fifthsof the strength. The spin singlet interaction is stronger than the spin triplet interactionwhich is the assigiment one would make from the observatioll that the Apn ground stateis .J= and notThe fact that the range of the hard-core AN potential comes out to be almost independent of the spin is a fortunate coincidence which makes the formalism much moretransparent than in the case of the soft-core potential. The method of developing a Adpotential from the AN potential will be given in detail with specific reference to thehard-core potential. The method for the soft-core potential is very similar but is ratherless illuminating and only the numerical results will be given for the soft-core case.Chapter 1. A Simple Model of the Hypertriton 11Table 1.1: AN Separable Potential Parameters.N /fi2“AN /fm’ a / fm r / fmAN -- Yamaguchi (HC) 0.1106 1.354 -2.29 3.17AN-- Yamaguchi (HC) 0.1017 1.359 -1.88 3.36NN - Yamaguchi- pp 0.2141 1.133 -17.1 2.83NN - 1S0 - Yamaguchi - np 0.2178 1.144 -23.75 2.75NN I5 - Yamaguchi - nn 0.2164 1.128 -18.45 2.83NN - 35 - Yamaguchi 0.2550 1.418 5.43 1.75AN- ‘S - Gaussian (HC) 0.1238 1.400 -2.29 3.17AN - - Gaussian (HC) 0.1129 1.402 -1.88 3.36AN - 15 - Gaussian (SC) 0.1312 1.452 -2.78 2.88AN- S1 - Gaussian (SC) 0.0915 1.597 -1.41 3.111.5 Lambda Deuteron Potential1.5.1 Spin AverageFor free AN scattering, the total spiii is either 0 or 1. With an unpolarized beam andtarget, the population densities for each value of the total spin and total spin projectionwill be equal and hence scattering occurs in the singlet state with probability and in thetriplet state with probability . For the A in the hypertriton, the relative probabilitiesare not the same as the free case as pointed out by Herudon and Tang [33] and in factdepend on the total spin S. If S-, ( which is the dominant part of the wavefunction ),then the AN interaction is spin singlet and - spin triplet. The small part with S =is all spin triplet.Consider first the S = state. In the following, the label of the particle is indicatedby the position of the spin vector. The A will be taken as particle #1.x = [2 -- ] (1.18)Now let us ask what the total spin of particles 1 and 2 is. This is found by writing statesChapter 1. A Simple Model of the Hypertriton 12of good total spin s and spin projection s, s, s ) for particles 1 and 2.=l,0)- 00)T] (1.19)By taking the overlap of this vector with itself it ca be seen that we have spintriplet and spin singlet. More formally, if the spin projection operators are definedthus, H=(—o1.2+ 1)/4 and ilt = (ui.u2 + 3)/4, thenX?lHX12? = 3/4xflx = 1/4 (1.20)The total 5pm of particles 1 and 3 follows the same pattern because the spin vectorsare symmetrical with respect to interchange of nucleons 2 and 3.Consider now the S = state. Let us take the projection to be for the sake ofargument. The conclusion will be independent of this choice. In the following equations,the state is decomposed into states of definite spin and projection for particles 1 and 2== [ + ( + T)TJ= [Ii, 1 ) + V2i,0)Tj (1.21)and this is clearly all spin triplet.These results can also be derived by using the theory of recoupling. When threeangular momenta are coupled together there are three choices of construction. Two mustfirst be combined to a subsystem total angular momentum and then this is combinedwith the remaining ‘spin’. There are three choices of the pair and the three resultingChapter 1. A Simple Model of the Hypertriton 13vectors are related by a linear transformation whose coefficient is proportional to theWigner 6-j symbol [34, p41].SM5 /1 1\i. 1. C1.K—ji 31V15=2 2 1 (1.22)S 1 S sJSo we have the result that the probability P that the AN spin is s, is/Ill iiP5 = 3(2.s + 1) 2 2 (1.23)1 S sJEquation (1.23) yields the results given in equations (1.20) directly and show that theyare independent of M.Since the ranges of the triplet and singlet AN potentials are so similar the effectivespin averaged potential was taken to have range 1.401 fm and the effective strengthwill depend on the spin state S. For S =eff = (3)AN+ = 0.1211 fm2 (1.24)For ‘ , eff is the triplet strength = 0.1129fm2 The raige was taken to be1 .401 fnf’ in both channels for the sake of simplification.To summarize,(k’VANk) = —Ag(k’)g(k) (125)where, g(k) exp[—(k/AAN)2]where the “eff” superscript indicates that spin averaging has been taken into account.The value of )i depends on the total spin of the AN pair and hence on the quantumnumbers of the NN partial wave. The subscript on reflects the orbital angular momentum of the NN partial wave and takes on the values 0 or 2. The value of AAN wastaken to be 1.401 fnf’.Chapter 1. A Simple Model of the Hypertriton 14For the case of the soft-core AN potential the following forms were used,(p!lVANeffp) A0go(p)go(p’) + g(p)g1(p’) for S =(IyVANep) = N1g(p)g(pf) for S (1.26)where, go(p) = exp[—(p/A°)2]where, gi(p) exp[—(p/A’)j (1.27)It is clear that the two potentials should give similar results because their strengths atlow momenta are the same. Consider the strength of the soft-core potential at q’ = q = 0when S = it is 3/4 x 0.1312 + 1/4 x 0.0915 0.1213fm which should be compared to0.1211 fm2. This is the most important effective strength because the S = stateforms the majority of the hypertriton which follows from the 95% s-state probability forthe nucleons in the deuteron.1.5.2 Momentum AverageThe following ansatz was made for the Ad potential. The subscripts 2 and 3 refer to thelabel on the nucleons so that V2 is the spin averaged potential between the A and N2 andV3 is the spin averaged potential between the A and N3.(‘ IVa) = fd3p’d (‘)(‘‘V2+V3d(p (1.28)This potential is a classical sum over the potentials from each nucleon and is a quantum mechanical average over the initial and final internal deuteron momenta weightedby the deuteron amplitude.The two body AN potential must first be embedded in the three body Hilbert space.This was achieved by multiplying the two body matrix element by a momentum conserving delta function. For example, V2 should not affect the momentum of N3 so the deltaChapter 1. A Simple Model of the Hypertriton 15function is 63(k — k3). The inverse transformation (K, ) “ (ks, k2, k3) is given by,(MA/s) 0 1 I’k2 = (MN/s) 1 —! j where = MA + 2MN (1.29)k3 (MN/s) —1 —and in the CM where I = 0 this simplifies to,qk2 = (1.30)k3=The two body potentials are thus,1 1 /(j’’V2) = 6(+-- (jr’ +)) ( - I ) g[k2]g ]2 2 \47rJ(‘‘V3) = 6(-_‘ - ()g[k3]g ] (1.31)In the above, k2 is the magiitude of the A momentum in the A-N2 CM. The vectork2 can be written in terms of and as shown below and this introduces kinematicalfactors and /3 which play an important role.— MNA—MA2—I2 MN+MA —k MA-MA3 = (1.32)where, MN+MA/2 = 0.7285= MN+MA 0.5430 (1.33)The contributions from N2 and N3 are the same except for a sign and so the Ad potentialcan be written as the sum of two terms, V+ and V.(‘VAd)= (-) [v’,) + V’,)] where, (1.34)Chapter 1. A Simple Model of the Hypertriton 16V’,q = (1.35)The pertinent value of ( either A or ), should be taken according to which partof the deuteron amplitude is contributing, ( either 1 = 0 or 1 = 2 ).The integral was performed by making the substitutions = and .‘= j7’ +and performing the trivial integration over ‘. Finally, only the s-wave part of theresulting expression was taken. It was found that with the s-wave part of the deuterononly, the maximum value of the next partial wave (1 = 2) was 1.4% of the strength of thes-wave, ( the 1 = 1 partial wave does not contribute to the JT = 1+ hypertriton whenthe only NN partial waves are 3S1 and 3D1 ).VAd(q’, q) (q’OO IVAdI qUO)= (—) [V(q’, q) + V(q’, q)] where, (1.36)V(q’, q)= f f ds )g[7’ + ]g[7+] (1.37)and y c + . Further, because of the even parity of the deuteron wavefunctionV+= V V so that,VAd(q’, q) = —2V(q’, q) where,V(q’,q) =‘d(+ )g[7’+]g[+](1.38)1.5.3 Evaluation of the Ad PotentialThis section describes the evaluation of V(q’, q) given by equation (1.38). The deuteronwavefunction for the sub-state with total angular momentum projection m, isbd(p)=1-02® (1.39)The spin part xi is coupled to the orbital part }‘() to give total angular momentum1 and projection m. The isospin part represents a neutron and proton coupled to zeroChapter 1. A Simple Model of the Hypertriton 17total isospin. The radial part of the deuteron was taken from the parameterization givenin [26, pp355,362]for the Bonn potential and was interpolated from a grid supplied byOelfke in the case of the Paris potential.The expression (1.38) should be spin averaged over projections rn according to thecoupling in the hypertriton. For a “spill-up” hypertriton we have 2/3 rn = 1 and 1/3m = 0. From the facts that V(q’, q) is real and only even values of l contribute, it canbe shown that the contribution due to rn = —i equals that due to m = 1 and hence thespin average peculiar to the spill 1/2 Ad states can be written as a generic spin averageyielding,2J+ ‘/2)/’d(+/2)?fl321}7+ /2) }i( /2) 7/0( ‘/2). (1.40)1,7lljThe expression for V(q’,q) given in equation (1.38) can thus be re-written as follows.V(q’,q) =21 1fd3sA7ffN(,qI)*m1(q) (1.41)lrO,2 +where, N1Th q’)= f /2 )1)( + /2 )g[+ ] (1.42)The angular dependence on was expressed using the following expedient trickemploying tile Legendre polynomials Pk (x).=11 (lXf(X)6(X-), (x - ) = 4 (1.43)—l k,mkTwo identities for spherical harmonics of two variables are needed to find that,- m1N1 (s,q) = Y (s)Ni(s,q) where,Ni(s,q) sh1(q/2)12__ x11+121 11.12.dx P12 (x) (s2+q2/4+sqx)1/ exp[-(2s+ 2q + 27sqx)/A]( 1.44)Chapter 1. A Simple IVIodel of the Hypertriton 18The identities are,ahlbI2 47r(2l + 1)’(a + b)+ [i + 1)!(2l2 i]Y1&, b) (1.45)and,= () [(2ll+2l ] r (l0j l0; 120> (1.46)—(21 + 1)1(212 + 1)! 1!1 47(s)4l + 1)! l1! 12! )Identity (1.45) follows from partial wave expanding = and regardingthe x— 0 limit. Identity (1.46) follows by notiig that y (., .) must transform like)7fl () and finding the coefficient by evaluating an integral of three spherical harmonics.Equation (1.47) holds only when l + l2 = l which is the case for all values entering intothe sum (1.44). The final form used for V(q’, q) is,V(q’q) = fs2A(sq!)Nj(sq) (1.48)1=0,2which follows from the trivial integration over d.. The integrations over s and x wereperformed using Gauss-Legencire quadrature with 11 equally spaced points covering x =—1 —# +1 and 100 points spanning the range of s which gave convergence at the 0.1%level.1.5.4 A Separable Fit to the Ad PotentialThe expression (1.38) gives the s-wave part of the Ad potential for all values of q and q’but is only reliable for low momenta, ( the relevant scale is the momentum space rangeof the AN potential !\AN ). A separable fit to this potential is desirable for two reasons.• The Schrödinger equation for a particle moving in a separable potential is easy tosolve and yields a wavefunction with a very simple form.Chapter 1 .A Simple Model of the Hypertriton 19• There is an appealing consistency in ending with a Ad potential expressed usingthe same formalism as the input AN potential. The level of sophistication used foreach potential is then the same.The form for the Ad potential was taken as,/ Ad / 2VAd(q , q) = — F(q )F(q) where, F(q) = exp[—(q/AAd) ] (1.49)As in the case of the AN potential, there are two parameters to find; the strengthA and the momentum space range AAd. The strength was found by setting q = 0 andq’ = 0 in equation (1.38). Thus,= 2 x I where,I fdss {I0) 2 (s)+ 2) 12 (s)eff}exp[9(s/)] (1.50)This is the normalization integral for the deuterori wavefunction in momentum spacemodulated by a form factor and constant multiplicative factor for each partial wave. Thestructure can be understood by studying two informative limits.1. Suppose that the AN interaction were of infinite range in momentum space. Thiscorresponds to a point-like interaction in configuration space. Then the modulatingform factor in the integral I would be one and the integral is easily performed.I = p0 + p2 (1.51)where P0 and P2 are the s-state and d-state probabilities in the deuteron. Thestrength of the Ad potential is thus twice the average )eff where the average is overthe AN spins found in a Ad s-wave system coupled to total angular momentum2. Suppose that the AN interaction were insensitive to spin. In that case, ) is thesame for both the s-wave and d-wave part of the deuteron wavefunction and canChapter 1. A Simple Model of the Hypertriton 20Table 1.2: Strength of the Ad Potential.Bonn A Bonn B Bonn C Paris 1r[fm21 (HC) 0.2222 0.2214 0.2205 0.2195 1,\Ad[frnl (SC) 0.2234 0.2226 0.2216 0.22i]P2 1% 4.38 4.99 5.61 5.77be factored out of the integral. The strength of the Ad potential is then 2 timesthe AN strength multiplied by I’, where1’ = fds 2{I (o) 2 (2) 12 ()} exp[-2(s/A)2J (1.52)This is simply the cleuteron normalization integral in momentum space modulatedby a form factor. The form factor is always less than or equal to one and therest of the integrand is positive semi-definite so the value of I’ is less than one.Thus, the strength of the Ad interaction is less than 2 times the strength of theAN interaction. How much less depends on the size of the deuteron compared tothe range of the AN interaction ( all sizes refer to momentum space of course ).If the size of the deuteron is much less than the range of the interaction then thestrength approaches 2 times the strength of the AN interaction. If however the sizeof the deuteron is much greater than the range of the AN interaction then the Adinteraction becomes very weak. This is a physically reasonable picture. The inputwas a AN potential which was weak at high momentum and this was averagedover the momenta found in the deuteron. If the two nucleons are found often tohave large relative momentum then the AN relative momentum will also be largeand hence the effective Ad potential will be weak. For Bonn potentials A,B and Cthe value of I’ was found to be 0.919, 0.916 and 0.912 respectively. For the Parispotential I’ was found to be 0.910.Chapter 1. A Simple Mode] of the Hypertriton 21Table 1.3: Ad Potential Range.q’ AAd/fnf1/ fm’ Bonn A Bonn B Bonn C Paris0.0 (HC) 0.94 0.94 0.94 0.940.0 (SC) 0.98 0.97 0.97 0.970.5 (HC) 1.01 1.01 1.01 1.010.5 (SC) 1.04 1.04 1.04 1.041.0 (HC) 1.15 1.15 1.15 1.151.0 (SC) 1.19 1.19 1.19 1.191.5 (HC) 1.33 1.33 1.33 1.331.5 (SC) 1.38 1.38 1.38 1.38Table 1.2 gives the values of ) found using equation (1.50) and various deuteronwavefunctions. The value of ) decreases slightly as we move from Bonn potentials A toC and this reflects the increasing d-state probability and the fact that .A < Alsogiven are results for the soft-core AN potential.The range Ad was found by fitting the ratio Vd(q’, q)/VAd(q”, 0) to the Gaussian formfor F(q), (equation (1.49)). The results are shown in table 1.3 for various values of q’. Ifthe potential really was separable then the value of AAd found would be independent ofq’ but this is not the case. Instead, the “slice” at q’ has a range which increases with q’and thus the potential (q’, q) flattens out at large q, q’. This behaviour can be shownto be reasonable by a model calculation which is given in appendix B. The fits are shownin figure 1.2.A good fit is required only where the potential is appreciable, which is in the rangeq 0 — 2 fm’. Further, if an intermediate value of q’ is chosen at which to performthe fit, then the resulting potential will be too large at low momentum and too smallat high momentum and the two effects will cancel to a certain c[egree. Let us take thesoft-core fit with Bonn C deuteron at q’ = I fm’ for the sake of a definite result i.e.AAd = 1.19 fn1, but remeniber that anything between 1.1 and 1.3 fin’ would also beChapter 1. A Simple Model of the Hypertriton--1.0 1.5 2.0 2.5 3.0a in fm—i22Figure 1.2: The ratio Vd(q’, q)/VAd(q’, 0) for various values of q’. The solid line is the fullcalculation and the dashed line is the fit for the soft-core potential and Bonn C deuteron.1.0-0.8-a,x 0.6->.Q4>0.20.00.\\q1 00 fm’fm1 \i0411.00.8CD& 0.6>- Q4.>0.200 —0.01.00.8) 0.5 1.0 1.5 2.0 2.5 3.q in fm—i1.00.5 1.5q in fm—i2.0 2.5 3.0.8a,b- 0.6>00.0:q in fm—iChapter 1. A Simple Model of the Hypertriton 23reasonable.It is to be expected that the calculation of is more reliable than the calculationof AAd since it depends on the more gross features of the AN potential and deuteronwavefunction, viz, the AN strength A, and also the relative size of the deuteron andAN potential. The range, however, is more sensitive to the detailed shape of VAN(q’, q)and the cleuteron wavefunction and the former may well be poorly represented.1.6 Solution of the Schr6dinger EquationThe Schrödinger equation for the lambda part of the wavefunction is,(+vAd) A) = —BAI’A) (1.53)The solution of this type of equation is standard and is given here for the sake of cornpleteness. Acting on the left with the bra (q I and letting a2 2BA one finds,(q2 +a2)(q) = 2F(q) j2(s)F(s) (1.54)The integral appearing in the above equation is a functional of /‘A and is thus a number.Writing this functional as K the wavefunction can be written,2dF(q)K F(q)‘‘A(q) = = (1.5)q2+a q2+awhere N is a normalization constant. The eigenvalue equation is found by demandingself-consistency using the first equality above to express K in terms of itself.2 [F(s12K = 2[tA / ds L K (1.56)Jo\Vith the Gaussian form for F(s) the integral is expressed in terms of the exponentiatedcomplementary error function cerfe(x) [35, p338,3.466(2)].= ( - cerfe() (1.57)Chapter 1. A Simple Model of the Hypertriton 242where, cerfe(x) = ex2[l— erf(x)J erf(x) = j e2 dt (1.58)Equation (1.57) links the A separation energy BA, the strength of the Ad potential,\Ad and the range of the Ad potential AAd. Given two of these quantities it can thus heused to determine the third. In order to know the wavefunction, equation (1.55), thevalues of a and AAd are needed. Since the calculation of ) is more reliable than that ofAAd it is desirable to use as an input parameter and to see if the value of AAd neededto satisfy equation (1.57) is reasonable by comparing it to the calculated value.The structure of equation (1.57) is revealed by rearranging in favour of the productAd AAAdA = () + (2 acerfe(a) (1.59)AAdThus, the values of )1d and AM which just bind the hypertriton, ( a —* 0 ), lie on thehyperbola “A1d () 2 = 0.225 fm. The amount of binding depends on how muchgreater than this constant the product of A and AM is.For the pertinent values, equation (1.59) is very weakly driven by a. This can beillustrated by taking ) to be 0.222 fm2 and using the measured value of the separationenergy to fiuid AAd. Equation (1.1) gives a = (6.8 + 1.3) x lO2fnf’. Equation (1.59)Was solved by iterating the right hand side until it converged with the result AM =1.17 + 0.03 fm1.The first term on the right hand side of equation (1.59) is 0.225 fm and at the solutionthe second term is 0.035 fm. The 20% error in a yields only a 2% error in AAd whichillustrates the J)oint that equation (1.59) is very weakly driven by a. Thus, in order tosuccessfully predict a a very fine tuning of ) and AM would be required. However,the aim here is not to calculate the binding energy of the hypertriton but rather to finda reasonable wavefunction. The experimental binding energy was used along with theChapter 1 A Simple Model of the Hypertriton 25calculated value of ) to yield the consistent AAd value which is l.17fm From theresults of the fit for F(q) it can be seen that this value is in reasonable agreement withthe calculated values.The theoretical uncertainty in AAd should be taken as roughly the same as the uncertainty in k’ because the solution of equation (1.57) lies near the hyperbola condition)AdAAd = constant. It is impossible to say what the accuracy of the calculation of Ais without performing a more sophisticated calculation with better knowledge of the ANpotential. However, when using this wavefunction it would be prudent to vary the valueof AAd by up to 10% to test for sensitivity to this parameter.To summarize, the lambda part of the hypertriton wavefunction /1(q) was found tobe,A(q) = N(QA)exp[q/Q] (1.60)1.17 fm1(+ - 10%) (1.61)(6.8 + 1.:3) x lO2fnf’ (1.62)N(Q)= {— [cerfe()(1 + -) - ()] } (1.63)[N(l.05)]2 = 0.106061fm’[N( 1. 17)j2 = 0.10391 1 fm[N(l.29)]2 = 0.102188fm’. (1.64)Figure 1.3 shows the wavefunction given in equation (1.60) where the value of QA istaken to be 1 .17 fm’. The peak of the momentum distribution is very near q = a =0.068 fni.0.0 0.2 0.4 0.6 0.8 1.0Lambda momentum q/fm126Chapter 1. A Simple Model of the Hypertriton7060-50-20-10 -0-0.Lambda part oftritonwcv ef unction— I I0.2 0.4 0.6 0.8Lambda momentum q/fm15- Lambda particlemomentum distribution1•Figure 1.3: Lambda part of the hypeitriton wavefunction in momentum representation.Chapter 2An Application of the Hypertriton Model2.1 IntroductionAs mentioned in the previous chapter, the experimental data on the hypertriton is limited.The lifetime r(H) has been measured in helium bubble chamber experiments [25,36,37,38] and also using nuclear emulsion techniques [39,40] with results varying from 0.95 xl0’° to :3.84 x 10b0 sec1 [25, see table 4]. However, this quantity is a poor test of amodel of the hypertriton since r(H) is relatively insensitive to the nuclear structure, thevalue being found to be near the free lambda particle lifetime T(A) both experimentallyand theoretically. Its insensitivity will be made manifest in this chapter.A better test is the branching ratio, R of two body r decays to all r decays.HF(H ,‘ + 3He)— 0.35 + .04 reference [25] (2.1)—* ir + all)The measurements of R from helium bubble chamber experiments are summarized intable 2.1.The branching ratio R. has been calculated using phenornenological H and 3He wave-functions by Dalitz [22] and Leon [23] and also by Kolesnikov and Kopylov [11] who usedH and 3He wavefunctions found from variational calculations employing five differentAN potentials all ‘consistent with fundameiital hypernuclear data”. The early work washampered by poor knowledge of the lambda decay potential but despite this, Dalitz wasable to conclude in 1958 that the spin of the hypertriton is one half. The results of thesecalculations are summarized in table 2.2. Note that the results of Dalitz and Leon should27Chapter 2. An Application of the Hypertriton Model 28Table 2J: Measurements of the Decay Branching Ratio R.R Reference0.39 + .07 Block et al. [38]0.:36 t Keyes et al. [36,37]0.30 + .07 Keyes et al. [25]0.35 + .04 [ mean valueTable 2.2: Theoretical Calculations of R..R Reference0.17 — 0.26 Dalitz [22]0.10 — 0.24 Leon [23]0.26 Vi Kolesnikov0.55 V2 & Kopylov [11]0.52 V30.50 V40.32 V5be multiplied by two due to an error in the application of the Pauli principle as explainedin Dalitz and Liu [24]. The range of values is clue to uncertainty in the lambda decaypotential. Given the better limits available today Dalitz and Leon would have foundR = 0.52 and R 0.48 respectively which corresponds to the top end of the range givenin table 2.2.2.2 Lambda Particle Decay AmplitudeThe A particle decay amplitude has the following general form a in the non-relativisticlimit [22].H(A.‘ p+) = +.Q/qo (2.2)‘Note that this definition of p differs by a minus sign from that of Dalitz. The definition here agreeswith the convention used by the Particle Data Group [41]Chapter 2. An Application of the Hypertriton Model 29where, . = s-wave strengthp-wave strengthQ = momentum of proton in A rest framemomentum of proton when the A is freeq0 = = 100.5 MeV.This form follows from the conservation of total angular momentum, the pion beingspinless and both A and proton having spin one half. Both s and p wave pious are allowedbecause j)arity is not conserved in the weak interaction. The ratio of these two strengthscan be found from the angular correlation a between the polarization of the A particleand the proton momentum. For a free A particle the decay amplitude is,(2.3)and so the decay rate for an ensemble of polarized A particles with density matrix p is,r(A p + ) Tr[MpMt] where, p = (1 + PA.) (2.4)The ratio of the polarized to unpolarized rate is thus,p += 1 + (2.5)p +—)2Re(.*j3)where, a = (2.6)s2 +Applying time reversal invariance to the decay amplitude we have, ( using . —* .,3Q,‘M — + j3.Q (2.7)and so .s and p must be real. However, the presence of the stroig interaction in thefinal state introduces extra phase shifts 6 and 13 for the s-wave and p-wave strengthsChapter 2. An Application of the Hypertriton Model 30respectively. The quantity A= 13 is measured to be 8°+4° [41] which is consistentwith results from low energy p scattering. In what follows the magnitudes of andwill be written as s and p respectively and their relative phase will be expressed in termsof A. The angular correlation a can be written in terms of the ratio r = p/s and theangle A thus,a= 2r cos(A)(2.8)1 + r2but remains unchanged when r — r1 because of the s —* p symmetry of expression(2.6). Its value is a 0.642+0.13 [41] which implies that r or r 0.368±0.010. Fromthe measurement of a alone it is not possible to say whether the p-wave is stronger orweaker than the s-wave. However, this can be found from the polarization of the protonP1 which is given by,— Tr[MpM]2 9— Tr[MpMt](210)l+aPA.Q2spsin(A)where, /3 (2.11)2 + p2(2.12)Thus, the quantity 7 15 the protons’ polarization in the direction PA when they emergeat an angle 7r/2 radians with respect to the axis of polarization of a fully polarizedensemble of lambda particles.If .s < p then‘= —0.77 while if .s > p then‘= +0.77. The experimental resultis 7 = 0.76 + .03 [41] and it is concluded that the s-wave strength is greater than thep-wave strength. In summary,p/s = 0.368 + .010(2.13)(p/s)2 = 0.1355 ± .0074Chapter 2. An Application of the Hypertriton Model 31The ratio R. can now be written in terms of matrix elements of the lambda particledecay potential and kinematical factors. Ignoring final state interactions between thepion and the nuclear fragments, the effective Hamiltonian is given by,Heff() = > M H ata) (2.14)i=l ,2,3where, M = s (2.15)i H ii ‘, ‘,‘) = exp[iJS3( — i‘)—‘)8(i— ‘) (2.16)The final state plane wave pion wavefunction has been incorporated into the decayamplitude operator using the operator H and the pion momentum is written Theeffective Hamiltonian acts on particles 1,2 and 3 as indicated by the superscript (i), a)being the destruction operator for a lambda particle with label (i) and a)t being thecreation operator for a proton with label (i). The evaluation of the matrix elements isdetailed below in two sections. First, the 3He final state and then the inclusive case.2.3 Two Body Decay RateThe exclusive decay rate for the 3He final state is,F(H + 3He)= I 2w(2)32(wq + 2M3He — A2) (2.17)J(He;mfHeff(H;mj) (2.18),rnjwhere, Wq Energy of pion with momentum = q2 + m2M3He = Mass of 3He = 2808.39 MeVA2 = M3H — M3He = 182.7 MeV= spin projection of Hrnf = spin projection of 3HeChapter 2. An Application of the HypertrI ton Model 32For a two body final state there is a unique value of momentum for which energy conservation is satisfied and in this case the CM momentum of the final state particles isq, = 114.3MeV. The decay rate is thus,F(H ,‘ + 3He) = x f I(3He; rnf Heff(qH; m) 2‘7U + Wq,r I“3He)(2.19)The nuclear states appearing in the H—3e matrix element must be antisymmetrizedwith respect to interchange of any two particle labels. The antisymmetrization of thehypertriton wavefunction is straightforward due to the fact that the inner product of theisospin vectors for nucleon and lambda particle is zero. Using the antisymmetrizationoperator A = 1 + P2 + P3 the fully antisymmetrized hypertriton wavefunction is,H) (1 + F2 +F3)) (2.20)where Ii/, ) is the vector represented in equation (1.6), is antisymmetric under interchange of particles 2 and 3 and is normalized ( ‘) = 1. Correct normalization ofI ‘IH) follows from A2 = 3A.In appendix C it is shown that the matrix element simplifies to,3He; rrzf IHeff(fl H; rn ) v( IM1fPI ) (2.21)where, l,(1)) = a1)ta5 ) (2.22)and also that the identity,( ‘1’3HeW11 1) ) /( 3HeIIH’’ILZ/’ ) (2.23)holds, due to the fact that the model hypertriton has only s = 1, L = 0,2 where .s is thetotal spin of particles 2 and 3 in the i/’ ) component and L is the total orbital angularmomentum of all three particles. The two body decay amplitude can thus be written,H(H — 7r3He) = F1(q) [s +p./3qoj (2.24)Chapter 2. An Application of the Hypertriton Model 33where is the Pauli spin vector of the three particle system and F1(q) is a form factorfor the H — 3He ED transition matrix element.F(q) v( 3He 1jjk ) (2.25)This form factor is related to that of Dalitz [22], Leon [23] and Kolesnikov and Kopylov[11] by,F1(q) Fother(q) (2.26)The “other” authors’ definition of the form factor arises naturally when using a purelyDerrick-Blatt S state wavefunction [42] for 3He which has the following form, ( note thatthis is the full antisymmetrizeci wavefunction ).(,2,rI3He;rnf) = (x—‘) He(7.2,12331) (2.27)The hypertriton wavefunction ‘(1) has the form,XiH(r233i) (2.28)and so the two body decay amplitude is(-)F(q) [ +p/3qo] (2.29)where F(q) is the overlap of the S states modulated by the pion wavefunction, ( see [22,equations (A3,A5)] ).The form factor F1(q) was evaluated usiig existing programs for the 3He ,‘ 3H transition occurring in muon capture by 3He. To facilitate this, the hypertriton wavefunctionwas expanded in the same type of Gaussian basis as used in the variational calculationof the trinucleon ground states by Kameyama et al. [18]. The details of the method ofthis expansion are given in appendix F.NinaxV-’A(Y) = AN(yj,j, 0) exp[—(y/yjv)2] (2.30)N=1Chapter 2. An Application of the Hypertriton Model 34Table 2.3: Parameters for the Expansion in Gaussian Basis.Xrnjn/ fm XIax / fm umax ymjn/ fm Yniax / fiu N11s-wave 0.05 20.0 20 1.00 100 20p-wave 0.10 18.0 20———flinax(1)=A1 x’I\’(x, 1) CXP[(X/Xn)2] (2.31)2where, N(x, 1)=f(l + 3/2) (2.32)The Gaussian ranges x7, are chosen to be in geometric progression as defined byXmax, 7max etc. The relevant parameters are given in table 2.3. The parameters forthe deuteron were taken to be those suggested by Kameyama et a! [18j. The valuesof y1, an(l Ymax follow from consideration of the lambda wavefunction in momentumspace. When Fourier transformed, the Gaussian basis has the same Gaussian form withmomentum space ranges qN 2/yN.Nmaxequation (2.30) (q) = AN(qN,0)exp[—(q/qN)2j (2.33)N=1The lower and upper scales of the lambda wavefunction in momentum representationare an(l QA respectively and the choices of Yiiii and Yiiax given in table 2.3 were madein order to cover the range —* QA.The expansion of the lambda wavefunction was checked by computing the normalization of the Gaussian fit. The input wavefunction was normalized analytically andso any deviation from one shows an inaccuracy in the fit.Nx [ 2YNYN’ ] (2.34)NN’1 (YN + YN’The value of ATt was found to deviate from one by 106 and it is concluded thatthe fit was satisfactory. The high quality 3He wavefunction of Kameyama et al [18] wasChapter 2. An Application of the Hyperl;riton Model 35used in the calculation of F1(q) and led to the values given in table 2.4. The variationin F1(q) from the value for QA 1.17 fm1 indicates the uncertainty due to the model ofthe hypertriton. The value at Qp, = 1.17 fnf’ is F1(q,) 0.57 which should be comparedto 0.75 [23, Leon], 0.73 [22, Dalitz] and 0.51—0.78 [11, Kolesnikov Kopylov].Performing the spin sums and angular integration in equation (2.19) with the formof the decay amplitude given in equation (2.24) it was found that,F(H 3He)2(1 + W:/M3He) +p2q/9q] x F1(q)2. (2.35)2.4 Total Decay RateThe total decay rate for all modes involving a r isF(H X) f 2wq(2)3 26(wq + 6MN + Ex — A3) (2.36)x I (X; rn Heff() H;m) 2 (2.37)?fl,rnj,XThe summation is over all states of two protons and one neutron which are energetically accessible, their internal energy with respect to a static proton plus deuteronbeing written as Lx. The kinetic energy of the CM motion of the three nucleon systemisq2/2(3MN) and A3 is mA —— BA 177.2 MeV.For the break-up channels Ex is positive semi-definite and Dalitz [22,24] has arguedthat it is reasonable to replace Lx by some mean value Ex corresponding to the peak inthe pion spectrum. When that is done, the energy conserving delta function yields onlyone value of q, namely i at which the matrix element need be evaluated and the sum overstates can he extended to a sum over all states regardless of energy conservation.F(H X)2(1 +WS3MN) X (X;rnfIHeH;mi)f2 (2.38)Chapter 2. An Application of the Hypertriton Model 36A correction must be made for the underestimate of the two body mode which occursat significantly different Ex from Ex. This is done directly by adding and subtractingterms equal to the two body rate at the correct and incorrect values of q thus,6F x 27r qF(q)2[s+p2q/9q] — F1()2[s+p22/9qJ (2.39)= {s6 + p22/qe} (2.40)2—2where, =— F1(q,) — F(q)=[()3Fl q 2- F)2] (2.41)When this is clone, the most appropriate value for is 96 MeV [43].Using closure X )( X = I and the antisymmetry of the hypertriton wavefunction, the matrix element can be written in terms of exchange integrals Q(q) and 1d(q)which take into account the effect of the Pauli principle on the outgoing proton. Inappendix D it is shown that,f(X;rnfjHeff(H;mi)27fl, flU1 5 1= s2[1+ i) — 7d@)] +p2(/qo)[1— 1s() ?]d(q)]. (2.42)The exchange integral (q) is due to the s-wave part of the deuteron and rj(q) is dueto the cl-wave part: there are no cross terms. The exchange integrals are overlaps of thelambda part of the liypertriton wavefunction and the deuteron wavefunction as shown inappendix D. Results for ij5(q) and 7jj(q) are given in table 2.4.Dalitz macic no estimate of () in the 1958 paper but later with Rayet [43] found0.31 in agreement with Leon [23]. This larger value corresponds to the useof B, 0.25 MeV which was the accepted value at the time. The lambda particlemomentum distribution found in the previous chapter peaked at = /iB and willChapter 2. An Application of the Hypertriton Model 37Table 2.4: Summary of H ,‘ 3He Form Factor Results.q/MeV F1(q) ?73(q) 7)(q) QA/fm’0 0.8041 0.5401 1.86 x iO 1.1796 0.6263 0.2116 1.36 x iO 1.17114 0.5729—— 1.170 0.7876 0.5280 1.60 x iO 1.0596 0.6093 0.1999 1.17 x i0 1.05114 0.5559—— 1.050 0.8169 0.5494 2.10 x i0 1.2996 0.6397 0.2211 1.55 x iO 1.29114 0.5863—— 1.29have a larger overlap with the proton in the final state cleuteron as BA increases sincethe deuteron wavefunction is larger than the lambda wavefunction in momentum space.In fact, s(q) should scale like and hence \/ due to the following argument.At low momentum, ( the deuteron scale is the pion mass = 0.7 fm ), the deuteronwavefun ction must look like,N(2.43)where iv is a normalization constant and j = /MNEB(d). The peak of the lambdamomentum distribution is at and so the overlap with the proton in the final statedeuteron is approximately equal to the the deuteron wavefunction evaluated at .cNi. e. f dpp2 A(p)d(p) @) 2 + (2.44)This scales with c since (c/cd)2 < 1. The numerical values are,work= 0.68 c.f. () = 0.72 (2.45)hls(q)other 0.2which support the hypothesis that the exchange integrals scale with \/.The final expression for the inclusive r decay rate is,F(HX) = q x2r(1 + wq/3MN)Chapter 2. An Application of the Hypeitriton Model 38{s2[1 +— d() +61+p2(/qo)[1 ——+ el} (2.46)2.5 ResultsThe final expression for R is,F 2 (q [s2+pq/9qgj x kR = 1(q) [s2(1 +)-) +6) +p2/q(1- m() - (q) + e)](2.47)where, k(1 + Wq/M3He) (2.48)(1 +/3MN)The kinematical factor 1 is 1 .003 and thus exceeds one by only 0.3%. The correctionsfor the underestimate of the two body decay in the denominator are 6 —2.5 x iOand€ 1.7 x 102 in the case 1.17 fnf’. The smallness of 6 shows that the effectof the extra phase space at q7, is almost exactly cancelled by the drop in the form factorF1.The error in this calculation was estimated by varying the value of QA as suggested inthe previous chapter and combining this in quadrature with the error in the ratio (p/s)2from experiment. Varying QA by 10% gives a change in R of 0.016, i.e. 5%. The changein H. induced by the error in (p/s)2 is 0.001 and the final result is,R = 0.33 + 0.02 (2.49)which is in agreement with the experimental value of 0.35 + 0.04.The total decay rate can also be calculated using the fact that the free A particledecay rate is given by,=F(A p+) x = (s2+p) x. (2.50)2 27r(1 +wq0/mp) 2Chapter 2. An Application of the T-Iypertriton Model 39The factor 3/2 arises from including the ir0n mode which occurs one half as often asthe 7rp mode clue to the I = rule aid the fact that the lambda particle has isospinzero. Since the isospin of the hypertriton is also zero the r0 modes will occur one half asoften as the modes and thus the total lifetime T(H) is given by,3 /qo\ [s2+p]kTT(AH) = T(A) -- 5-q [s2(l + is(q) — (q) + 8) +p2/q(1—i/s(q) — 17d(q) + f)](2.51)(1 +w/3MN)where, kT =. (2.52)(1 + wq0/mp)The structure of the hypertriton affects the value of r(H) only via the exchange integrals i() and which enhance the free rate by approximately 6%. The kinematicalfactors also increase the rate by 6%, the overall effect being a reduction in the lifetimeof 12%. With the values given in table 2.4 the total hypertriton lifetime due to mesonicdecay is found to be,r(H) = 0.88r(A) = 2.32 x 10_lU seconds (2.53)The experimental results for r(H) are not clear. The values found from bubblechamber experiments are shown in table 2.5. Keyes et al. [25] argues that the resultof Block et al. [:38] could be in error due to a misclassification of true decays at rest asdecays in flight and also that the results of nuclear emulsion experiments are unreliable.Taking an average b of the bubble chamber results from Keyes et al. [25] and Keyeset al. [36,37] it was found that T(H) = (2.44) x 10_lU seconds. The limits that canbe set on i() from this result are () = 0.1 + 0.2 which provide no real test of thehypertriton wavefunction.In summary, the ratio R of two body to all pionic decay rates is found to agree withbThis average was found by transforming r toy where y exp[—Ar] with .X chosen as O.80x 1O’°sec’.The value of ,\ was found by demancliiig that the skew errors on T became symmetrical. A weightedaverage could then be performed assuming that the variable y is normally distributed.Chapter 2. An Application of the Hypertriton Model 40Table 2.5: Bubble Chamber Results for T(H).r(H)/1010sec Reference0 95 0J9 Block et al. [38]•• 0.152 64t0.84• --0.52 Keyes et al. [36,37]2 28 0.46•Keyes et al. [36,37]2 46 0.62. +4 Keyes et a!. [25]2.20+1.02 Keyes et al. [25]—0.532.64+092 Keyes et a!. [25]0.54experiment which lends confidence to the wavefunction. Although the lifetime is alsofound to agree with experiment, this is a poor test of the wavefunction.Chapter 3Muon Capture by 3He: Rate and Spin Observables in the ElementaryParticle Model and Their Sensitivity to the Pseudoscalar Form Factor.3.1 IntroductionThe Elementary Particle Model (EPM) was first used to calculate the muon capture rateby 3He in 1965 [45].. It was reviewed with emphasis on the the value of the pseudoscalarform factor in 1968 [46] and compared to an impulse approximation result by Phillips etal in 1975 [47]. The results are concisely reviewed by Kim and Primakoff in [48, p.88].The essential aspect of the EPM is the way in which the 3H and 3He nuclei aredescribed. They are treated as elementary particles with respect to the number of degreesof freedom used in the wavefunction, while their structure reveals itself through non-elementary couplings and q2 dependent form factors. It is important to realize thatthese are nuclear form factors and so represent the full structure of the nucleus includingcontributions from the nucleons and meson exchange.Hwang [49] has applied the EPM to spin observables and noted that the triton asymmetry is sensitive to the pseudoscalar form factor F. Just how sensitive the asymmetryis and also how sensitive other spin observables are to F was investigated using accuratevalues for the trinucleon form factors. An analysis of the theoretical uncertainties in thespin observables was also made.41The Elementary Particle Model 42Figure 3.1: Feynman diagram for muon capture.(k’)3He(k)3.2 Definition of Elementary Particle Model and KinematicsIn the EPM, the triton and helion are assumed to be members of an isospin doublet.This can be written formally as,He) = 3N:m,k)®1/2, 1/2) (3.1)j3H ) :3N:mJ) ® 1/2, — 1/2) (3.2)where k is the trinucleon centre of mass momentum and rn the spin projection. Thelatter vector is in the two dimensional isospin space I =-, 13 ) and the space-spin vectorsare the same for 311e and 3H, ( for the same k and mi). Given that both the spin andisospin of the trinucleon bound states are one half, five degrees of freedom are needed todescribe the state, three continuous labels (k), the spin projection m, and the isospinprojection 13.The muon capture amplitude is found by evaluating the Feynman diagram shown inThe Elementary Particle Model 43figure :3.1, writing free spinors for 3He and 3H. This diagram defines the kinematic 4-vectors i/v. k and k’ whose contravariant components are written = }. Themuon momentum and the helion momentum k are set to zero. The neutrino momentumii is 10:3.22 MeV and the four momentum transfer squared is —0.954m. Thesefigures are found using four momentum conservation and taking the initial state energyto be the mass of the muonic atom Mat0111.Matom = M3He + — 11 keV = 2914.039 MeVM3He 2808.392 MeV [51] (3.3)105.658 MeV [41]3.3 The Hadronic CurrentIn the absence of second class currents, the hadronic current has the following form,‘hadroic (k’) [FV7a + FMi + FA75 +F75] u(k) (34)which defines the form factors Fv, FM, FA and F. In the above,q a = —— 7137a)/275= j7Ol23= 75M3 = 2808.66 MeVufu = 1In equation (3.4), M3 is an arbitrary parameter with dimensions of energy. It is introduced to make all the form factors dimensionless and its value is taken to be the averagetrinucleon mass for the sake of convenience. In accordance with ecluations (3.1) and (3.2),the same space-spin function has been written for 3He and 3H. The spinor u(k) obeysthe Dirac equation for a free particle,( — M3)u(k) = 0. (3.5)The Element arv Particle Model 44The M3 denominators in equation (3.4) have no physical significance but the M3 appearing in equation (3.5) should represent the trinucleon bound state degenerate mass. Ofcourse, 3He and 3H are not quite degenerate but the average mass is a reasonable choice.The values of the form factors are set as follows. By invoking the Isotriplet VectorCurrent Hypothesis (IVC) [50], one can relate the vector form factors to the electromagnetic form factors of 3He and H. Specifically,Fv(q2) = 2F(q)— F’(q2) (3.6)FM(q2) = HeFIe(q2) TrF’r(q2) (37)where i is the anomalous magnetic moment and F1, iF2 are electromagnetic form factorsanalogous to F and FM in equation (3.4). F1 and F2 have the value one at q2 = 0. Thevalues of the anomalous magnetic moments are,He—8.:3689 (3.8)Tr= +7.9173. (3.9)These values follow from the measured magnetic moments of 1-1e = —2.127624(1) n.m.anl 1t’ 2.978960(1) n.m.[5l] and the relationships between t and tc which are,11He= (2 + KHe)/31 n.m. (3.10)Tr (1 + Tr)/3I n.m. (3.11)where, 3’ = M3/m = 2.99344 (3.12)and m1, is the proton mass. At q2 = 0, equations (3.6) and (3.7) give F = 1.000and FM= —16.286. At the pertinent q2, world averaged values are taken from electronscattering experiments, which yields [52],Fv(—0.954 in2) 0.8:34 ± 0.011Ii (3.1:3)FM(—0.954 m) —13.969 + 0.052.The Elementary Particle Model 45The value of FA at q2 0 is measured by the 3H beta decay half-life [53]-[561.FA(O) —g(O.96l + 0.003) = +1.212 + 0.004 (3.14)where gA is the nucleon axial form factor at q2 = 0, having the value —1.261 + 0.004[41]. The q2 dependence of FA is based on impulse apl)roximation expressions for thetrinucleon form factors as expounded by Delorme [57] and used by Klieb and Rood [58].These expressions give the nuclear form factors in terms of nucleoii form factors andreduced matrix elements between the 3H and 3He states. The form factors are given by[58, equations :3.1-3.3]:Fv(q2) gv(q2)[1j° (3.15)FM(q2) :3[gv(q2)+ gM(q2)][] — gv(q2)[1]° — 32g(q2)[j]11 (3.16)FA(q2) gA(q2)[j (3.17)Fp(q2) = 9gp(q2)[5]+ 63g(q2)[]21 (3.18)where [1]0, [j_, []+ [j21 and [iP]1’ are the reduced matrix elements and are q2dependent ( see [58, equations 3.7,3.8 and appendix G] for their definition ). The nucleonform factors gv, gri, gA and gp are defined by an equation for the nucleonic charged weakcurrent analogous to equation (3.4) with M3 replaced by MN, the average mass of thenucleon. It should be noted that it is more usual to write the mass of the muon and not2MN in the denominator of the gp term: that convention will not be used here.The q2 dependence of FA is thus given by,U ( 2\. I 2\ [1—( 2‘Aq ) — gAq I L° q (3 19FA(0)-gA(0) []-(0)The usual argument [45,46] is that the q2 dependence of [6] can be found from the q2dependence of FM. This will be true if the last two terms in equation (3.16) are negligible.The Elementary Particle Model 46At q2 —0.954m, F is about 0.8 which fixes the second term in equation (3.16).The [iP]” term is small because of the negligible p-wave support in the trinucleonwavefunctions and so the expression is dominated by the first term. ( Alternatively, onecan appeal to a smooth and small change in [öJ over this q2 region. At q2 = 0, FAis about equal to (—)g so that [6](0) is about —1. The value of gv + g at q2—0.954m is about 4.7 so the first term in the expression for FM is about —14 . As FM is—-14.0, equation (3.16) is dominated by the first term.The nucleon form factors gv and have the same q2 dependence [59) so we have,[](q2)— gv(O) Fj(q2) p320[5]-(0)- gv(q2) FM(0)Writing dipole form factors for gv and g and substituting the above in equation (3.19)gives,FA(q2) — (1—q2/Mv) FM(q2)3 21FA(0) (1 — /M)2 FM(0) (“FA(—0.954m) = 1.050 + 0.004 (3.22)where, M = 0.7 10 CeV2 [59)M 1.08 + .04 GeV2 see appendix IA better method is to use the q2 dependence of F1 + FM which is the same as the q2dependence of [] in the limit that [iP]” is zero. In that case,F(q2)— (1—q2/Mv) (Fv(q2)+ F(q2)3 23FA(O) — (1 — /M)2 Fv(0) + FM(0)FA(—0.954nI) 1.052 + 0.004 (3.24)The value of FA given by equation (3.24) was used.The dependence of the various observables on F will be shown by plotting themagainst the ratio Fp/Fp’ .“ Fp’ “ is the value of F obtained from the simplest form of thepartially conserved axial current hypothesis (PCAC) [60] as shown below.The Elementary Particle Mode] 47The PCAC links the divergence of the axial current A(x) to the pion field ,(x):the relationship being,ãA(x) a1,rn3(x) (3.25)where a7, is the pion decay constant and a1, = 0.9436 + 0.0011 [41]. The aim is to link F’Aand F to the pion trinucleon coupling constant F1, defined by,(3HJj(x) 3He) = —F1,(q2)(k’)7su(k)e (3.26)m7rTo do this one other ingredient is needed, namely the Klein-Gordon equation.[aa. + m1,] (x) = j(x) (3.27)The field operators j(x) and (x) operate on elements of Fock space supporting onlyinitial 4-momentum k and final 4-momentum k”, so that &L acting on (x) in such amatrix element can be replaced by iq. This allows the replacement of j(x) in equation(:3.26) by (—q2 + rn7,2) (x). At this point, the PCAC relation, ( equation (3.25)), isused to replace the pion field, which leads to the axial current aHcI finally to the axialform factors.(:3H j(x)I3He) = (-q2 + rn7,2) (Hl(x)jHe) (3.28)(—q2 + m7,)d(3HA(x)Ie (3.29)a,, rn7,= (—q2+m7,)2M3(FA + 42 Fp)(k’)5u(k)e (3.30)q2 —a7,rn2A + 4M32= (rn7,2— q2) F7,(q2) (3.31)Evaluating equation (:3.31) at q2 = 0 yields the Goldberger-Treirnan relation.FA(0) —a7,F(0) (3.32)The Elementary Particle Mode] 48By using this and rearranging equation (3.31) in favour of F, the following relationship between F, FA and F1. can be derived.Fp(q2)= m1.2_q FA(q2)[i + €(q2)] (3.33)with,2) — m2 1F1.(q2)/F0) (3 34€ q- (-q2) FA(q)/FA 0)The above procedure is due to Primakoff [61]. If it is assumed that the q2 dependence ofF1 is the same as the q2 dependence of FA, € vanishes. The q2 dependence of F is thendominated by the pion pole. The quantity F’ is defined as,I 2 4M32 2F (q ) = 2 2 FA(q (3.3o)rn7,—qand will be referred to as the “ PCAC value “ of F.Klieb and Rood [58] provide two rough estimations of the size of €.1. They evaluate FA using equation (3.17) and F using equation (3.18). Theythen fit€over the range —m < q2 < rn1.2 using equation (3.33). The resultis € —0.052. They evaluate FA using equation (3.17) and F1. with an expression of similarform and fit them to dipole expressions over the range —m < q < rn1.2.These fits can he used to evaluate€ directly from equation (3.34) and theresult at q2 —0.954rn is € —0.063.4 Rate and Spin ObservablesAt q2 =—0.954n, the weak interaction is well apl)rOxirnated by a point-like interactionso that evaluation of the Feynman diagram in figure 3.1 yields,M= /2Vd J°hadroiiic (3.36)The Elementary Particle Model 49where the leptomc current is,jiePtOfliC= (1’){7(1—‘5)}u(t) (3.37)CF is the Fermi constant and Vd is a Cabbibo-Kobayashi-Maskawa matrix element[62][63] linking the up arid down quarks. M can be written in two component formthus,M V (1 — iL){C + Gi+ GA.} (3.38)where the two dimensional spinors have been dropped and,N’(w+M3)Fv (i 11 ) — F4 (1’ + w — M3) 0.854 + 0.011w+M3 2M3 w+M3(Fv + FM + FA — F) 0.6027 + 0.00098w+M3 2M3CA = FA— ‘ (Fv + FM) 1.29:3:3 + 0.0041w+M3w = /(,,2 + M32)lepton Pauli spin matrixtrinucleon Pauli spin matrixThe numerical values for G ,Cp and CA are at. q2= —O.954m and the value for Op onlyholds when F takes on the PCAC value, Fe’. N’ is the normalization constant for the3H spinor and w is close to the energy of the triton. Despite the appearance of equation(3.38), it is not a non-relativistic reduction.If a 2x2 matrix, M, is defined by,M (1—{Gv + Cp + GA.} (3.39)then the rate is proportional to T 2, whereT 2 Tr [pj MtM] (3.40)The Elementary Particle Mode] 50where p is the initial state density matrix and MtM is given by,MM Go2 + gL + + + (3.41)where, ç2 = G2 + 3GA2 + O2—ri 2 ri 2 ri 2 on ri‘-‘ A ‘—‘P ‘- v H- ‘‘—-‘ A ‘-‘ iohr-i fri ri ‘ n-i 2= “—‘V’--’P ‘—‘A) — ‘—‘A7 = 2GA(Gv+Gp—GA)—20p(Gv + GA)The rate is given by,F= V 2 N’2C I ) 2 ,2 (_ )f IT 2 (3.42)where q(;) is the spatial atomic wavefuiiction and C is a correction factor which takesinto account the non-pointlike nature of the nucleus, ( the value C=0.9788 was chosen -see chapter 5.) and \/s is the mass of the muonic atom ( 2914.039 MeV ).The initial state density matrix is written in terms of the hyperfine populations forthe four lowest lying atomic states. These are all associated with the iS state and thepopulation densities are written as N(f,f), where f is the projection of the grand totalangular momentum f, on the quantization axis . The quantum number f equals 0(1)and f may take the values 0(-1,0,i) giving four states in total.p N(f,f,)If,f2) f fj (3.43)f,fzThe density matrix is taken to be diagonal in the f,f space rather than the ,5I-Ie space because j,f are good quantum numbers. This follows from the fact that thewidth of the IS state, ( due almost entirely to the muon lifetime ), is much less than thehyperfine splitting.F(1S) = 3.0 x 10° eVThe Elementary Particle Model 51E(hyperfine) 0.2 eV (3.44)The values taken by N(f,f2) is the subject of chapter 7. Using,Tr [ f,f2)( .f,fz (A + + + + + x= A + B[2f(f +1) -3] + [ + F.jf2+ [f(f +1) - -11+ [-f(f +1) + 3f2j (3.45)one finds,T 2 0o2 [N(1,i) + N(l,0) + N(1,-i) + N(0,0)]—(a’ + ) cos0 [N(1,1) — N(1,-1)] (3.46)+ [N(1,1) + N(1,0) + N(1,-1) — 3N(0,0)]+ 6’ [N(1,0) — N(0,0) + cos29 (N(1,1) + N(1,-1) — 2N(1,0))]where 0 is the angle the triton makes with the z axis.cos 0 k’.. (3.47)The expression (3.46) was rearranged as follows, so that its structure is more clearlyseen.T 2 02(1 + APPi(cos0) + AtPtP2(cos0) + AP) (3.48)where P, P and P express the deviation of the hyperfine populations from the statisticalvalues and A, A and A are the “Analyzing Powers”. The factors P1 and P2 are theLegendre polynomials.P = N(i,i)— N(1,—1)= N(l,l) + N(1,—1) — 2N(1,0)= N(1,l) + N(1,0) + N(1,—1) — 3N(0,0) = 1 — 4N(0,0) (3.49)The Elementary Particle Mode] 52Using the Madison convention [64] to define polarization of the f = 1 state, the vector polarization Pz is 4P/(3+P) and the tensor polarization p is 4P/(3+P). Theanalyzing powers are given by,A = —(+/3)/G02= 26/3G0= (‘ + 6/3)/G2. (3.50)There are thus three spin observables. The fact that there are three corresponds tothe four states in which the muonic ion can find itself prior to capture. These four statesgive four rates which can be measured. For a statistical population one arrives at anisotropic rate given by G02. For iiou-statistical populations, the rate is anisotropic theangular moments being given by A and A. The total rate i.e. integrated over all angles,differs from the statistical rate and this deviation is given by A which measures thedifference between the f 0 and f = 1 total rate.rf-1-statFf=o= 1— 3A (3.51)FstatThe unpolarized rate was found by setting all of the N(f,f) to 1/4.F0 = G2 Vud 12 N’2 C I (O) 2 ,2 (i—a2 (3.52)3.5 ResultsThe value for the unpolarized rate, with F set at its PCAC value, was found to beF0 = 1497 + 11 s which should be compared to the experimental results of AuerbachThe E]ementaiy Particle Model 53et al [65] and Clay et al [66].11505 +46 s_i [65]F0 = (3.53)1465 + 67 [66]The value found herein agrees with the weighted average of the above results, whichis Fo 1492+:38 srn’. Phillips [47] finds F0 =1425 s1, and Frazier and Kim [46] findFo = 1449 s, the difference between the two being solely due to the values of Fv, FMand FA used. The enhancement of the present calculation with respect to the resultof Phillips, is 5.1 %. The corretion factor is 1.4% larger and the form factors give anenhancement of :3.7%.The values of the spin observables, with F set to the PCAC value are,A = 0.5243 + 0.0057 (3.54)A1 = —0.37933 + 0.00064 (3.55)A = —0.0959 + 0.0060 . (3.56)It is pertinent to ask which of the four observables A, A1, A, Fo is the most sensitiveto the value of F. The sensitivity to F is shown in figure 3.2 where the observables aredrawn scaled to their PCAC values i.e. the curve labelled A is actually A divided byits value at F = F’ etc. It can be seen that A is more sensitive to F than A1 whichis more sensitive than A which is more sensitive than Fo. Quantitatively, the pertinentfigure of merit for an observable A is the slope of log(A) with respect to log(Fp). This isgiven for F0, A, A1 and A in table 3.1.When “measuring” F it is necessary to combine an experimental error with a theoretical error. This should be done in the following manner. Let be the fractionalerror in the theoretical prediction of F0 and be the fractional error in the measuredvalue of F0. The fractional error in a quantity A is the error divided by the value of A.The Elementary Particle Model 541;2.0-1.5-1.0-0.5-0.0 0.5 1.0 1.5Figure 3.2: Sensitivity of observables to F.2.00.02.00.0 0.5 1.0 1.5pFigure 3.3: Unpolarized rate versus F.The Elementary Particle Model 55Table 3.1: Sensitivity of Observables 0 to Fp.0 dF F,Fo 0.11A 0.38A 0.75A 0.89(F)Define a quantity % by,2 2/ (P) \ / (F) ‘\((F))2= ( J -b ( FF J (3.57)F dFp F J dFp F IThis quantity was concocted as follows. Suppose the experimental error was zero.Then, the error in F would be due solely to the error in the theoretical 1)rediCtiofl of Fand would equal the absolute error in F0 divided by the slope of F0. This is equal tothe fractional error divided by the slope of the log of F0. When both an experimentalerror and a theoretical error contribute, the probability density function for F is theconvolution of two normal distributions, ( variances o and u ), which is itself normaland has variance equal to o+o. Thus, is the total error in the value of Fp resultingfrom the measurement of Fo. Evaluating this expression one finds,( (F)2— (°•°25N (0.0072Fp)o.ii) +o11}= 0.24 (3.58)By evaluating C02 for different values of F it is easy to find what value reproduces theexperimental result for F0. The dependence of F0 on F is shown in figure :3.3. If all theassumptions made in the EPM are correct then one has the following “measurement” ofF.______________= 1.03 + 0.24 (3.59)The Elementary Partide Model 56What precision is needed in a measurement of A, A or A in order to better theabove measurement of Fp? By evaluating expressions analogous to equation (3.57) it wasfound that,(Av))2= (8))2+(1)2 (3.60)(At))2= ((A)2 (0M017)2(3.61)())2= ())2 + (03)2(3.62)which lead to the following requirements for the fractional errors in the measurements ofA, A and A.(Av) <0.24 if qj) <0.09 (3.63)<0.24 if j) <0.18 (3.64)<0.24 if j) <o (3.65)Also, it can be seen that the minimum is 0.061, the minimum is 0.0023and the minimum is 0.071.These results indicate that all the spin observables A, A and A offer a betterdetermination of F than does the rate. Because F is strongly dependent on gp, equation(3.18), it follows that the spin observables will be sensitive to gp and this motivates themicroscopic calculation detailed in the next chapter.Chapter 4Muon Capture by 3He in the Impulse Approximation4.1 IntroductionThe elementary particle model (EPM) describes quasi-elastic a muon capture by 3Hein a compact and useful way. However, it is unsatisfactory in that it is a convenientparameterization of the 3He — H transition rather than providing a complete theoreticalunderstanding. A more fundamental approach is at the ‘microscopic’ level which meansusing nucleons and mesons to describe the iiitial and flial nuclear states and the muoncapture interaction. The microscopic approach is applicable to muon capture by othernuclei and also muon induced break-up channels in a uiifiecl way, whereas the EPMintroduces a new set of form factors for each nucleus and final state and thus has limitedpredictive power.The impulse approximation (IA) is the first step in a complete microscopic calculation.The term “impulse approximation” will he taken to mean that only one-body currentsare included. Other currents clue to the virtual mesons also present in an interactingsystem of nucleons are necessarily two-body or in general n-body (n > 1) currents.There are two ingredients in this calculation; the description of the nuclear states andthe effective muon capture Hamiltonian. Highly sophisticated and reliable wavefunctionsfor 3He and H were used aid are described in detail in section 3. The muon captureHamiltonian was derived using the on-shell IA which is expected to be a reasonableaThe 3He 3H channel is termed quasi-elastic rather than elastic because the transformation of amuon into a neutrino involves a release of energy.57Chapter 4. Muon Capture by 3He in the Impulse Approximation 58first approximation for the nuclear current. This calculation is intended to eliminateany uncertainty clue to the description of the nuclear states so that ai unambiguousconclusion regarding the quality of the IA can be made.An early calculation by Peterson [106] used a simple phenornenological based formfor the wavefunction and found that the rate fell short of the experimental value byapproximately 13% in the IA.Phillips [47] has also calculated the total rate in the IA using wavefunctions for 3Heand 3H based on separable NN partial wave potentials. The result fell short of theexperimental value by approximately 10% and it is unclear whether this discrepancy wasdue to the wavefunctions or the IA.The calculation by iKlieb and Rood [58] employed a wavefunction which is a solutionof the three body Schrödinger equation with the Reid soft core potential constrainedto act only in the I0 and 3S1-D channels. These authors found the rate to be 1268seconds which is 15% short of the experimental value. Again, the wavefunction usedis deficient in that it does not represent a complete solution of the three body problemand also does not reproduce the correct binding energy.4.2 The Trinucleon Bound StatesThe triton (3H) and helion (3He) form a good isospin doublet and have spin one half andpositive parity. To find wavefunctions for these nuclei it is necessary to solve the threebody Sclirödinger equation. In solving the three body Schrödinger ecluation, it is usualto separate it into the Fadcleev equations [17] as shown below.If Vc. is the potential acting between particles and-y, where (c/3y) is a cyclicpermutation of (123), then the Schrödinger equation for the wavefunction Ji is,(T+V+V2+—E))=0 (4.1)Chapter 4. Muon Capture by 3He in the Impulse Approximation 59where T is the kinetic energy of the three particles. The Faddeev equations result fromwriting the full wavefunction as the sum of three Faddeev components ,L’1 /‘2 andThe Fadcleev components are found by solving the following coupled equations.(T+V- E)I1) = —V1(j2)+ ))(T+V2-E))=-V(I3)+)(T+V3- ))=-+lb (4.2)For three identical particles the Faddeev components‘2) and ‘) are simply permutations of /‘ ) as can be seen from the equations (4.2). The total wavefunction ‘P isthus,= (+P23)) (4.3)where P2, P3 are permutaton operators as defined in appendix E. The solutions of theFacldeev equations for the trinucleon bound state have attracted much attention over thelast twenty years [67]-[93] but the most important property of the resulting wavefunctionsis the phenomenon of scaling as exposed by Friar, Gibson, Chen and Payne [:30]. Theseauthors clearly show that various low energy observables of the trinucleon bound state alldepend on the binding energy of the solution obtained according to a simple power lawirrespective of the details of the NN potential used in the Faddeev equations. In orderfor wavefunctions to provide good descriptions of 3He and 3H then, it is a necessary ( butnot sufficient ) condition that they have the correct binding energy.Most NN realistic potentials underbind the triton by ‘--1 MeV. This is not a haddisagreement since the binding eiergy is a balance of roughly 45 MeV kinetic energy and—53 MeV potential energy. It has been shown [82]-[85] that the inclusion of a three bodyforce enhances the binding energy. The amount of enhancement depends sensitively onthe irNN form factor cut off A. It is thus possible to tune A in order to reproduce theexperimental binding energy for 3H. This procedure is perhaps the most natural wayChapter 4. Muon Capture by 3He in the Impulse Approximation 60to ensure that a :3N bound state wavefunction satisfies the condition of possessing thecorrect binding energy.4.3 The Kamirnura WavefunctionsThe wavefunctions for the helion and triton used in this calculation were developed byH.Kameyama, M.Karnimura and Y.Fukushima and their properties are described in detailin [18]. These wavefunctions represent accurate solutions of the Schrödinger equation forthe 3N system, include the Coulomb repulsion in 3He exactly ( i.e. non-l)erturbatively )and use a tuned three-body force to make the small adjustment necessary to reproducethe experimental binding energies. The Argonne V14 NN potential [69] was used as well asthe Tucson-Melbourne 2r exchange model [70,71] of the three body force. The ArgonneV14 potential gives an excellent fit to the deuteron properties and NN scattering phaseshifts up to the pion production threshold.The wavefunctions are found from the variational principle. This method has theadvantage over regular solutions of the Faddeev equations that the potential need notbe partial wave expanded ( and hence truncated at some point ) and so a solution“feels” more of the potential than the solution of the Faddeev equations. The variationalapproach leads to fast convergence of the binding energy with respect to the number ofchannels in the Faddeev component, a statement which will he explained below. Previousvariational calculations [94]-[100J were not as precise as the Fadcleev solutions but thisdeficiency has been overcome by Kamirnura et al.by an astute choice of basis functions.Here follows a summary of these wavefunctions.The Faddeev component is written as a sum of channel wavefunctions ) where eachchannel is defined by the quantum numbers l, A, L, s, S, and the Russell-SaundersChapter 4. Muon Capture by 3He in the Impulse Approximation 61(LS) coupling scheme has been used.= Ik) (4.4)Mj 1() = i (x, y) (, ) ® x1] (4.5)In equation (4.4), particles 2 and 3 ( the ‘pair’ ) have been coupled to spin sa, SOSJflfli and have orbital angular momentum 1a in their CM. Particle 1 ( the ‘spectator’ ) hasorbital angular momentum with respect to the CM of the pair and is coupled to thepail to give total spin S, total orbital angular momentum L, total angular momentumand total isospin . The spherical harmonic )‘, spin vector X and isospin vector ij aredefined precisely in appendix E (see equations (E.8)-(E.13)). The chaiinel wavefunctionsi/(x, y) are written as a sum of Gaussian basis states whose ranges are in geometricrogressioILThrnax ,]V,nsx(x,y) = ANq(x)q(y) (4.6)n,N=lq(x) N?IlGx1exp(—x2/ ) ql(y) = NNy exp(—y2/y) (4.7)n—i ( N—Ip ( ) tNmax—imax max— jInax== (4.8)YminThe distribution of ranges is dense at small distances which allows the description ofshort range correlations. At large distances the basis functions add coherently so thatwith a suitable choice of Yiiiax the long tail of the wavefunction can be described.The normalization coefficients Njn and N, are chosen so that,2]2= 1j dyy2[( )] = 1 (4.9)The non-linear variational parameters (fliiiax, Xiji Xmax) and (IVrnax, Ymin, ymax) canbe chosen independently for each channel and thell the expansion coefficients AN andChapter 4. Muon Capture I)y 3He in the Impulse Approximation 62Table 4.1: Properties of the Wavefunctions.3H(22) I H(8) 3He(22) I He(8)EB 8.43 8.34 7.76 7.67EB (exp.) 8.48 7.72P(L=0) % 90.61 90.70 90.64 90.73P(L=1) % 0.16 0.14 0.16 0.14P(L=2) % 9.23 9.16 9.20 9.14K.E. MeV 49.07 48.64 48.21 47.78V MeV —57.50 —56.98 —55.97 —55.45binding energy are found from the Rayleigh-Ritz variational principle which yields ageneralized eigenvalue equation for the AN. bIt has been demonstrated [18] that these wavefunctions are as accurate as the Fadcleev solutions and further that the binding energy of the solution converges with respectto the number of channels faster [101, see figure 5]. This convergence is especially important when a three-body force is included since then the Faddeev solutions show poorconvergence because of the strong dependence of the three body force on the angularorientation of the three particles.Two wavefunctions were used comprising of an 8 channel or 22 channel expansionof the Faddeev component. Some properties of these wavefunctions are given in table4.1. The label 3H(8) indicates that the properties listed pertain to the 8 channel tritonwavefunction. E is the binding energy (the experimental data is taken from [102]). P(L)is the total probability for L, = L. K.E. and V are the kinetic and potential energies ofthe three nucleons. The parameters defining these wavefunctions are given in tables 4.2and 4.3.bThe values of the basis coefficients AN were provided by M.Kamirniira.Chapter 4. Muon Capture I)y 3He in the Impulse Approximation 63Table 4.2: Channel Specifications for the 8 Channel Wavefunction.Channel —— umax X1j X11-lax Nmax Yniin Ymal, ,\. L s Sa a fin frn frn frn1 0 0 0 0 1/2 1 15 0.05 15.0 15 0.3 9.02 0 0 0 1 1/2 0 15 0.05 15.0 15 0.3 9.03 2 2 0 1 1/2 0 15 0.10 15.0 15 0.3 9.04 2 2 1 1 1/2 0 15 0.10 15.0 15 0.3 9.05 2 2 1 1 :3/2 0 15 0.10 15.0 15 0.3 9.06 2 0 2 1 3/2 0 15 0.10 15.0 15 0.3 9.07 0 2 2 1 3/2 0 15 0.10 15.0 15 0.3 9.08 2L 2 1 3/2 0 15 0.10 15.0 15 0.3 9.0Table 4.:3: Channel Specifications for the 22 Channel Wavefunction.Channel ————— umax Xmjfl XnIax Nmax Ymin YmaxCi ,\ La 5a Sa fin fin fin fm1 0 0 0 0 1/2 1 15 0.05 1.5.0 15 0.3 9.02 0 0 0 1 1/2 0 15 0.05 15.0 15 0.3 9.03 2 2 0 1 1/2 0 15 0.10 10.0 15 0.3 9.04 1 1 0 1 1/2 1 10 0.10 10.0 10 0.3 6.05 1 1 0 0 1/2 0 10 0.10 10.0 10 0.3 6.06 2 2 0 0 1/2 1 10 0.10 10.0 10 0.3 6.07 :3 3 0 1 1/2 1 10 0.10 15.0 10 0.3 6.08 2 2 1 1 1/2 0 15 0.10 15.0 15 0.3 9.09 2 2 1 1 3/2 0 15 0.05 10.0 15 0.3 9.010 1 1 1 1 1/2 1 10 0.05 10.0 10 0.3 6.0ii 1 1 1 1 3/2 1 10 0.10 10.0 10 0.3 6.012 1 1 1 0 1/2 0 10 0.10 10.0 10 0.3 6.01:3 2 2 1 0 1/2 1 10 0.10 10.0 10 0.3 6.014 :3 :3 1 1 1/2 1 10 0.10 10.0 10 0.3 6.015 :3 3 1 1 :3/2 1 10 0.10 10.0 10 0.3 6.016 2 0 2 1 3/2 0 15 0.10 15.0 15 0.3 9.017 0 2 2 1 3/2 0 15 0.05 15.0 15 0.3 9.018 2 2 2 1 3/2 0 15 0.05 15.0 15 0.3 9.019 1 1 2 1 3/2 1 10 0.10 10.0 10 0.3 6.020 :3 1 2 1 :3/2 1 10 0.10 10.0 10 0.3 6.021 1 3 2 1 3/2 1 10 0.10 10.0 10 0.3 6.022 3 3 2 1 3/2 1 10 0.10 10.0 10 0.3 6.0Chapter 4. Muon Capture by 3He in the Impulse Approximation 644.4 The Effective HamiltonianThe effective Hamiltonian Hoff for muon capture was first derived by Luyten, Rood andTolhoek [103]. The basic scheme is to sum the contributions of each proton iu the nucleusaccording to a non-relativistic reduction of the 1r + p —* v, + n amplitude. For a protonof 4-momentum k’ and neutron of 4-momentum k this amplitude is,M(p + p v + n) = VudJaJadroic (4.10)where,jlePtornc= Ti(V){7a(i —75)}U([t) (4.11)= (k’) [gv7a + gMia2 +g5+ gp75 2 ] u(k) (4.12)and, u(k)tu(k) = 1 and, qa = — (4.13)The hadronic current is discussed in appendix I where the values of the nucleon formfactors gv,gM,gA,gp are also given (gv 1.0, grvi 3.6, g. —1.2, gp —145).In equation (4.12) n(k) and (k’) are the 4 component Dirac spinors for the initialproton and final neutron respectively and obey the free Dirac equation. The current inequation (4.12) is that of a free proton —> free neutron transition. There are two problemswhich arise when this current is applied to bound nucleons.First, the current of a bound proton bound neutron transition is not the same asthat of a free proton —+ free neutron transition. In general there are extra terms and theform factors gain a k’2, P dependence as well as the more familiar q2 dependence [104].Second, relativistic wavefunctions for 3He and 3H are not available and so even if onedid know the current for a bound p(k) —> n(Ic”) transition there would still have to besome choice made of (k°, k’°) as functions of (k, k’).‘With regard to the first problem the approach that was taken was to neglect the extraterms and also the k2, k’2 dependence of the form factors.Chapter 4. Muon Capture by 3He in the Impulse Approximation 65With regard to the second problem let us consider what choices could be made forko.c One choice is to 1)Ut the struck proton on its euergy shell i.e. take,= (P + M. (4.14)However, there is an inconsistency in that the sum of the energies of the three nucleonsin 3He is less than the sum of the masses and the above assignment gives k° > MN.Another view is to take the struck proton to be as much off shell as the other twoparticles and thus,= MN — Eu(3He)/3. (4.15)Yet another view is to notice that the muon capture process is a low q2 process and3He is asymptotically like a bound deuteron-proton system so that the energy of thestruck proton should be,MN — EB(3He) + EB(d). (4.16)There is certainly a certain uncertainty as to what to take for k°. Of that we can bequite sure. The approach that was taken was to put both proton and neutron on-shell.This can he termed strict impulse approximation in the sense that the nucleon current isexactly that of a free proton —* free neutron transition. This approach is standard andsince the emphasis in this work is on the nuclear structure, it is desirable not to cloudthe issue with an unusual nucleon current.It is necessary to expand the hadronic current in powers of k/MN, k’/MN so that theintegrals appearing in the matrix elements of the effective Hamiltonian can be evaluatedanalytically. An alternative approach is to evaluate the integrals numerically using momentum space wavefunctions but that approach has not been pursued. After expansionto some order in k/MN, k’/MN the identification Ic’ = Ic — 17 was made. Finally, theresulting current was contracted with the lepton current and simplified.dHaviig made this choice k’° follows from energy conservation at the vertexChapter 4. Muon Capture by 3He in the Impulse Approximation 66Using this procedure, it is important to have some idea of the various sizes of termsso that a consistent expansion can be made i.e. an expansion which includes all termscontributing at a given level. A useful expansion parameter is m,L/MN = 0.11. Theneutrino energy ii in any muon capture reaction is constrained to be between zero andthe mass of the muon so that quantities like i//MN can safely be taken to be first order inthe expansion parameter. The matrix element of k can be identified with that of i/3 aswill be explained in the next section and so taking k/MN to be first order in the expansionparameter is a very safe identification. A very crude estimate of(k2/M)fo1lowing fromthis is v2/9M 0.1% and this value is in accordance with the size of the k2/Mcontributions found by Friar [105]. The contributions of these terms will be neglected. Inconclusion, by keeping terms second order in il/MN and first order in k/MN an effectiveHamiltonian was found which should be good to about 1%.One further point concerns the size of the terms involving the pseudoscalar form2m2factor gp. In appendix lit is shown that g which implies that —-gp g. Itis therefore necessary to keep extra terms which involve gp because of its large size andthis is affected by assigning gp order minus two and the other form factors order zero.The result for hadronic expanded to order two is shown below where the identificationq= k’ — k has been used.___‘2 iu.(i’ xhac1ronic = gv 1—_ __— Q1,12 + AT/12 + ATK2 +()IVIN °‘‘1N ‘tIVIN ‘ILVINiu.(’x)__-4M + 4M. +A9.(k + k) -iq° .k 1 3/c2 ‘2 “\ tji ( ,2 3k’22MN — 8M — 8M)— 2MN — 8M — 8M} (4.17)_k+k’ iX iX q° ( (7 ix(+’)J1fr01— v 2MN+2MN+gM2MN — 2MN 2MN+2MNChapter 4. Muon Capture by 3FIe iii the Impulse Approximation 67L (____2 \____- 4M 8M - 8M) + 4M ( + + ‘.)] +q E ot, / 3k2 e5tI / 3k’2 “1P2 L2MN c\ — 8M — 8MN) — 2MN — 8M — 8M)j (4.18)The leptouic current in two component form is,1= (l —1Jieptonic = -(1 —I,3±L)L) (4.19)Setting k’ = k — i7 and q° = m,, — v in equations (4.17), (4.18), contracting theleptonic and hadronic currents and including the neutrino wavefunction the followingeffective Hamiltonian was found.CFHeff =V11dMIJexp(_ii7.ij) (4.20)I—(1 iL){GN + Gi + +(1)-.L-. [G1 + + + GLNa (v)+MN(i).GvLN11(Ji) + GNv(z’.Jj) + Gi x J} (4.21)J)2 \ V2 / — (4.22)= gv (i + 2MN — 8M) — M4iN____I________ ____, r (i — q° +—/ 3,,2=— ‘gv+gM2MN [ 2MN) P2 1 — 8M)] (4.23)v2 iiq°iN_____ _ETA g( — 8M) — (v +M)2 +M42 (4.24)ii “ rnv2rl(1)__ _(4.25)= gA(1+4+gP163=— (4.26)11(4.27)Chapter 4. Muon Capture by 3He in the Impulse Approximation 68ii q°GLvN = (4.28)4MN 2V1N(1) V q°= gA — gM, (4.29)=gp2N(4.30)G= gv4 +M2 (4.31)The Pauli Spin operator 3j acts on the nucleon and the effective couplingshave the superfix N to indicate that they are nucleon couplings which distinguishes themfrom the effective couplings appearing in the EPM formalism.‘T is the isospin loweringoperator which has non-zero matrix element oniy for an initial state proton and a finalstate neutron in which case its matrix element is one.This result was compared to that of Luyten et al.[103] who made an expansion to“first order in v/2MN and ks/MN”. All the terms of order zero and one agree. Theterms of order two are absent of course in that result. A comparison was also made withthe result of Friar [105] who used a Foldy-Wouthuysen reduction to obtain an effectiveHamiltonian including “corrections of order (1/MN)2”. The differences between thatresult and the oiie found here are of two types. First, extra terms have been includedfor gp which was a result of the realization that gp has order minus twod. Second, thereare differences which are like iiq°/M = 3 x iO. This quantity is of the same order ofmagnitude as the error one would make by identifying q° with k’° — k°. It is stressedthat this identification has not been made at any point in the derivation of the effectiveHamiltonian where q° = m1 — ii was used.dWjtI the more conventional definition of gp it has order minus one. Of course, the conclusionsregarding which terms to keep involving gp are independent of the choice of convention.Chapter 4. Muon Capture by He in the Impulse Approximation 694.5 Matrix ElementsThe effective Hamiltonian has terms of order (k/iW)° and (k/M)L The matrix elementof the latter terms can be simplified by making the approximation that the 3He and Hwavefunctions form an exact isospin doublet. i.e.I3He) =3H)= )II=,I3=—U (4.32)In that case the matrix element of Ic can be written in terms of the matrix elementof I by making an integration by parts following the identification Ic1 = —ivy and alsoi=2’/3.-(H ZIj exp(—i)3He)=H 1j exp(—i)j3He) (4.33)This procednre is due to Peterson [106]. The quality of this approximation was testedby evaluating the matrix element of the isospin lowering operator 1 = 1j + 1j + 1between 3He and 3H. If 3He and 3H formed an exact isospin doublet then this matrixelement would eqnal one. In fact it was fomid that,(3HIHe) = 0.9998 (4.34)which indicates that the error made in nsing equation (4.33) is approximately 0.02%.Consider the non spin-flip matrix element ( H I exp(—i.i) 3He ). Since 3He andH have total angular momentum the total angular momentum of any operator whichhas a non-zero matrix element can be only 0 or 1. A further selection rule is providedby the parity of intial and final states which implies that operators with negative parityhave zero matrix element. Expanding the plane wave in the matrix element the followingsimplification was arrived at.3(H; ?flj I exp(—W.i)j3He;in = [1]°6(rnf,rn) (4.35)Chapter 4. Muon Capture by 3He in the Impulse Approximation 70where, [110 = (3H ijo(vr1){3He) (4.36)For the Spin operator there are two multipoles which contribute.:3( H: m1 exp(—i)3He;rn) = xt {[]o — 3J))1[121}(4.37)where, []01= (3H11 ITjo() 3He)/ (4.38)[5]21 (H Ij2(vr) [}) ® jj 3He)/ (4.39)The matrix element between 3He and 3H has been written in terms of the PauliSpin vector for the trinucleon states, two-dimensional Pauli spinors X7 and X forthe final and intial states and also the reduced matrix element of the vector operatorbetween two spin one half states which equals By using this form and writing(mf, 7n) in the non spin-flip matrix elemeit the total matrix element of theeffective Hamiltonian between the nuclear states can be written:—(RH; mj 11eff3He; rn) = XtMIAX (4.40)where, M1A = Vud (1—3’) {G + (;IAL} (4.41)which should be compared to the expression for the matrix element .41 in the elementaryparticle model (EPM) given by equation (3.38). The effective form factors G, G, Gare given by,= {c;+ 3MN- G)] []0= + 3M (G + GvvN - GN)] - 3M GLN] [6121=+ 3MN[6] (4.42)Chapter 4. Muon Capture by 3He in the Impulse Approximation 71where, []+ [8J°” + \/2[6J21 (443)[6]— = [ë]°’ 1[8j21 (4.44)These effective form factors may be directly compared to the effective form factorsG, Op and GA arising in the EPM. The calculation of the reduced matrix elements[1]°, [6j°” and [6]2 is detailed in appeidix 0. The results are shown in table 4.4. Thesmallness of [6j21 is a result of the low three momentum transfer ii in the process.Table 4.4: The Reduced Matrix Elements.[110 [6]01 []218 channel 0.851 —0.808 0.001522 channel 0.853 —0.809 0.0015Table 4.5 shows the value of G, G and G which follow from the values of thenucleoii form factors given in appendix I and the matrix elements given in table 4.4. Thefirst row gives the effective form factors from the EPM along with their experimental Uncertaiuty. The second and third rows give their values in the IA along with the deviationfrom the EPM value both with and without relativistic corrections.Table 4.5: The Effective Form Factors.GAEPM 0.85 + 0.01 0.603 ± 0.001 1.293 + 0.004IA O(/MN)1 0.836 —2% 0.551—9% 1.205 —7%IA O(k/MN)° 0.866 +1% 0.506 —16% 1.202 —7%Using the values for G, G and G given in table 4.5 (i.e. those including k/MNcorrections ) and the expressions developed in chapter 3 for the following results for thetotal rate and analyzing powers in the IA were found.= 1325 seconds’Chapter 4. Muon Capture by 3He in the Impulse Approximation 72= 0.55A = —0.372A—0.0774.6 Summary and ConclusionsA calculation of quasi-elastic muon capture by 3He has been performed using sophisticated and reliable wavefunctions for the nuclear states. Care was taken to include allnecessary terms with the pseucloscalar form factor gp in the effective Hamiltonian wherethe PCAC relation was used to give the order of gp in the expansion parameter mJMN.The total rate falls short of the experimental value by 11%. The values of the analyzing powers differ from those found in the EPM by 6% (A), 2% (At) and 23% (As).Regarding table 4.5 it is seen that the IA reproduces the value of Gv found in theEPM whereas Gp and GA fall short of their EPM values by about 10%. Since Gand G are proportional to [6101 and G is proportional to [110 this can be expressedas a lack of magnetic strength in the IA. A possible remedy is the inclusion of mesonexchange currents (MEC) which are likely to enhance the magnetic strength in the muoncapture hamiltonian. This is because MEC contribute at low order to isovector magneticnioments as has been shown by many authors [107j-[11 1].Chapter 5Muon Wavefunction Overlap Reduction Factor5.1 IntroductionAt the values of q2 relevant to muon capture, the weak interaction is very close to beingpointlike. The initial muon and proton must be coincident with the final muon neutrinoand neutron as is made manifest in the matrix elements for the process,M (3H I exp(—i)(r)M 3He) (5.1)where the s-wave muon wavefunction ,(r) and neutrino wavefunction exp(—i.P) areevaluated at the initial/final proton/neutron coordinate. M() is the muon captureamplitude for capture by the particle labelled (i) and has the form.M 7(1— UL.V) (Gv + + GA.) + smaller terms. (5.2)The standard procedure in evaluating the matrix element in equation (5.1) is toremove the muon wavefunction and replace it by an average value () where the averageis over the nuclear transition density appearing in the matrix element M. At this pointno approximation has been made. The value of will be of the order of a fermi becausethat is the size of the trinucleon systems. Since ,1(r) varies slowly at small r, iswritten in terms of the value of the Bohr muon wavefunction p(r) evaluated at r = 0thus,M 2 C (0) 2 (3H I exp(—i)Mj 3He) 2 (53)where, (0) 2 4(Zmred)3 (5.4)73Chapter 5. Muon Wavefunction Overlap Reduction Factor 74and 11red is the reduced mass of the muon-helion system. The reduction factor C expressesthe reduction in p,(r) as r increases from zero. For small nuclei such as 3He C is closeto 1. The evaluation of C is the subject of this chapter.Consider the evaluation of M 2 ignoring the r dependence of the muon wavefunction. The rate receives contributions from the spin-flip and non spin-flip reduced matrixelements giving,J MJ2= G[1]2+ G)[6]2 (5.5)spinsi ri2 rI 2wnere,‘-(1)2GA + (GA — Gp)2 (5.6)and, [1] = (H Ijo(vr)3He)[6] = (3Hf J(i)j0(vrj)3He). (5.7)In general, the r-dependence of the reduced matrix element [1] will not be the sameas that of [6] so that C is a combination of a spin-flip correction factor C() and a nonspin-flip correction factor C(1).C= C(flG1)[1]2+ )G[6j2 (5.8)G)[1] +The exact definitions of C(1) and C(a) are,= jdrr2p()(r,zI) (5.9)where takes on the values 1 and u and the density is the pertinent density arisingin the reduced matrix elements.[1]=dr rp(l)(ri, ii) x constant (5.10)[6]= f dr VP() (ri ii) x constant (5.11)1 = I drr2p(*)(r, ii) (5.12)Chapter 5. Muon ‘vVavefunction Overlap Reduction Factor 75The constants appearing in equations (5.10,5.11) are determined using the normalization condition, equation (5.12). The densities p)(r, ii) are thus trinucleon isovectordensities, not charge densities, modulated by a factor jo(I’r) from the zeroth multipoleof the neutrino wavefunction, the higher multipoles making iegligib1e contribution. Theisovector density is related to the point proton charge densities of 3He and 3H thus,3He 3HPisovector(r) = 2p () — p (r) (5.13)where both p3He and 3H are normalized to one.Explicit forms for p(1)(r, v) and p()(r, ii) are given below for the sake of clarity. Theyare found by making the choice i = 1 in equations (5.10,5.11) and using the relation=2y/3 where y is the usual Jacobi spectator co-ordiiate ( see appendix E ).fd3xdy8’12( 3H; in )I’j(2iíy/3)(j3He; in)p(l)(rl, ii)= ()3f d3xd3y( H; m )Jjo(2vy/3)(3He; in)(5.14)fd3xd3y8/2) ( 3H; in I)IL1u’)jo(2vy/3)(£I3He; in)P(g)(71,11) ()3f d3xd3y(H;mjg)Jl)ul)j0(2iiy/3)(3He;m)(5.15)In the above equations in is the spin projection of the trinucleon and can either take thevalue + or —.Table 5.1: Previous Calculations of the Reduction Factor C.C Reference0.965 Kim and Primakoff [112]0.9704 Peterson [106]0.965 Donnelly aid Walecka [113,114]The results of previous calculations are given in table 5.1. Kim and Primakoff useda square well nuclear density with radius fixed to reproduce the r.m.s. electric chargeradius of 3He. Their calculation is criticized for several reasons.Chapter 5. Muon Wavefunction Overlap Reduction Factor 761. The nuclear density should be an isovector density and not an electric charge densitythe latter being the sum of isoscalar and isovector densities.2. The charge deisity measured in electron scattering experiments includes the finiteextent of the nucleons. The density appearing in the matrix elements [1] and [8]is a point nucleon density.”:3. The dominant contribution to the rate comes from the spin-flip matrix elementand so it is the magnetic radius rather than the charge radius which is the morepertinent parameter.4. The effect of the neutrino wavefunction is ignored.5. The density used is uiwealistic for the 3He-3H system which has a long tail inconfiguration space.6. The pertinent parameter for the nuclear density is not the second moment (r2) butrather the first moment (r).The last point follows from the fact that the nuclear density is much smaller than themuon orbital. The muon wavefunction is approximately a Bohr type orbital exp(—r/a,)with a, = 1:32.78 fin and for small values of r behaves as.exp(—r/a) = 1 — v/a1, +r2/2a +... (5.16)so that the value of is given approximately by,1- (r)/a. (5.17)aThe spatial distribution of the nucleons has already been included via the nucleon form factorsappearing in the effective couplings Gv,GA and Gp. To include this spatial distribution in the p()(r, ii)would be inconsistent.Chapter 5. Muon Wavefunction Overlap Reduction Factor 77Since (r) < a, the higher order terms in the expansion (5.16) will give oniy a smallcorrection to C so it is important for a model nuclear density to reproduce the firstmoment of the real nuclear density. The density taken by Kim and Primakoff was notrealistic and so has a poor first moment. This point can be illustrated quantitatively.A square well of root mean square radius 1.88 fm has a first moment of 1.82 fm. Thecharge distribution of 3He measured by McCarthy, Sick and Whitney [115] has a rootmean square radius of 1.88 fm and a first moment of 1.70 fm. An exponential distribution,which matches the asymptotic form of the trinucleon system ), with a root mean squareradius of 1.88 fm has a first moment of 1.63 fm.The calculation of Peterson used a realistic 3He charge density to find the non-Bohrmuon wavefunction numerically but it is not stated whether the overlap of this wavefunction was taken with an isovector density or a 3He density. However, taking the overlapof an unperturbed Bohr orl)ital with the 3He charge density given in the paper givesC = 0.9749. This result suggests that the overlap was indeed taken with the 3He chargedensity since the perturbed muon wavefunction will be diminished near r = 0 and thisleads to a slightly smaller value of C. The effect of the neutrino wavefunction was notincluded. The calculation of Peterson can thus be criticized for the reasons (1),(2),(3)and (4) given above.The calculation of Donnelly and Walecka usedl a realistic 3He charge density to finda relativistic muon wavefunction and took the overlap of the muon probability densitywith the pertinent nuclear transition density i.e. exactly that density arising in theirmatrix elements. However, their work can be criticized on the grounds that the nucleartransition density is based on a (i.s) shell model configuration of the trinucleon boundstate. Apart from lacking any two particle correlations, the model neglects the d-stateof the bound state which represents 9% in probability.Our knowledge of the trinucleon bound states has increased greatly over the last 15Chapter 5. Muon Wavefunction Overlap Reduction Factor 78years and this allows a better calculation of the correction factor C. The method thatwas used to calculate C is given in the next section.5.2 MethodAn accurate trinucleon isovector density for the case v = 0 was found from the 3He and3H wavefunctions of Kameyama et al [18]. This density yields a coarse result for C whichwas fine tuned by applying the following corrections.• Perturbation of p(r) due to non-pointlike nature of the 3He nucleus.• The effect of relativity on p(r).• The effect of the neutrino wavefunction.The coarse result employs a Bohr atomic wavefunction and ignores the presence ofthe neutrino wavefunction which is equivalent to setting the magnitude of the threemomentum transfer to zero. The corrections were calculated using a model transitiondensity which allowed systematic study of the corrections. The model density which issupposed to represent the p(*)(r,O) was taken to be,p7n(7) = exp(—r/a). (5.18)The value of a was set by exact calculations of the first moment of the isovectortransition density at zero momentum transfer. Since there are two matrix elementswhich arise in muon capture, there will be two values of a. They correspond to thenon-spin flip amplitude [1] and the spin-flip amplitude [5]. Using the antisymmetry ofthe nuclear states to simplify the matrix elements, the values of a(1) and a() are givenChapter 5. Muon Wavefunction Overlap Reduction Factor 79by,(3Hjj A3)UH(r) fdrr3p( ) = :3a(1) (HIHe) 1.661 fm(5.19)3a(a)—— 1.536 ITE(3HWJ3He)where the particle label (j) on the operators can be 1,2 or 3 and the matrix elements arereduced in spin space but not in isospin space.Since the Bohr muonic wavefunction is of the same exponential form as the modeldensity, there arises a dimensionless scale parameter ‘s’ which is the ratio of the size ofthe nuclear density to the size of the muonic orbit.= a(4)/a (5.20)From equations (5.19) we have,S(i) 4.17 x i0and, 3.86 x l0 (5.21)The scale parameter s is small and expansions in this parameter converge quickly.For example, consider the overlap of the Bohr wavefunction with the model density.= f dr2p(r) = 1 — 3 + 6s2 — lOs3 +... (5.22)0The term 692 contributes at the 1 x iO level. Since the model density has a reasonably realistic sha1)e the second moment will be well reproduced. This implies two things.First, the modiei density can be used to arrive at the coarse result for C and the levelof accuracy will be 2 x iO at the very worst. Second, all the corrections to the coarseresult depend on the nuclear density at small r i.e. r the size of the nucleus 1—2 fm.If the model density reproduced all the moments (rn), ii 0, 1,2 ... then it would equalChapter 5. Muon Wavefunction Overlap Reduction Factor 80the real density. Although the model density does not do this, the important momentsare for small values of n since we hope to reproduce the small r behaviour. Fitting themodel density to the smallest non-trivial moment not only gives an accurate coarse valuefor C, but also ensures that it is realistic in the iml)ortant region.The coarse results for C(1) and C(0 at zero momentum transfer, using a Bohr muonwavefunction and realistic trinucleon isovector densities to set the values of (i) and 8(u)are,C(1) 0.9753 (5.23)C(j) 0.9772. (5.24)5.3 Corrections5.3.1 Perturbation of the Muon WavefunctionThe effect of the perturbation of the the muon wavefunction should lessen the value ofC. This is because the potential from an extended charge distribution is less attractivethan that from a point charge. The muon is thus more likely to be found further awayfrom the nucleus which corresponds to a greater amplitude for large r. By normalizationconstraints the perturbed wavefunction is smaller than the unperturbed wavefunction atsmall r and this implies a smaller value of C, ( remembering that the correction factorC is defined with respect to the value of the unperturbed wavefunction at 7’ = 0 ). The3He charge density is taken as,p3He(7.) =exp(—r/a) (5.25)with ac = 0.543 fm fixed by the r.m.s. charge radius of 311e which is 1.88 fm [77, tableII]. The scale factor for the charge density is thus,s = = 4.09 x i0 (5.26)a1,Chapter 5. Muon Wavefunction Overlap Reduction Factor 81which is close to both S(i) and S). The potential for such a charge distribution is foundby solving Poisson’s equation analytically.V2 = (5.27)where, V = electric potentialp = charge density = (Ze)p(r)= electric permittivity of free space (5.28)The method of solution is as follows.1. Fourier transform the potential and charge density.2. Solve Poisson’s eciuation algebraically.3. Fourier transform the potential back again.The result that was found is,V(r)=[1- CXIJ(V/dic)] - exp(_r/ac)}. (5.29)The potential entering into the Schrödinger eciuation for the muon wavefunction is—eV(r) which has a piece like 1/i’ and also other pieces which express the non-pointlike nature of the nucleus and ca be treated as a perturbation. The perturbing potential is then,AH (Zn) exp(—r/ac) ( + (5.30)and the perturl)ecl wavefunction is expressed as,i)=(r)°(i) + a(nlm)i(r)Y7) (5.31)(nim) (100)Chapter 5. Muon Wavefunction Overlap Reduction Factor 82(nlrnIAH 100)where, a(nlm)= (5.32)—Ejm = unperturbed energy eigenvalue for state I nirn)(‘7 \2 (‘7—— L) “red—— (• 33)2n— 2anSince H is not a function of i, only the S states are mixed. The general form of thes-wave Bohr orbitals is [116],noN(n)exp[—r/(na)], F1 (1 — n; 2; 2r/(na1)), N(n) = (4/n3) (5.34)(a)2where 1F (; 1; z) is the confluent hypergeometric function. By evaluating the matrixelement in equation (5.32) it was found that the coefficients a(n) a(nOO) are secondorder in the scale parameter s.N(n)8 535) c(ii/2)The change in is thus,= a(n)N(n) J dr rp(,)(r, ii) exp[—r/(na)JiFi (I — n; 2; 2r/(na))0a(n)iV(n)[1 + O(s)]. (5.36)n1Ec1uation (5.36) follows from the normalization condition, equation (5.12), after expansion of tile muon wavefunction and the confluent hypergeometric function for smallr. Thus, to second order ill,= —8s 722 = —8s = —0.0001 (5.37)n=2The sum over n was performed by expanding into partial fractions and observing thecancellation of all but two of the terms. The effect of the non-pointlike charge distributionon the muon wavefunction thus affects C at the 2 x i0 level and corrects= 1_3s+62to = 1 — 3s — 2.s2 (5.38)Chapter 5. Muon Wavefunction Overlap Reduction Factor 83where use of the proximity of .s to (i) and has been made.The above calculation was repeated with a charge density of slightly different formr2 exp(—r/a) as used by Peterson [106]. In that case the perturbing potential is,= (Z)exp(-r/a2[1+ ( + + 18)] (5.39)with a2 0.34:3 fin which is fixed by the r.m.s. charge radius of 3He. The perturbationcorrects by —20a/a —0.0001 which is the same as for the exponential chargeclensi ty.5.3.2 Relativistic EffectsThe size of the effect of relativity on the muon wavefunction should be roughly (Za)2 =2 x iO since Zn’ is the order of magnitude of the speed of the muon in its IS orbit.This corresponds to a contribution of order 2 and, depending upon the exact details ofthe calculation, may or may not change C appreciably.The solution of the Dirac equation for a particle moving in a Coulomb 1/v potentialwill be used to calculate the overlap with the nuclear transition density. The form of thewavefunction is [116],g (r) Y( )Xcp(r) = (5.40)if(r) [Y ® XL]where g(r) is the large component, f(r) the small component and X’ is the lepton spin-ispinor with projection rn. Standard practice [117,118,114] is to redefine (7 thus,C’ — 1 dr 3He1(7)t( 15 41Pc \ I (O)2where p’(r) is the charge density of the initial 3He state. However, the overlap of thenuclear density should be taken with a muon amplitude and iot a probability density i.e.without making any assumptions or approximations. The pertineilt amplitude can beChapter 5. Muon Wavelunction Overlap Reduction Factor 84found by considering the construction of the muon capture effective Hamiltoiiian whichis the contraction of a leptonic current with the non-relativistic reduction of a hadroniccurrent.Xx M XLXN = Jj (5.42)where,•‘pto1l = u(ii)[y(l —75)]U(t) (5.43)The usual procedure is to take the muon Dirac spinor to be that of a free muon at rest,(XL\\= f (5.44)\O)which leads to,leptoIl = XL 7(1 — .i) XLJepton = xL {(1 — LL)} XL. (5.45)The matrix element of the resulting effective Hamiltonian is evaluated thus,i exp(—i )y(r)M 3He) (5.46)where contains the spinor structure and nucleon form factors etc. This is equivalentto taking the spinor for the muon to be the product of and the spinor in equation(5.44) and not inserting the muon wavefunction into the matrix element since it is alreadypresent in the leptonic current.i.e. take= XL {(i L)NR} XLeptofl XL {(i — L L)YNR} XL. (5.47)Replacing the product spinor by the relativistic spinor,( g(r)°()X1’i.e. i —f = i i (5.48)0 ) \\ if(r) [}‘ c XL]im IChapter 5. Muon Wavefunction Overlap Reduction Factor 85the lepton current becomes,‘ePton x {(i - L.,)[g(r)y() — if(r)Yi()]} 0 XLJiepton 4 {(i .i)(_)[g(r)Yo() — if(r)Y)]} ® XL (549)where the angular dependence on is understood to couple to total angular momentumone half. By comparing the al)Ove with equations (5.47) it is seen that the effect ofrelativity can be taken into account by making the substitution,NR()y0()0 XL [g(r)Yo() — if(r)Y1()] 0 XL. (5.50)This leads to the following relativistic definition of C.CREL—j, 2 (0)2 + (ã)2 (5.51)where,= j dr2p(r)jo(vr)g(r)Cf= j dr7(r)ji(iir)f(r) (5.52)The density appearing in the definition of C has the factorj1(vr) rather than thejo(vr) because the small component of the muon Dirac spinor has orbital angular momentum I . Conservation of total angular momentum along with a parity selection ruleimply that the zerot.h order multipole of the neutrino wavefunction makes no contribution to Cj and the first non-trivial multipole is of order one. The functions g and f are[116],g(r)= () (r2 ‘ 1)) exp(-r/a)(2r/a’= C g(r) (5.53)where, 71 = [1 — (Z)2]= [1 + (Za/7l)2j_= 7i• (5.54)Chapter 5. Muon Wavefunction Overlap Reduction Factor 86A useful small parameter is 6 = (Z)2 2.13 x iO. Expressing everything in termsof 6 we have.g(r)2 i/4)exp(_r/a)(2r/a2f(r) (-)(6/4)g(r) (5.55)1—6/2I— 6/2. (5.56)Consider the size of c compared to that of 0g First replacej1(z’r) byj0(vr) in theclefiuiition of C1. This will give an upper limit for Gf since the factor ji(vr) will reduceG1 considerably.)2 < (1_—6/4 = 5 x 10 (5.57)(2 \1 +€JThus, C’1 can safely be neglected since its contribution to C is less than 1 x i0.Evaluating C’q using the model density prn(r), the solution of the Dirac equation for apoint-like iucleus given by equation (5.53) and neglecting terms smaller than or equal to6/4 it was found that.C (2s)(1— 3s + 6s2). (5.58)The factor (2.s) comes from the logarithmic divergence of g(r) at small r. This enhancesC and the effect is of the order —6 ln(2s) = +1.0 x 10 since,(2s)8 = 1 — 61n(2s) + [61n(2)j2/2— ... (5.59)However, it is not clear that equation (5.58) is the correct result due to the followingargument. The enhancement of C is due to the 1/rS/2 enhancement of the relativisticwavefunction g(r) over the non-relativistic Bohr wavefunction(v)= ()2exp(r/a)Chapter 5. Muon Wavefunction Overlap Reduction Factor 87() 2 exp(—r/a) x (2r/a)6/(1 — 0.;346 + 0(62)) (5.60)In arriving at equation (5.60) use was made of the gamma function expansion,F(:3— 8) = F(3)(1 — 6’(3) +...) (5.61)where i/i is the digamma function having the value,=—‘y + 3/2 0.92, = Euler’s constant = 0.57722. (5.62)Is the singularity at r = 0, ( and hence enhancement at small r ), peculiar to therelativistic nature of the Dirac equation or to the singular nature of the Coulomb potential1/r? i.e. when a more realistic non-singular potential is used in the Dirac equation, isthere still an enhancement of g(r) over cp(r) at small r’? If not then the result (5.58)will not be correct.The answers to the above questions were found by numerically solving the Diracequation for the electric potential given in equation (5.29). The calculation is detailed inthe next section.5.3.3 Numerical Solution of the Dirac Equation for an Extended ChargeDistributionThe coupled differential equations for the functions g(r) and f(r) are,= [m + F + eV(r)jf(r)—[mre — F — eV(r)jg(r)— 2f(r)/r (5.63)where E is the energy of the solution equaling the reduced mass minus the bindingenergy. The exponential decay of the functions g and f was factored out by defining newChapter 5. Muon Wavefunction Overlap Reduction Factor 88functions G and F thus,g(r) = exp(—r/a1L)G( )f(r) = exp(—r/a)F(r) (5.64)and the equations that were actually solved are,= + E + eV(r)jF(r) + C(r)/a= [mred - E - eV(r)]G(r) + F(r)(1-(5.65)7 7with boundary conditions,G(r) —* r’1 as r —* 00F(r) (—)r’’ ()2 as r 00. (5.66)Fourth order Runge-Kutta extrapolation b was used to solve these equation starting atr = 20 fm and using variable step size down to r 0.02 fm. The method was checked bysolving for the Coulomb J)otefltial 1/r and comparing the result to the analytic solution.Agreement was found at the i09 level for G and the i0 level for F at the finalr = 0.O2fm point.When solving equations (5.65) with the non-singular potential the energy of the solution was taken to be,E/mreci = E1 + (Zo’)28s (5.67)which is the Coulombic solution (ti) plus an estimate of the change in binding energyfound from first order perturbation of the non-relativistic solution.AE = ([HI) (5.68)= 4 j dr r2 exp(—2r/aL) exp(—r/ac) ( + (5.69)fl7red(ZQ)8.Sc[1 + 0()] (5.70)bSee Numerzcal Recipes by W.H.Press, B.P.Flannery, S.A.Teukolsky and W.T.Vetterling page 550 foran explanation of this technique.Chapter 5. Muon Wavefunction Overlap Reduction Factor 89B taking the overlap of the solution for G(v) with the model density pm(r) thecorrection factor C() was found to be,REL-‘perturbed — 0.. 0 5. 1which is equal to the non-relativistic result (0.9772) including the efFect of perturbation(—0.0002). It is concluded that the effects of relativity on the value of C can be neglectedat the i0 level.5.3.4 Neutrino WavefunctionThe nuclear transition density includes the factor jo(iir) stemming from the neutrinowavefunction. The presence of this factor gives a value of C which depends on the valueof the three momentum transfer v. In this section, the result for C is calculated as afunction of i-’. Using the model density the dependence of C on v is,= f dr r2 exp(—r/a)p7(r)jo(iir)(1 +v2a). (5.72)The (1 + ii2a) factor comes from the renormalization of the total density p(4)(r, v)according to equation (5.12). After expansion of the muon wavefunction for small r theintegral is easily performed and it was found that,=— 3(1 — 2/.3)+ 6s2(1 — 2)(1 + 712) (1 + 7/2)2where, 1 = vu. (5.73)The apparently paradoxical result that the correction factor becomes larger than onefor large v is understood by noting that jo(vr) becomes negative for ir < zir < 27r so thatthe effect of the decreasing muon wavefunction in this region is to reduce in magnitudethe negative contribution to the matrix element thus increasing its overall value.Chapter 5. Muon Wavefunction Overlap Reduction Factor 90For the 3He —* H transition the neutrino momentum i is 103.22 MeV which gives= 0.29 for a(1) and = 0.27 for a(). The correction to the term of order s2 corrects Cby less than I x i0 and this correction will be neglected.The changes in C due to the neutrino wavefunction are,C()(0) = 0.9753 C(l)(i) = 0.9778C()(0) = 0.9772 C()(i) = 0.9792 (5.74)and thus the effect of the neutrino wavefunction is to increase C(i) by 0.0025 and C()by 0.0020 which are significant changes. When combined with the other corrections, thiscorrection will be taken into account by modifying the coefficients of .s thus,(1) = 1 — 3st(l) + U(s2)= 1— 3st() H- Q(2) (575)where, t(i) = 0.897t() =_____= 0.911. (5.76)5.4 SummaryIncluding relativity, neutrino wavefunction and muon wavefunction perturbation the expression for C(*) i5,C() (1— 6s(&)t() + 5%)) (5.77)Inserting the numerical values,C(1) = 0.9776C() 0.9790. (5.78)Chapter 5. Muon Wavefunction Overlap Reduction Factor 91Table 5.2: Summary of Corrections to Coarse Result.Effectneutrino wavefunction +0.0020muon perturbation —0.0002relativity 0.0000In the impulse approximation, the matrix elements [112 and [8]2 contribute to the totalcapture rate in the ratio 1792:8208. The elementary particle model however suggeststhat the ratio of spin scalar to spin vector contributions to the rate is 1602:8398. Thepertinent combinations of C() and C() are then,= 0.1792C(I) + 0.8208C() = 0.9787eff= 0.1602(7(1) + 0.8398C() 0.9788 (5.79)and the latter result will be taken for the sake of definiteness. The sizes of the variouseffects are given in table 5.2.In comparing this result to the other calculations the relevant ciuantity is the deviationof C from one, ( since this is what is calculated ), even though it is the absolute value of Cwhich affects the muon capture rate. The deviations from one of the 1)revious calculationsare 0.035 [112], 0.0:30 [106] and 0.035 [113,114] which should be compared to 0.021. Thepresent result is thus significantly different from previous calculations. In heavier nucleiwhere the deviation of C from one is large, the theory used here may yield a significantchange in the absolute value of C itself.The difference in the value of C obtained in the present calculation compared withprevious work, is due mainly to the fact that previous calculations have overestimatedthe first moment of the pertinent density. There are two important points which led tothe small value of (r) used in the present work.Chapter 5. Muon Wavefunction Overlap Reduction Factor 92• It is the spin-flip density which makes the dommant contribution to Cs’. The spin-flip ( or magnetic ) radius is smaller than the non spin-flip ( or charge ) radius.• The density used herein is a point nucleon density which is smaller than a realdensity available to experiment. Thus, in previous calculations of C which basedthe nuclear density on measured charge densities, the smearing of the charge dueto the structure of the nucleons was implicitly included and this led to a densitywhose first moment was too large.Further, the effects of the neutrino wavefunction had not been included in previouscalculations.While it is the isovector densities which are relevant for the 3He— 3H transition,other transitions will involve different densities. For example in calculating the inclusiverate in the closure approximation, the relevant density is the electric charge density ofthe initial state and this should be taken with the muon probability density and not themuon amplitude.To conclude, the muon wavefunction overlap reduction factor is found to he 0.9788which is larger than previous calculations.Chapter 6Muon Induced Break-up: the Deuteron Channel6.1 IntroductionAs well as the quasi-elastic reaction +3He — ii, -b 3H, there are two break-up channelswhich are open to muon capture by 3He. They are,j+3He ,v + p+n+n (6.1)and are referred to as the cleuteron and proton channels respectively, the label beinggiven by the electrically charged, and hence easily observed, particle in the final state.Phillips et al.{47] have calculated the total rate for each channel in the impulse approximation using the Arnado model for the final states with s-wave NN interaction onlyand a phenomenological 3He wavefunction. Those authors found,f(dllV)tiieor = 414 s_IF(Pllh1l1)theor = 209 (6.2)Until recently the only experimental results available were for the total break-up rateand total neutron rate.F(dnii1.+ pnnzI,)x1). = 660 + 160 s reference [119]= 665 reference [65] (6.3)F(dni, + 2pnnviL)exp. 1200 + 170 s reference [120]93Chapter 6. Muon Induced Break-up: the Deuteron Channel 94While the total break-up rate agrees with theory, the total neutron rate does not.Experiment E569 conducted at TRIUMF in the summer of 1990 has measured energyspectra of 1)rOtOflS and deuterons with high momentum [121]. There are eight data pointsfor the proton channel which span a range of proton momentum from 180 MeV to 320MeV. There are only three data points for the deuteron channel. One is at deuteronmomentum 275 MeV and the other two are at deuteron momentum :325 MeV. Theintegrated deuteron and proton rates cannot be deduced from these measurements sincenot enough of the spectrum is covered.The calculation described in this chapter is of the deuteron spectrum in the planewave impulse approximation (PWIA). The formalism used is similar to that used inelectron induced l)reak-up reactions and the present work represents the first step inraising the level of sophistication of muon induced break-up calculations to that of theelectron case.6.2 KinematicsTo kinematically define the final state requires two variables. This result is arrived at asfollows. There are three particles each with three independent momentum componentswhich gives nine unknowns.• The 3 particles must be in a plane. Set one coml)Onent of each momentum to zero6 unknowns.• There are now 2 momentum conservation equations and I energy conservationequation 3 unknowns.• There is a trivial rotation in the plane 2 unknowns.Chapter 6. Muon Induced Break-up: the Deuteron Channel 95The following list defines the kinematical variables appearing in the formalism.= helion 4-momentum = [M3He, 0]= muon 4-momentum = [m,,0]= neutron 4-momentum = [Es, 17]= deuteron 4-momentum = [Ed, d]= neutrino 4-momentum = [ii, 17]= 4-momentum transfer = — = [Qo, Q]pa= 4-momentum of struck proton = [E,y7] (6.4)The two kinematical variables that will be used to define the final state are the magnitudeof the deuteron momentum d and the cosine of the angle between the deuteron and thethree momentum transfer x = d.Q.Using energy and momentum conservation we have,Matom = Ed + E11 + Q(6.5)where, Ed (j2 + inj2)(2+ m112) (6.6)Equations (6.5) lead to the following expressions for the ma.gmtude of the three momentum transfer Q and neutron energy E1, in terms of d and x.QQI,x)= (1 — d.x//3) (6.7)=— (1 — 1/3)] (6.8)where, /3(d) = Maton,— Ed (6.9)(d) = [1- +miñ] (6.10)Chapter 6. IVIuon Induced Break-up: the Deuteron Channel 96Figure 6.1: The four momentum transfer squared q2 as a function of deuteron momentum.1.00.50.0—0.5—1.0Figure 6.2: Figurative diagram of the PWIA.LIw±0 100 200 300 (MeV) 400Chapter 6. Muon Induced Break-up: the Deuteron Channel 97The deuteron momentum is constrained to be less than :356.35 MeV, the upper limitcorresponding to the situation where deuteron and neutron are emitted back to backand the neutrino has very small momentum. When the cleuteron and neutron havezero relative momentum the neutrino has its maximum energy of 97.18 MeV. The fourmomentum transfer squared q2 is given by,q2 = m12(1 — 2Q/m1j (6.11)and ranges from —O.84m12 to +m2. Its value as a function of d is shown in figure 6.1.The range of values of q2 for each value of d corresponds to the variation of x from —1to + 1. Unlike the quasi-elastic chaimel or any kinematical region available to electronscattering, q2 may become time-like, (i.e. q2> 0 ). This allows the interesting possibility,as in radiative muon capture, of finding observables sensitive to the value of gp whichaccording to PCAC should rapidly increase in value as q2 approaches +m2.6.3 PWIAThe formalism of PWIA is a great. simplification of the full problem l)ecause of thefollowing three approximations.1. The strong interaction between the neutron and cleuteron in the final state is ignored. The final state wavefunction is then a product of a plane wave for theneutron and the free deuteron wavefunction.2. The weak current is taken to be that from the nucleons only i.e. meson exchangecurrents are neglected.:3. Only the direct nucleon knock out amplitude is included. The direct deuteronknock out amplitude is ignored.Chapter 6. Muon Induced Break-up: the Deuteron hanne1 98A figurative diagram which describes the PWIA is shown in figure 6.2. To put thePWIA in perspective it is useful to make a diagrammatic expansion of the matrix element.Let us represent the fully antisymmetrized iiitial 3He state by the symbol,1i)= 2 He (6.12)3where the numbers label the nucleons. The final state must also he fully antisymmetricand using the permutation operators P2 and P3 can be written,1) = (1 + P2 +P3)f)(1 (6.13)where f )1 is the Faddeev component and is antisymmetric under interchange ofparticle labels 2 and :3. The symbolic representation of the Fadcleev component is,1 1(i)(f I= H + (6.14)where the first symbol represents the disconnected part of the wavefunction ( planewaves ) and the second the connected part.In general, the current has one-body two-body and three-body parts where the n-bodycurrent depends simultaneously on the coordinates of n particles. These currents arerepresented symbolically by a wavy line attached to one, two or three nucleons accordingto the nature of the current.Using the permutation symmetry of the current and also the final and initial wavefunctions, the matrix element can be simplified to,M= (fIJI) (6.15)= :3()( fIJIi). (6.16)Chapter 6. Muon Induced Break-up: the Deuteron Channel 99This matrix element is represented in figure 6.3. The PWIA corresponds to includingoniy the top left diagram (la) in evaluation of the matrix element and thus representsoniy the very first stage of a complete calculation.Diagrams lb and ic represent direct deuteron knock out and diagrams 2a-2c representthe connected one-body amplitude which includes the effects of final state interactions.Diagrams 3a-3c and 4a-4c represent the disconnected and connected amplitudes for thetwo-body current and diagrams 5d and 5e represent the disconnected and connectedamplitudes for the three-body current.It is shown in appendix H that, including only diagram la, the deuteron spectrumtakes the following form.=7dp2(d)j dx M 2 (d, x) (6.17)In equation (6.17) p2(d) is the two body break-up momentum distribution for 3He andequals 3 times the probability of finding a deuteron of momentum d and proton withequal and opposite momentum when observing a 3He nucleus at rest. aThe factor y is,G2Vud2 C2 (o)I2. (6.18)0F is the Fermi coistant and Vd is a Cabbibo-Kobayashi-Maskawa matrix element[62][63] linking the up and down quarks. C2 is the muon wavefunction overlap reductionfactor for two body break-up and is a function of d.If the x dependence of M 2 is ignored then the integral over x can be performedanalytically. In the final calculation this integral was performed using Gauss-Legeidrenumerical integration including the x dependence of M 2 but the deviation from theaThe factor three arises because of the normalization of the bound state to one. The probabilityof observing a proton in 3He is thus 2/3 and this must be multiplied by 3 to find the total number ofprotons. This factor of three arises naturally when properly antisymmetrized states are used and is notinserted by hand.Chapter 6. Muon Induced Break-up: the Deuteron Ohannel 100+ +34 H0d+e5 H H°(a b C+12+Figure 6.3: Diagrammatic expansion of the matrix element.Chapter 6. Muon Induced Break-up: the Deuteron Channel 101approximate analytical result was found to be 5% at the most. The analytical result is,7d2p(d)/3k M 2 (d, 0). (6.19)where k is a kinematical factor close to one.k — 1ic(1 +d2/3,9)6 20- (1-d2/)-.As the deuteron momentum ranges over zero to its maximum value, tc varies from0.09 to zero, k varies from one to 1.14 and /3— m, — m varies from —5.5 MeV to —-39.1MeV.6.4 Two Body Break Up Momentum DistributionThe two body break-up momentum distribution p2(d) is a theoretical construct centralto the PWIA formalism. It is defined exactly by equations (H.18) and (H.16) but can bethought of simply as the square of the overlap of a proton of momentum —d and deuteronof momentum d with a 3He nucleus.p2(d) H ( proton(—d), deuteron(d)l3He) 12 (6.21)A calculation of p2(d) was performed using the 3He wavefunction of Kameyama et al[18] and the deuteron wavefunction from the Bonn Potential C which was expressed inthe same Gaussian basis as the 3He wavefunction, ( see chapter 2, section 3, equation(2.31) and also appendix F ). The final expression used for p2(d) was,p2(d) = {E(°)(d) + 2E(2)(d)} (6.22)Chapter 6. Muon Induced Break-up: the Deuteron Channel 1026(la, l)(Aa, (ia, )6(Sa, 1)2a,1=O,2la Lawhere, E(d)= x[6(2Sa + l)(2La + 1)] 1 Sa . (6.23)1 12 2X f dp 2(l) (P)’He(P dl)The above expression follows from equations (11.16) and (11.18) in appendix H and canbe shown to equal that of Meier-Hajduk et al [122] by making the following transformationfrom jj to LS coupling.= ((ic a)La,(saa; XS L,la ‘\a La[(2ja + l)(2ta + 1)(2Sa + 1)(2La + 1)] a Sa (6.24)Ja kaThe form of I1(p, d) is given in section 5 of appendix C. The results of the calculation for 8-channel and 22-channel 3He wavefunctions are shown in figure 6.4. Also shownare results of electron scattering coincidence measurements 3He(e, e’p)d which have beenused to infer the value of p2(d) in the context of PWIA theory. It should be noted thatthe ‘experimental’ values for p2(d) rely entirely on the validity of PWIA and thus are inerror due to the presence of final state interactions (FSI) and the contributions of mesonexchange currents (MEC). The calculations of Laget [125] indicate that the inferred valueof p2(d) is expected to be up to 50% too small because of FSI and 10% too large because of MEC in the kinematical range of the experiments. These numbers indicate thesize of the uncertainty involved in inferring p2(d) from electron scattering experimentsrather than being precise corrections that could be applied. Bearing this is mind, theexperimentally inferred p2(d) agrees reasonably with the theoretical calculation. TheChapter 6. Muon Induced Break-up: the Deuteron Channel 103Deuteron momentum d [MeVIFigure 6.4: Two body break-up momentum distribution for 3He. Data are taken fromtwo experiments performed at Saclay [123,124] in 1982 and 1987. The straight line is afit to the data.101020 1982• adata‘1987 data1fit to data8 channel 3He.22 channel 3He10°ND>0CS.101 -o(NQ1020 100 200 300Chapter 6. Muon Induced Break-up: the Deuteron Channel 104enhancement of the theoretical value over the fit to the data ranges from one, i.e. noenhancement at the zero deuteron momentum point d 0 to 2.4 at d = :350 MeV. Theform of the fit was,p2(d) exp[—ad + b— 25 d in MeV (6.25)and a least squares analysis led to the following values for the free parameters a, b and c.(1 = 3.40 x l02MeVb = 7.5c = :30 MeV (6.26)The theoretical results for 8 and 22 channel 3He wavefunctions deviate from eachother for d > 200 MeV. This is due to the fact that the higher partial waves are absentin the Faddeev component of the 8 channel wavefunction. Although these higher partialwaves are present in the fully autisymmetrized 8-channel wavefunction, they are poorlyconstrained by a Schrodinger eciuation with a Fadcleev component containing only thelower partial waves. The 22-channel wavefunction thus represents a better solution ofthe nuclear three body harniltonian for large momentum.The theoretical value of p2(d) is well reproduced by the form,pl1(d)= exp[—at1’d+ bt1’— d + 25 (6.27)where, = 3.23 x l0_2 MeV= 7.9cthl 37 MeV. (6.28)Chapter 6. Muon Induced Break-up: the Deuteron Channel 1056.5 Muon Wavefunction Overlap Reduction Factor for Two Body Break UpThe muon wavefunction overlap reduction factor for two body break-up C2 is not thesame as as the reduction factor for the quasi-elastic channel C as discussed in chapter 5.This is because the spatial distribution of the matrix element for the3He(t, vd)n processis not the same as in the3He(t,v1)3H case. The neutroll produced in the former case fliesaway from the recoiling deuteron and, in plane wave approximation, is equally likely tobe found in any fixed volume of space. In the quasi-elastic process however, the neutronis captured by the strong nuclear forces of the other two nucleons and a triton is formedwhich confines the neutron to be within a fermi or so of the centre of mass of the Hnucleus. Thus, C2 can be expected to be less than C since the nuclear density has largerspatial extent for the deuteron channel.The matrix element which appears in the PWIA is,M= ( ns; dJcj j(l)I() exp(—i 1)(rj)3He;J). (6.29)The final state has the following representation in configuration space using the decoupledspin/isospin basis {€, } introduced in appendix H.( dJ(j 7y) = eXp(_iri*.y) x (1Jd irni; 17fls(X)1(X)x6(.s,s11)(i€,—)6(s237, 1)(i237O)6(mj, 0) (6.30)where i is the neutron momentum in the nd CM.9 1—= (i7— d) (6.31)Combining the 7 dependence of the fluial state with that of the neutrino wavefunctiongives an overall exponential modulating factor which was found to be a function of d onlyafter the momentum conservation condition given in equation (6.5) was used.exp[i(-i7.1- j*)] = exp[i( - + = exp[icJ (6.32)Chapter 6. Muon Induced Break-up: the Deuteron Channel 106Following a calculation similar to that for p2(d) and averaging the matrix elementsquared over all deuteron angles it was found that,1 dd 2f M2= 4 x f dyy2E(A)(y)L(y)jA(dy) . (6.33)JsflJa S11) A—0,2This implies the following definition of C2.C012 +2C1= 12 1212(0) (2)where,‘(A) f dy2E(y)j(dy) (6.35)f dy E(A)(y)j(dy) exp(—2y/3a) 2C(A) =°° 2A (6.36)10 dyy (y)JA(dy)The functions E(A)(y) are given by equation (6.23) with the variable q replaced byand the dummy variable p replaced by x. They are equal to the cl-3He overlap functionsu(y) which are used to determine the 3He asymptotic normalization constants Cs andC, ( see [18, equation 4.7, figure 71b and [126] ). In calculating E(x)(y) the deuteronwavefunction from Bonn l)otential C was used.Results for E’() are shown in figure 6.5 ( y = 3r/2 in the three body CM ).According to the theory of chapter 5, the pertinent properties of these densities are theirfirst and second moments (r) and (r2).(r) — f dr r3E(A)(r) (72)(A) f dr r4E(A)(r) (6 37)—j dr2E(>’)(r)— J° dr2E(”)(r)The first moments (r)(°) and (r)(2) were found to be 3.42 fm and 3.14 fm respectively.The value of (J(2)/I(o))2 was found to he 7 x iO and given the proximity of (r)(2) to(r)(o) the safe approximation C2 C(o) was made.bTIe author is grateful to M.Kamirnura who provided numerical values of the d-3He overlap functionsfor the sake of corriparison to this calculation.Chapter 6. Muon Induced Break-up: the Deuteron Channel 1070.350.300.25- 0.200.150.100.050.00—0.05Figure 6.5:dash-clottedA fit to E(o)(i) was made so that the integrations in equation (6.36) could he performedanalytically. The form of the fit was,u(r) N[exp(—r/e) + (r/f) exp(—r/g)] (6.38)where ‘u(r) reflects the shape of E(°)(r) but is normalized to one over the measure r2 dr.The free parameters d, e, f which gave the best fit are,e = 1.295f = 0.316g = 0.5423r=2y/3 (fm)cl31-1e overlap functions E(>). The solid line is the 22-channel and thethe 8-channel 3He wavefunction.(6.39)Chapter 6. Muon Induced Break-up: the Deuteron Channel 108Figure 6.6: The muon wavefunction reduction factor for two body break-up as a functionof the deuteron momentum.100 200 300 0Deuteron momentum d [MeV]A single exponential was found to be inadequate for values of the deuteron momentaabove 1.0 fm1 clue to the influence of the jo(3dr/2) factor. The correction factor C2 wasevahated by finding the first and second moments of the distribution p(r) defined by,p(r) =r2u(r)jo(3dr/2) (6.40)and then using,C2(d)=(()(2))2(6.41)The expressions for (r) and (7.2) are,(r) = 6e [] + 24f5/g {(i+)41 (642)2e [(1,)2] + 6f4/g [](.2) - 24e[(14] + 120f6/g {1+2]2e [()2] + 6,f4/g []where. i = (3da/2)2 = (3dc/2)2 (6.44)The value of 02 ranges from 0.95 at d = 0 to 1.02 at d = 356 MeV as shown in figure6.6.Chapter 6. Muon Induced Break-up: the Deuteron Channel 1096.6 Trinucleon Structure Functions6.6.1 General ConsiderationsThe hadronic tensor W describes fully the structure of the 3He —÷ dn transition withrespect to its charge changing weak interaction.2 (ilJf)J (3 - P - q)(fjJi) (6.45)A general expression for the tensor can be written in terms of five structure functions as shown below where the hypothesis of time reversal invariance has been made toeliminate a contribution like i(Pq— qP).W We(g) + + qP8M3He 2M3HeMN1P + (P+He 2 +Wie‘ I (6.46)M3HeMNThe numbering of the structure functions follows that of Llewellyn-Smith [127] andfactors 1/M3He and 1/2MN are consisteit1y associated with P and q respectively.In PWIA, W is simply related to the nucleon tensor W as shown in appendix H.W= () fd3pp2(II)W (6.47)W= () (p,sp{j n,s1)J (2 )36(fl —p —q)(n,s’jp,sp).Sn,Sp(6.48)The nucleonic tensor lends itself to the same type of expansion in terms of structurecn4functions as WHe= —Wg + +1+W + W2M2 (pq + q9) (6.49)Chapter 6. Muon Induced Break-up: the Deuteron Channel 110Let us define nucleonic structure functions T to be the W without some of thekinematical factors thus,W = T () 6(E - - Q°)/(2ir)3. (6.50)The Lorentz invariance of the T can be inferred from the following expression forthe W whose Lorentz invariance follows from equations (6.48) and (6.49).W = T2MN(q + 2p.q)/(2ir)3 (6.51)Equation (6.51) was found by making the following identification.6(E— E11 + Q°) = 2E116(q + 2p.q) (6.52)The T will turn out to be simple combinations of the nucleon form factors and thekinematical quantityq2/4MN and lead to an illuminating definition of the matrix elementsquared in contraction with the various Lorentz invariant parts of the lel)ton tensor L.Defining the trinucleon tensors Te by,W = 4 f dpp2p(p)T6(E + Eci — M3He — Q°)/(2)3 (6.53)we have the link valid in PWIA,T = fT(. (6.54)The relations between the Te and the T which follow from equation (6.54) are,( pI2 ( M‘He—j 1N + 92 ‘N I I i)lVl J \ —‘n1--pT2— T2He N\ -1—’n’---’pT3 — 3(MNHe — TNLJ/ 2( IVIN‘He ‘Nj1\ J—Jp Jinm5 m5 (MN‘He ‘N j“ ‘—‘nChapter 6. Muon Induced Break-up: the Deuteron Channel 111where the following identifications have been made.[dj3p°— E PJ 4rM— MNM3He,.j3pYp,/— + + [l72)P (6 56)J 4ir M— 3M9 M MHeThe T are found by evaluating the following sum of Dirac spinor contractions.T= () (6.57)N SpSnIn equation (6.57),/3= gv + gMiu91 +g5+ gp275 (6.58)=gv7a— + gA77—P25 (6.59)which follows from equations (6.50) and (H.6). The spinor u(p’, Si)) describes the struckproton and a priori is not known. Two choices will be made for this spinor as describedin the next two sections. The resulting theory will be referred to as on-shell and off-shellimpulse approximation.6.6.2 On-shell Impulse ApproximationThe usual piactice is to set the energy of the struck nucleon by its three momentumaccording to,= (72+ Mx2) (6.60)and to take the spinor of the bound to be that of a free particle.ju(p) = MNU(p) (6.61)Following this practice, the structure functions T were calculated using standardtrace techniques [128, p123]. These structure functions will be referred to as “on-shell”Chapter 6. I4uon Induced Break-up: the Den teron Channel 112since the four-momentum of the struck nucleon is constrained to lie on a three dimensionalsurface defined by equation (6.60) commonly referred to as the energy shell. It was foundthat,2ml(ON) 2 q \2 2=— Aff2 [gv+gM) +gA‘IIVI N2(ON) 2 2 q2 2TN = gv +gA —gM±IV1N3(ON)TN = 2gA(gv + gM)24(ON) 2 q 2 2TN = —[g +2gvgM + 2gAgp H- 2 (g + gp )j41v1Nm5(ON)— m2(ON)1N— ‘N6.6.3 Off-shell Impulse ApproximationThe struck proton is iot on its energy shell. It is a bound particle and thus does not havea single well defined value of energy because of its interaction with the other particles inthe system. Further, the presence of the binding potential implies that the total weakcurrent will consist not only of one body operators but also two body and in generalmany body operators. The vector part of the one body current is not conserved which isin violation of the conserved vector current hypothesis (CVC). This is because the CVCapplies to the full current and the one body curreirt is just part of the total current.The off-shell nature of the struck proton and the non-conservation of the vector currentare both symptoms of the same phenomenon, namely the fact that the proton whichcaptures the orbiting muon is under the influence of a strong ( with respect to thecapture potential ) binding potential.In this section, an estimate is made of the correction due to the off-shell nature ofthe struck proton. The term impulse approximation will be used to mean that onlythe one body part of the current is included with no specification of how the current isChapter 6. Muon Induced Break-up: the Deuteron Channel 113constructed so that this calculation is an off-shell impulse approximation calculation.The energy of the struck proton is set to a single value by using energy conservation.This is a consistent approach in PWIA since the absence of any final state interactionallows the following identification for the energy of the struck proton.= M3He — Ed. (6.63)For small values of d i.e. d < md the kinetic energy of the struck proton is,E— m1—EB(3He) + EB(d) —d2/2md (6.64)so that there is “missing kinetic energy” even at d 0 of 5.5 MeV. A useful dimensionlessparameter which measures the amount by which the struck proton is off-shell is definedby,= 1— 14//MN where, W (E — d2). (6.65)The effective mass W has been introduced and is a function of the three momentum ofthe struck proton 7 —d. The concept of effective mass is familiar from the theory ofelectron motion in solids. In that case the effect of the strong binding Coulomb potentialfrom the lattice on the kinematical properties of the electrons is taken into account byusing an effective mass found from the dispersion relation. Energy conservation has beenused to give an effective dispersion relation for a nucleon inside a 3He nucleus, ( equation(6.63) ) which in turn defines the effective mass. The parameter varies from 0.6% to12% as d varies from its lowest to highest values as shown in figure 6.7.To calculate the current of off-shell particles requires a dynamical model for thebinding potential. In this work, the off-shell current is not calculated but rather estimatedusing a reasonable prescription as follows.The paper of Naus et al.[104] identifies four choices that must be made in any prescription for estimating off-shell curreiits. They are,Chapter 6. Muon Induced Break-up: the Deuteron Ohannel 114Figure 6.7: The off-shell parameter 6..14.12.10.08.06.04.02.00• Wavefunction: The first step is to choose a wavefunction for the initial struckproton. If the particle was free then this would be the spinor obeying the free Diracequation. Without a relativistic nuclear wavefunction some arbritrary choice mustbe made.• Vertex Operator: For an off-shell particle, the vertex operator for the currentmust be generalized and this introduces extra free parameters which have to befixed.• Kinematics: In the usual approach, the energy of the struck particle is set by itsthree momentum to the on-shell value. There is thus a problem of energy imbalanceat the hadronic W± vertex which needs to be corrected.• Current Conservation: The full vector current is conserved. The one-bodyChapter 6. Muon Induced Break-up: the Deuteron hanne1 115part is not. Current conservation is usually put in by hand in calculations ofelectron scattering cross sections and this corresponds to adding an approximatecontribution from meson exchange currents.The four choices that are made in this calculation are given below.Wavefunct ionThe idea of using a quasi-free spinor is introduced. After finding the energy of thestruck proton by energy conservation, ( assuming no final state interaction ), and thecorresponding effective mass W() the spinor is taken to obey the following equation.6Up(p) = Wu(p) (6.66)This is similar to the free Dirac equation but the mass MN has been replaced by theeffective mass W. Multiplying equation (6.66) on the left by yields,E _2 14/2 (6.67)which shows that the replacement of MN by W is consistent. The equation (6.66) corresponds to that of a nucleon moving in a scalar potential and the time componentof a vector potential Vo where Vs and Vo are local in momentum representation and aregiven by,Vo((7) = (2 + W(q)2) - (2 + MN2)-6(MN+ 2MNVs((7) = W()— MN = —6MN (6.68)Vertex OperatorThe current is half off-shell. The term half off-shell means that only one fermion atthe vertex is off-shell rather than two. In general, the half off shell vertex operator isgiven by [104],OFF =(g7a +gZ+g2)AChapter 6. Muon Induced Break-up: the Deuteron Channel 116(g7a+ gEa+(g7a+ g q + gEa)7sA+2MN(g7a+g2 +gZa)75A_ (6.69)W+ pl a______where, A= 2Wand Z= 2MN(6.70)The form factors g are functions of three scalars which are the four-momenta squaredof each of the particles at the vertex. When both the fermions are on shell their four-momenta squared equals their masses squared which are constants. Hence, the dependence on three scalars simplifies to the familiar q2 dependence. In the half off-shell casethere is dependence on two variables which are q2 and 1472.The operators A± project out positive and negative energy solutions of the quasi-freeDirac equation and with the choice of spinor made above we have,A_u(p) = 0 Au(p) = u(p) (6.71)so that half of the terms in equation (6.69) vanish. There are six terms left, the two withcoefficients gs and g’ being second class. In the on-shell case the vector second classcurrent can be set to zero by current conservation.q20 = üqaI’iu = gs üu (6.72)2MNwhere, F=gv7a + gMza + gs (6.73)The conservation of the on-shell current follows from the gauge invariance of theinteraction which yields a Ward-Takahashi identity and eventually to the above equation.For the half off-shell current, the Ward-Takahashi identity is,= u(n’) 0. (6.74)Chapter 6. Muon Induced Break-up: the Deuteron Channel 117With the quasi-free spinor this implies that,g(q2,W)= 2MN 6[l g(q2,W)]. (6.75)It is noted that the same condition which implies the absence of second class currents inthe on-shell current also implies their presence in the half off-shell current.The choice that will be made for the value of gs is zero. The validity of this approximation is estimated by considering the contraction of q with the leptoii tensor L.Using equation (11.2) we have,qL rn—. (6.76)Any contribution from gs will thus be of order gsm/2MN compared to terms like gvor g whose size is about one. Ignoring the W2 dependence of gv and taking its q2dependence from appendix I, the size of the gs contributions are estimated to be,61Mx7’v<3%. (6.77)2MN 6A similar argument will apply to axial second class current using an extension of thePCAC hypothesis. The choice g = 0 was made.Finally, the W2 dependence of the form factors is ignored i.e. the following approximation is macIc,g(q2,W2) g(q2,MN2). (6.78)Naus [129] has calculated the W2 dependence of the electromagnetic form factors gv andg for the proton. Using the results therein [129, Figures 2.4,2.5 pp 46,47] the followingestimate of the deviation from the on-shell values of gi and g was made.= g(q2,W2)— g(q2,M)<0.1%, A(gM) A(gv) (6.79)gv (q, N)Chapter 6. Muon Induced Break-up: the Deuteron Channel 118KinematicsBy using the value of E implied by energy conservation the usual problem of energyimbalance at the hacironic vertex is circumvented.Current ConservationForcing current conservation is equivalent to making an estimate of meson exchangecurrents and only applies to the vector part of the current. In this work such an estimatewill not be made and the results can said to be in “strict impulse approximation”. Thelargest meson exchange currents contribute to the axial part of the current and a simpleestimate cannot be made since the axial current is not conserved.Now that the four choices have been made, the off-shell structure functions can becalculated. A modified Gordon identity and modified spin summation are needed.13 13= ü(n,s11)[(l— 6J2)7 — (6.80)-7%+W 16+MN(1—)u(p ,.Sp)U(p ,s)= 2E = 2E (6.8flsp p pRetaining only terms of first order in 6, the structure functions were found to be,T = [T° + 6Tj where, (6.82)2 q2= —gA +ffgM(gv+gM)‘[IVI N=0= —gAgM= 2[gi(gv + gM) + ggp}= [gjvi(gv + grvi) + gAgPj. (6.83)Chapter 6. Muon Induced Break-up: the Den teron Channel 1196.7 Matrix Element SquaredThe quantity that will be referred to as the matrix element squared is defined by equation(H.20) and is given by,T maPlvi== 2Te + Te — Te(1 — it) + Te + TheW (6.84)where the dimensionless kinematical variables it and in are,‘a = q2/m,?— 0.88 ‘a 1 (6.85)in = mJM = 0.11. (6.86)The are given by equations (6.55),(6.82),(6.83) and (6.62). The values of the nucleonform factors gv, gM, gA and gp which were taken are given in appendix I. In short, gv, gand g are measured in the spacelike q2 region and gp is inferred from g using the PCAChypothesis. Their values in the timelike q2 region are found by analytic continuationwhich can be thought of as the smoothest extrapolation possible.The value of M 2 as a function of the denteron momentum is shown in figure 6.8for both the on-shell and off-shell impulse approximation. The off-shell contributions areseen to hold the value of M 12 roughly constant and enhance its value over the on-shellcase by a factor of up to 3.8 at the high energy end of the deuteron spectrum.6.8 ResultsThe deuteron spectrum obtained is shown as a function of deuteron momentum in figure6.9 along with data from experiment E569 [121]. The off-shell impulse approximationIAOFF — solid line ) gives a larger value than the on-shell impulse approximation( IAON — dashed line ) for all values of d and agrees reasonably with the data althoughChapter 6. Muon Induced Break-up: the Deuteron Channe1 1204--....-off—shell3 on—shell2--1- Deuteron -Mo me nt u mMeV0- I I-0 100 200 300 400Figure 6.8: The matrix element squared.there is room for further enhancement at the high d points. The higher result for IAOFFis clue solely to the larger matrix element for IAOFF as shown in figure 6.8. The differencebetween the off-shell and on-shell result is greatest at large deuteron momentum and thisis because the amount that the struck proton is off-shell increases with d, ( see figure6.7).The total integrated rate was found to be 1627 s1 for IAOFF and 1583 s1 for IAON.The modal cleuteron momentum is 67 MeV. The total is roughly a factor four higher thanthe total i-ate found by Phillips [47j who included final state interactions employing theAmado model.How might the results for the deuteron spectrum change when the effects of finalstate interaction (FSI) and meson exchange currents (MEC) are included ? The followingqualitative statements can he made.The effects of FSI will be largest at small d. This follows from the fact that FSI areChapter 6. Muon Induced Break-up: the Deuteron Channel 121-oU)>0-o-o16128U)>00Deuteron0 100 200momentum300d [MeV]10_i102Deuteron momentum d [MeVIFigure 6.9: Deuteron spectra for variousimpulse approximation. The dashed linedata are taken from Cummings et al [121].ranges of d. The solid line is the off-shellis the on-shell impulse approximation. TheChapter 6. Muon Induced Break-up: the Deuteron Channel 122400I:300014/322 4/3kx 1 2k11(d, x) = [d2 + 9(1 — dx/)2— 3(1— dx/)jFigure 6.10: The neutron/deuteron momentum in the nd CM as a function of the LABdeuteron momentum.200100largest when the neutron and deuteron have small relative momentum in their CM andhence small relative speed in any other frame. Their relative momentum in the nd CMk1d is given by,k1d j. (6.87)By using the momentum conservation condition, equation (6.5), k11d can be writtenas a function of d and x = d.Q thus,(6.88)The value ofk11(d,x) is shown in figure 6.10. The effects of FSI will be greatest whenk11 = 0 which occurs when d = 66 MeV ( and x = +1). At this kinematical point, theneutrino travels at r radians to both neutron and deuteron which are likely to recombineand form a tritonc thus robbing the deuteron channel of flux and feeding the quasi-elasticC Note that the in-flight nd—*3H reaction does not violate energy momentum conservation since thelow k1d neutron deuteron state is a virtual state in this case.Chapter 6. Muon Induced Break-up: the Deuteron Channel 123channel. An example of this type of reduction is provided by the work of Van Meijgaard[130, p78,figure 4.7] on electron induced two body breakup of 3He where the cross sectionis reduced by a factor two when final state interactions are included. A counter exampleis provided by the proton-proton pion production reaction pp In this caseattraction between the final state proton and neutron causes peaked enhancement inthe cross section as shown by Dubach et al.[131]. In summary, final state interactionsare expected to affect the differential rate considerably and especially at low deuteronmomenta. An overall reduction in the rate is expected although there may also be apeaked enhancement near d = 66 MeV.In order to assess the probable MEC contributions to the differential rate, the resultsof Goulard, Lorazo and Primakoff [132] and Dautry, Rho and Riska [133] are noted. Theformer authors found that the matrix element squared for muon capture by deuteriumreceived an extra 200% from MEC contributions at the zero neutrino energy kinematicalpoint. The latter authors found that the total deuterium muon capture rate was enhancedby only 5%. The total rate is dominated by the large neutrino energy values since thephase space goes like the neutrino energy squared and so we may assert that the effects ofMEC are large at small neutrino energy Q and small at large Q. This assertion is verifiedby a calculation by Doi et al.[134, see figure 3] which showed that the MEC contributionsto the differential muon capture rate by deuterium steadily increase with the neutronenergy in the nn CM. The kinematics of the 1i3He — v,dn process are similar to thoseof the td —* vnn in that high deuteron/neutron momenta correspond to low neutrinoenergy. Figure 6.11 shows the neutrino energy as a function of d.In summary, MEC are expected to increase the differential rate, especially at highdeuteron momenta, by factors of up to three.Chapter 6. Muon Induced Break-up: the Deuteron G’hannel 124100>0)LJ 75E500100 200 300Deuteron Momentum d [MeV]Figure 6.11: The neutrino energy Q as a function of LAB deuteron momentum.Chapter 7Muon Depolarization and Hyperfine Populations7.1 OverviewIn chapter 3, the spin observables A, A and A were defined for the muon captureprocess + 3He ,‘ v + 3H. The rate F was written as,(iF= (F0/2)(1 + APP1(cosO) + APtP2(cosO) + AP) (7.1)d(cos 0)where F0 is the unpolarized rate. The analyzing powers and A depend only onthe nuclear Prol)erties of the 3He —* 3H transition. However, in any experiment intendedto measure the analyzing powers, the actual 1)lysical quantity observed will be a productof an analyzing power and a population parameter or P. For example, the totalrate averaged over all angles is,= f d(cos 0) dcos = F0(1 + AP) (7.2)which depends on the product In order to use the measuremeilt of such anobservable to find the analyzing powers it is necessary to know the population densitiesN(f,f) and hence Ps., Pt and P. This chapter describes a calculation of the N(f,f).7.2 IntroductionThe life history of muons which eventually form muoiic atoms has been comprehensivelytreated by Rose and Mann (for the case of 12( ) in 1961 [135]. The life history for thecase of 3He is similar and may thus be divided into five parts.125Chapter 7. Muon Depolarization and Hyperfine Populations 1261. The muon life begins when it is created in the decay of a charged pion. A negativemuon produced thus is 100% polarized along its axis of motion in the pion restframe. This is because the accompanying anti-neutrino is right handed which bythe conservation of angular momentum forces the muon to have right helicity. ( Anyorbital angular momentum has zero z-component because the momenta are backto back ). Polarized muon beams retain this polarization by selecting ‘backwardmuons’, i.e. those which are produced in the CM at an angle r with respect to thelab pion momentum.2. Upon entering the target, the muon is decelerated from an energy 30 MeV to--200 eV [136, figure 9]. This proceeds by ionization of the target material andsmall-angle elastic scattering.3. The muon is then captured into an atomic state via an Auger transition.4. The remaining electron is ejected in a prompt Auger transition.5. The muon cascades through the atomic orbitals emitting real photons until itreaches the 15 state.The processes of atomic capture ( part 3 ) and cascade ( p’ 5 ) lead to substantialdepolarization of the muon. The depolarization factor should be about 1/6 according tothe following simple argument.Assume that the muon is initially spin up. Upon capture, the muon spin g couplesto the orbital angular momentum 1 to form the total angular momentum . Suppose 1points at an angle & to the z-axis and 1 > s. 1 and.precess around j and ..3 cosO.The time averaged vector will point along j and so its z-component, ( as measured bya weak magnetic held ), will be proportional to cos2 0. Averaging aver all 0 shows thatChapter 7. Muon Depolarization and Hyperfine Populations 127the muon polarization has dropped by a factor three. Averaging over all 0 correspondsto taking the direction of the muon momentum completely randomized at capture.Immediately after capture, the populations of the two fine structure levels are givenby their statistical weight 2j+1, so for large 1 about half the atoms are in the jl+level and half are in the j=l— level. Consider the allowed values of Aj in the cascade:by restricting the transitions to Al —1, it is clear that only Aj = 0, —1 are allowed. Infact, the atoms in the j=l+ state cascade exclusively via Aj = —1 transitions and theatoms in the j=l— state have just one Aj = 0 transition, all the rest being Aj = —1.Considering these two types of transition: one has,T+ J-f (T-f)+’_(7.3)j = (l—1)+’ Aj=0In the first case, the change in j is accommodated by the change in 1. However, in thesecond case we need . = ‘ + i in order to preserve the magnitude of j. For an initialspin up rnuoi it is possible to conserve the component of spin by ending up with a spinup muon and a spin sideways photon or a Spin down muon and a spin up photon. Bothof these possibilities are equally likely and so the muon completely loses its polarization.This complete depolarization is the fate of about half the muons so the cascade causesthe muon polarization to drop by a factor two. Combining the effects of atomic captureand cascade one arrives at a total depolarization factor of 1/6 as claimed above.Since 3He has nuclear spin there is a hyperfine interaction between the nuclear spinand the atomic total angular momentum j. To include the effect of this hyperfineinteraction, the atomic cascade is further subdivided into two distinct parts although inreality this distinction is not so clear cut.• The first stage is defined by setting the hyperfine interaction between the nuclearspin, i and the total angular momentum of the muon atomic orbital, jto zero.YChapter 7. Muon Depolarization and Hyperfine Populations 128Figure 7.1: Term diagram.50 1 2 3 4....• In the second stage i and j are coupled to form f, the grand total angular momentum and the atomic states have good.1 and f . This is the strong hyperfinecoupling limit and it is assumed that transitions to states of different f could bedistinguished by the energy of the photon emitted i.e. the width of the state ismuch less than the hyperfine splitting.7.3 Cascade Calculation7.3.1 In the Absence of Hyperfine CouplingIn calculating the depolarization in the cascade three assumptions will be made.• The angular correlation of the muon momentum just before atomic capture is zero.i.e. the muon (lirection is completely randomized.• The cascade transitions are due to electric dipole radiation, ( El transitions ).• The route to the ground state is purely “right to left” omi the term diagram.Chapter 7. Muon Depolarization and Hyperfine Populations 129Rose and Mann [135] have calculated the angular correlation (kE.kE0)of a deceleratingmuon in carbon. They find that it drops with energy loss like (E/E0)1/’with,2m M(M +m,1)2 (7.4)M is the mass of the scattering atom and E/E0 is the muon energy divided by its initialvalue. For carbon, the correlation has dropped to less than 0.4% when the muon still has90% of its initial energy left. hmerting the 3He mass into equation (7.4), it was foundthat the muon angular correlation has dropped to less than 1% by the time its energy is72% of the original value. According to this calculation, the assumption of zero angularcorrelation is a good one since E/E0 at capture is approximately 1% for 3He.With regard to the second assumption it is noted that higher order electromagnetictransitions will be weaker in rate by a factor of the order of a, the fine structure constant,which is less than 1%.To explain the third assumption, the term diagram is introduced in figure 7.1. Thisdiagram shows atomic states positioned according to energy, with a non-linear scale toocoarse for hyperfine or even fine structure to show. Each line represents a state of fixedn, 1. If the scale were expanded so that the fine structure could be seen, two states withj = 1 + - would emerge from each single line except for 1 = 0.By restricting transitions to “right to left” only Al = —1 is allowed but An can takeany value < 0. This will tend to over-estimate P,, since the extra transitions, ( Al = +1 ),will cause extra depolarization. The left to right transitions are extra because of theAl = +1 selection rule. Suppose we start at 1: the minimum number of transitionsthat will take us to the ground state is then 1. If there were to he just one left to righttransition en route then there would also have to be one right to left transition to getback to the original 1, plus all the others which gives a total of 1+2. Every spontaneoustransition causes depolarization in accordance with the second law of thermodynamics.Chapter 7. Muon Depolarization and Hyperfine Populations 130Hence, the presence of left to right transitions implies extra depolarization.As an example of the quality of this approximation, consider the fate of an atomfinding itself initially with n=lo. This value is chosen for the sake of argument but itis suggested by the results of Haff and Tomnbrello [136] who have calculated Auger ratesfor the capture process and find a peak around ii = 15. The work of Rose and Mann[135] suggests also that the circular or nearly circular orbits are preferred, ( an orbit isdubbed ‘circular’ if 1 = n — 1). Bohr [137] has suggested that the high 1 states would bepreferred because of their higher statistical weight although it is clear from the results of[135] that the situation is not so simple.Consider first the circular orbit with n = 15 and 1 = 14. Its oniy option is to decayto a circular a =14, 1 = 13 state and subsequently via circular orbits only, all the waydown to the iS state. Thus, every transition is right to left and the total left to righttransition probability is 0%. This statement is also true for the a = 15, 1 = 13 statesince its first transition must be to either an a = 14, 1 = 12 state or an a = 13, 1=12state. The latter state is circular and an atom finding itself in such a state can decay viacircular orbits only. The former state is equivalent to the a = 15, 1 = 13 state and mayonly induce other right to left transitions.Consider now the a = 15, 1 = 12 state. There is one possible left to right transitionand there are three possible right to left transitions. The transition probabilities Pg/aregiven by the simple dipole formula [116].Da’l — 3’ - 2— j-aw L1 Tmm’rn’= 1/6Z4mi-ecia5( —)3f11 (Rf)2 (7.6)where, f11= l+for 1—> 1+11+1= forll—121 + IChapter 7. Muon Depolarization and Hyperfine Populations 131Table 7.1: Transition Probabilities for the 3He-1i Atom.Transition Rate( n,l) —+ (n,l ‘) s(15,12) — (14,1:3) 3.43x i0(15,12) —* (14,11) 3.62x i0(15,12) —* (13,11) 2.15x io(15,12) —* (12,11) 7.35x 106(15,11) —* (14,12) i.06x i0(15,11) —* (13,12) i.53x iO(15,11) — (14,10) 3.13x i07(15,11) —p (13,10) 2.49x io(15,11) (12,10) 1.52x iO(15,11) — (11,10) 5.62x 106(5,2) —* (4,3) 1.61x 108(5,2) (4,1) 4.74x iO(5,2) —* (3,1) 1.08x i0°(5,2) — (2,1) :3.01 x 10’°and,n’i’ (—1 )_1 (n + i)!(n’ + l — 1)! (4nn+i (n — 711)fl+n’_2l_2R71i 4(21 — 1)! (n — 1 — 1)! (n’ — 1)! (ii + n’’’ x (7.7){2Fl(_nr —nr’; 21; ()2)— (: ni) 2F1(—nr — 2, —nr’; 21; (i;2 )}with, = n—i—inT = n —The quantity ‘11recl is the reduced mass of the muonic atom, w is the energy of the emittedphoton and 2F1 (a, 3; y; x) is the hypergeornetric function. The transition probabilitiesfor the n = 15, l = 12 state are given in table 7.1 as well as those for the n = 15, i = 11state and the n 5, l = 2 state.For the n 15, 1 = 12 term, the probability of a left to right transition is only 0.05%while it is 0.2% for the n = 15,1 = 11 term. The preference for the (15, 12) —* (14, 11)Chapter 7. Muon Depolarization and Hyperfine Populations 132over the (15, 12) — (14, 13) transition is due solely to a larger dipole matrix elementbetween the states. As the atom decays to lower n states, the transition to the stateof lowest energy becomes favoured as can be seen from the final series of entries intotable 7.1. At low n, the enhancement of the /n = —1 dipole matrix element is thesame as it was at large n but here the (energy)3 factor takes over. For the 5d state, leftto right transitions represent 0.35% of the total transition probability.One can summarize the physical l)icture thus. The initial population is mostly athigh ii and high I and it decays right to left on the term diagram with more and moreatoms falling into circular orbits as n decreases.In calculating the cascade depolarization the statistical tensor formalism, as has beenexpounded with refereice to this problem by Nagamine and Yamazaki [138] as well asKunioa [1:39], will be used. To describe the relative populations of sub-states of a given(ii, l,j) state requires 2j + 1 scalar quantities and the simplest choice would be thepopulations themselves However, one could choose any linearly independentcombination and a statistical tensor is one such combination with the peculiar propertyof transforming simply under rotations, ( see [34, 1)109] ). The statistical tensor Bk(n,l,j)is defined by,Bk(n,l,j) = 2j + 1 (_1)3’ThiP,,j,(mi)(k0 jmj; j —mi) (7.8)‘n.jwhich can be inverted to give the populations as a linear combination of the statisticaltensors.(1)_J+nhi= /2j + 1 Bk(n.,I,i)(kO jm; j —rn) (7.9)kaThe author is grateful for helpful discussions with Y.KunoChapter 7. Muon Depolarization and Hyperfine Populations 133With this choice of normalization, Bo(n,l,j) equals the total population of the statesIn,l,j,mj ). Also,3 2B1(J)= i(i + 1) mP,1,i,(m)3.Bo(J) x Polarization (7.10)where the I)olaIizatiofl is defined asPolarization = mj (7.11)jThus, the first rank statistical tensor is proportional to the vector polarization of thestates.In calculating the muon depolarization, it is necessary to consider first the atomiccapture process. According to the first assumption, a given 1 orbital must be filledwithout prejudice towards any sub-state, m1. This is accomplished by combining themuon statistical tensor Bk(S) with an orbital angular momentum statistical tensor Bk(n,l)describing a level n, 1) which is statistically populated. By setting P,,i(7n1) = l/(21 + 1)in equation (7.8). it was found that the only non-zero Bk(n,l) has k=0. The probabilityof finding the state n, 1, j, rn. ) is given by,P5(rn3)P,j(m1I (jrnj 1m; sin) 2. (7.12)fl1 ,7TiBy inserting this expression in equation (7.8), using the inversion formula (7.9) toreplace P(m), P7,,(m1) and performing some Racah algebra, a general equation forcombining two statistical tensors was found. It is,Bk(n,l,j) [(2,sIl 1)] Bk1(s) Bk2(l) x1’Chapter 7. Muon Depolarization and Hyperfine Populations 134S ij[(2k1 + 1)(2k2 + 1)] (kO I k10; k20) $ 1 j (7.13)k1 k2 kThis equation does not agree with equation (5) of [139]. That equation is inconsistentwith the normalization conventions used in the paper and is missing the Clebsch-Cordancoefficient.From the Clebsch-Gordon coefficient in equation (7.13) and the fact that k2 is restricted to zero it is clear that k call only take on the values zero or one. One then hasthe result for the levels n, 1, j ), that only the population and vector polarization arenon-zero irrespective of the value of j i.e. all tensors Bk(n,l,j) of rank greater than one,are zero. Since k2O, equation (7.13) simplifies to,Bk(n,l,j) = (-l)1JW(jj; ki) Bk(s)Bo(l) (7.14)Next, the evolution of Bk(n,l,j) during the cascade will be traced. Consider a transition from the state ii, i, j, rnj ) to the state n’, 1’, j’, rn ). Using the second assumptiongiven at the start of this section, the transition operator can be written as a first ranktensor operator in orbital angular momentum space T1. The relative transition ratefor the various rn.1, rns is calculated taking the same reduced matrix element for all thetransitions. If is the rate for a transition from the state n, l,j, 7n to the staten’, 1’, j’, rn then,(l’,j’, rn T1 1, j, rn)(j’rn l’77i; rn) (jrnj irni; m)nil ,rn’nr8 ,br; un1)[(2j + 1)(2j’ + 1)j W(jj’ll’; 1)(1m -rn j’m; j -mi) (7.15)Chapter 7. Muon Depolarization and Hyperfine Populations 135The new populationP1i,i(m) is given by.P1,(in) = P1,(nr) x Normalization constant) (7.16)1,J ,?njBy inserting this in an expression for Bk(n’, j’, 1’), replacing the populations using theinversion formula (7.9), performing some Racah algebra and finally normalizing, one findsthat the statistical tensor Bk(n,l,j) suffers the following change.Bk(n’,j’, 1’) (21 + 1)(2j’ + 1) W(jj’ll’; 1 )2 Uk(jlj) Bk(n,l,j) (7.17)j=l+with,Uk(jlj) = (_l)kJJ’[(2j + 1)(2j’ + l)}W(jjj’j’; ki) (7.18)The change in Bk(n,l,j) depends only on 1, j, j’ and not on n, so all allowed “rightto left” transitions emanating from a particular term are equivalent with respect todepolarization.Further, it is clear that only the 1 dependence of the initial distribution is important.If, for the sake of simplicity, all transitions are taken to be Al = —1 and An=max.then after one El transition all the muons are in circular orbits with 1 =— 1. Thisroute is illustrated in figure 7.2.A question exists as to the dependence of the residual muon polarization on the initial1 distribution. To answer this, the depolarization of a 100% polarized ensemble of muonswas calculated for distributions supporting a single 1 value and the results are shown intable 7.2, (the route for 1 = 0 is (n,l) —* (2, 1) —* (1,0)). The following salient featuresare noted.• For large 1. the value of p residual is near 1/6 as expected from the simple classicalargument, (1/6 = 16.7% ). The depolarization follows the pattern of 1/3 at captureand 1/2 clue to the cascade.Chap tei 7. Muon Depolarization and Hyperfin e PopulationsFigure 7.2: Effective cascade route.136Table 7.2: Muon Depolarization with 1 = 1 Populated Only.presidual% p after caPture% p residual/ p after capture >< 100%25 16.9 33.4 50.623 17.0 33.4 50.921 17.1 33.4 51.219 17.3 33.4 51.817 17.5 33.4 52.415 17.8 33.4 53.313 18.1 33.4 54.211 18.5 :33.5 55.29 19.2 33.5 57.37 20.2 33.5 60.16 20.9 :33.7 62.05 21.9 33.9 64.64 23.4 34.2 68.43 25.7 34.7 74.12 30.1 36.0 83.6I 40.7 40.7 100.00 40.7 100.0 40.71 >Chapter 7. Muon Depolarization and Hyperfine Populations 137Table 7.3: Muon Polarization During the Cascade.1 P%9 33.58 33.17 32.66 32.05 31.24 30.03 28.22 25.11 19.20 19.2• This picture is reasonable down to low values of 1, ( 1 5, 6 say ).• The 1 = 1 — 1 = 0 transition causes no depolarization.• The 1 0 — I I transition causes extensive depolarization, ( 2/5 ).The values of p residual for low 1 are calculated in the An =max. approximation. Areal population starting entirely with 1 = 2 would have PILresidu less than 30.1% becauseof the presence of transitions to n S with n >1 which then give rise to extra left to righttransitions. However, the purpose of table 7.2 is to indicate the depolarization due tospecific paths rather than to make any definite statements about real muon depolarization.The cascade depolarization is found to occur maiiily at the latter stages of the cascade.An example is shown in table 7.3, where the muon polarization is given at various stagesof the particular cascade starting at 1 = 1 = 9. This general feature is found for all valuesof 1.The Rose and Mann 1 distribution is shown in figure 7.3. It is clear from table 7.2 thatany distribution of this basic shape will give presicluall/6.In other words, the residualChapter 7. Muon Depolarization and Hyperfine Populations 138Figure 7.3: Rose and Mann 1 distribution.2500 I I I I -2000--1500- U-C .c1000-500--0- I • 1• • -0 2 4 6 8 10 12orbital angular momentum quantum number Imuon polarization is rather insensitive to the initial distribution of the muonic atomsamong atomic states. Any reasonable initial distribution will lead to presidua11/6Experimental results for I = 0 nuclei are listed in Table 7.4 [140]. The anomalousvalue for 4He found by Souder et al [141] has been confirmed by Glaclisch [142] who found= 4.4 + 0.3 % over a pressure range of 9.8 —* 20.6 atmospheres. The result of Souderwas at 7 and 14 atmospheres. It should be noted that 4He was the only gaseous targetused in the experiments reported in [140]. Since 3He and 4He have the same atomicnumber one would expect similar physics in the absence of the hyperfine interaction, ( atleast at the atomic level ), and the above mentioned anomalous result must be taken intoaccount when making predictions for the N(f,f) in 3He.7.3.2 With Hyperfine CouplingGenerally speaking, the effect of coupling between the spin of polarized nuclei and theatomic total angular momentum is expected to increase the value of the residual muonChapter 7. Muon Depolarization and Hyperfine Populations 139Table 7.4: Residual Muon Polarization P, for Spin Zero Nuclei.Element P%4He 6+1i2C 14+ 416Q 15+ 424Mg 19+ 5285i 16+ 4S 15+364Zi 19+ 5Cd 19+ 5Pb 19+6polarization. This will be true when the nuclear spm is pointing in the direction of theincident muon’s spin. Of course, when these spins point against each other there is thepossibility of a cancelling out effect.This effect can be realized by the following simple argument which uses the vectormodel for spiis. Suppose the nuclear spin i and the muon spin are coupled, formingthe total spin.1. For a l)olaIize(l nucleus, i will tend to point ill a certain direction ( letus call this direction up ). The resultant f will also tend to point up in the case that anunpolarized spin is added. The coupling interaction causes i and . to precess around fand when we then project onto the ‘up’ axis it will be found more often up than downbecause it is following 1 which itself is more often up than clown.The hyperfine splitting of levels goes like the expectation value of 1/r3 multiplied by1(1+ 1)/j(j ± 1). l3ethe and Salpeter [116, page 17] give a result for hydrogen-like atomicwavefunctions,Z3\/ anl(l + 1)(l + ) (7.19)where a!. is the muonic Bohr radius = 132.8 fm. In general, the hyperfine splitting getssmaller as n grows. Thus it is expected that at some value of n, (u1) say ), the hyperfineChapter 7. Muon Depolarization and Hyperfine Populations 140splitting is so small as to be smeared by the natural width of a level. Although there maybe a region where the hyperfine splitting is comparable to the natural width, the physicalsituation will be modeled by assuming zero hyperfine coupling for ri > 71hyp and strongIiyperfine coupling for ri < fl1)• The theoretical description consists of coupling thenuclear spin i to the atomic total angular momentum j to form a state of good f,f, at= ni. This is done by combining the populations of the nuclear levels, ( described byBk1(i) ), with the populations of the atomic levels, ( described by Bk2(n11,l,j) ), usingthe following equation which is equivalent to equation (7.13). The statistical tensorBk(ri,l,j,f) will be non-zero for k=0,l.2 since both the atoms and the nuclei have vectorpolarization, (k1,k2 = 1).Bk(n,l,j,f)= [2i +1)(2+ 1)] B1 (i) Bk2(n,l,j) xk1 ,k2i if[(2k1 + i)(2k2 + l)] (kO k10; k20) j i f (7.20)k1 k2 kThe evolution of Bk(n,l,j,f) down to the iS state is then traced assuming El traiisitions as before. The following equation describes the change in Bk(n,l,j,f) due to thetransition 1, j, f) — 1’, j’, f’).Bk(n,l,j,.f) = [(21 + I)(2j + 1)(2j’ + 1)(2f’ + l)]uk(f if’) xW(j, j’, 1,1’; 1 W(f, 1’ j, j’; 1)2Bk(n,l,j,f) (7.21)This is derived in the same way as equation (7.17). It is helpful to consider the casewhere illlyp 1, ‘He =1 and P = 0 to understand the basic idea. In this case, thenuclei are always spin up T but the muons are found half of the time spin up ‘ and theother half half spin down .. Half the time one finds the configuration fl which is allf=1,f=1. The other half, one finds the configuration fl, which is half f=1,f=0 andChapter 7. Muon Depolarization and Hyperfine Populations 141half f=0,j=0. The latter two states contribute nothing to the polarization of the f=1state which takes the value one half. In this sirnj)ie case it equals the muon polarizationafter coupling so we started with a totally unpolarized ensemble of muons and now theyhave polarization .The 15 hyperfine statistical tensors B(f) will be parameterized l) tile four constantsA,B,C and D which are defined by tile following equations.B(0) = 025— AP11P3He/4B(1) = 0.75+AP,tP3He/4B(i) BP+CP3He\/B1S(i)= DPP3He (7.22)This form is found by noting the following.• Tile only statistical tensor contributing to Bk(n,1,j ,f) in tile cascade transitions isof tile SfllC rank k.• Tile Clebsh-Gorclon coefficient in equation (7.20) limits the values k1 and k2 cantake on for a given value of k.Take B(l) for example. Only tile tensors Bk(n,l,j,f) with k 1 contribute and toform these (k1,2) can take on the values (0,1) or (1,0). Thus, B(i) has two terms: oneproportional to PR aildI tile other prOI)ortional to PHe.Tile equations (7.22) imply the following identifications for and P.1v BPR+CP3NePt DPRP3HeP = AP1jP3 (7.23)Chapter 7. Muon Depolarization and Hyperfine Populations 142Table 7.5: Residual Muon Polarization per Unit Target Polarization for Various Valuesof1 2 3 4 5 61presiclual% 59 28 2:3 21 20 19]The value of p residual is given by,p residual N(1, 1) — N(i, 1) P = B P + C PHe• (7.24)Table 7.5 lists p residual in the case where P, 1 and PHe = 1. The population densitiesat are taken from the case 1 = 13. This choice is motivated by the results of Haffand Tombrello [136] who found an initial distribution peaked for n = 10— 15.It is clear that the effect of the hyperfine interaction is to re-polarize the muon asexpected, an effect which decreases as n1 increases.7.4 Attempt to Include Effects of External CollisionsAs mentioned earlier, the anomalous residual muon polarization for 4He indicates thata simple cascade theory is inadequate. The extra depolarization is presumably due tocollisions between the muonic ion and the surrounding 3He atoms which induce Starkand external Auger transitions. The effects of external collisions will be included bymaking the fol1owiig two assumptions.• All the extra depolarization occurs while the atom is in a state with n>• The polarization of the state n, 1, j) is given by its cascade value multiplied by adepolarization factor Ddepol. This depolarization factor is the same for both j=l+and jzil—.Chapter 7. Muon Depolarization and Hyperfine Populations 143If it is true that the extra depolarizatiou is due to Stark transitions the first asslimption should be valid. Placci [143] has measured muonic x-rays in 3He at 7 atmospheresand found a fit to the intensities using a rate Vv’1 for complete Stark mixing at principalquantum number 71 given by,= w13 (f3)5 (7.25)with W13 1010 s. This rate drops rapidly with n. and at n = 5, W 8 x iO s whichshould be compared to the radiative rates given in table 7.1. For the n= 5, 1=4 term,the radiative rate is 1.4 x 1010 —I which is much greater than W5. The Stark transitionprobability goes like the dipole matrix element squared which is why it decreases with n.The second assumption is rather less justifiable and should be treated as a crudeapproximation rather than the feature of some particular model.The calculation begins at n = ni at which point we need to decide what values of Bk(n11,l,j) to use. The Bk(n11,l,j) should reproduce the experimental value ofp residual= 4.4±0.3 % if they were simply cascaded down to the iS level with no hyperfine interaction. There are four quantities of interest, the two populations and two vectorpolarizations of the j=l+ states at n = fl11). Remember that the higher rank tensorsare zero because of the assumption of zero angular correlation for the muon at capture.The I)oPulatiofls can safely be taken as statistical, i.e.Bo(n,l,j)= 2(21+1) (7.26)Using the second assumption, tile value of the vector polarizations are set by Ddepalwhich equals tile experimental residual muon polarization divided by the value that tilevector polarizations would have given in the absence of the hyperfine interaction.Having set tile Bk(n11,l,j) tile muon population is then cascacledi down to the iS stateusing equation (7.21). The values of A,B,C and D thus found depend on tile value of nibut also on the initial 1 distribution. However, it is expected that the latter dependenceCli ap ter 7. Mu on Depolarization and Hyperfin e Populations 144Table 7.6: Values of the A,B,C,D Parameters with External Depolarization.n1 1 A B C D1 14,20 .0440 .0220 .5000 .04402 14 .0440 .0173 .2083 .02832 20 .0440 .0169 .208:3 .02853 14 .0440 .0170 .1609 .02013 20 .0440 .0165 .1609 .02014 14 .0440 .0170 .1396 .01684 20 .0440 .0165 .1396 .01695 14 .0440 .0170 .127:3 .01505 20 .0440 .0165 .1273 .01516 14 .0440 .0171 .119:3 .01396 20 .0440 .0166 .119:3 .0140will be small since p,sidua1 itself is rather insensitive to the initial 1 distribution. Resultsfor 1 14, 20 are given in table 7.6 for values of up to 6. The values of Dol are0.246 for 1 = 14 and 0.255 for 1 = 20.The results of this method can be tested by comparing the residual muon polarization with a recent result of Newbury et al.[144j. This experiment measured the residualpolarization of an originally unpolarized muon captured by a 1)olarizecl 3He target at 8atmospheres. It was found to be 7.2 + 0.8% per unit target polarization. In this experimental situation. p residual is given by the CP3He so that the residual muon polarizationper uiit target polarization is C. Looking at table 7.6 it is clear that there is some depolarization due to collisions after the hyperfine interaction becomes important and so theclear distinction made in the first assumption above is not valid at this pressure.7.5 Summary and ConclusionsThe intra-atomic processes prior to nuclear muon capture have been understood and it isclear that the anomalous residual muon polarization found in 4He is an inter-atomic ( orCli apt er 7. Mu on Depolarization and Hyperfine Pop ulat ions 145rather ionic-atomic ) effect. Any reasonable initial distribution amongst atomic stateswill give a residual polarization of 1/6 for a 100% 1)olarized muon i)eairi.Results have been found for the hyperfine populations in the limit that all externaldepolarizmg effects occur before a point where the hyperfine interaction becomes important. By comparing the predictions for p residual with a recent measurement usinga polarized 3He target and an unpolarized muon beam it is concluded that this simpleapproach is umeliable.In order to further understand the hyperfine population densities, the procedure usedby Landua and Klempt [145,146] could be applied. These authors calculated the effect ofatomic collisions on x-ray intensities for muonic and pionic 4He and were able to reproducethe density dependent anomalous pionic x-ray intensities with few free parameters. 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Lett. 22 (1969) 1014.[163] H.PrimakofF, Nuci. Phys. A317 (1979) 279[164] A.W.Thornas aud K.Holinde, Phys. Rev. Lett. 63 (1989) 1025.Appendix ANatural UnitsIn nuclear and particle physics it is convenient to adopt the natural system of units. Inthis system all quantities have dimensions of energy ( or alternatively length ) raised tosome power and cumbersome scale factors which often occur are eliminated by settingthem to one. The two scales used to define the natural system are the speed of light invacuo c and the reduced Planck constant h. The speed of light is an upper limit set byspecial relativity and sets the scale for speeds. The Planck constant indicates the size ofquantum mechanical effects as in, for example, the uncertainty relation for energy andtime observables,/Et > h/2. (A.1)The mechanics of the natural system can be understood by considering the followingtwo examples of conversion from natural units to Système Internationale (SI) units.Suppose we are given a momentum in MeV. The ( non-natural ) dimensions of momentum are [M.L.T’] and those of energy are [M.L2.T] It is therefore necessary toinsert a factor of the speed of light in order to convert from natural to SI.1 “MeV” = e x 106 “Joules” (A.2)= e X106Kg.m.s1 (A.3)In the above equations e is the magnitude of the electronic charge and the units inquotation marks are natural units.The second example concerns the equivalence of energy and inverse length in the154Appendix A. Natural Units 155natural system. What is the conversion factor which takes us from inverse length toenergy ? The dimensions of c and h are needed to perform the following manipulation.They are [L.T’] and [M.L2.T1jrespectively.1 “fn” = 1015 “m” dimensions [L’l (A.4)= l015ic “nit” dimensions [M.L2.T] (A.5)= 10’5hc Joules (A.6)1 o’ h= C MeV = 197.327 MeV (A.7)106 eAppendix BModel Calculation of VAd(q’, q)This appendix details the model calculation of VAd(q’, q). A simple form of the deuteronwavefunction is used which generates the general behaviour of VAd(q’, q) and explainswhy the value of AAd found in section 1.5 increases with q’.The deuteron wavefunction is taken to be s-wave oniy and to have gaussian form.= —(q/d)2 (B.1)00 3/2N2 = j dqq2 H/(qH2=F(3/2) (d2/2) (B.2)The only free parameter in this function is d, the size of the deuteron in momentumrepresentation. This parameter is fixed by demanding that the model calculation ofgives the same result as the full calculation which used a realistic deuteron wavefunctionwith both s-waves and d-waves. The wavefunction given by equation (B.1) has the rightshape for the s-wave part of the deuteron i.e. it is a monotonically decreasing functionof q and the s-wave represents about 95% of the deuteron in probability.Using equation (1.38) and the deuteron wavefunction given in equation (B.l) thefollowing expression was found for the s-wave Ad potential. The range of the AN potentialAAN is written simply as A for the sake of notational clarity.VAd(q’,q) = —2J(q’,q) (B.3)where, J(q’, q) = _q/A’f dss2e_22K(s,q) K(s, q’) (B.4)and J( q) _2sx/A d([3 + q2 + sqx]) (B.5)156Appendix B. Model Calculation of VAd(q’, q) 157Performing the integral over x yields the sinhc function which is the hyperbolic equivalentof the sinc function: sinhc(x) = sinh(x)/x. One then finds,(q’, q) = q2/’aN2 ths2e_28sinhc(sq/g2) sinhc(sq’/g2 (B.6)1 21-. .L— .1_ j__W‘ 2 — 2 2cr1 A 4d+ (B.7)g2 — A2 d2The value of .1(0, 0) can be found by noting that Iim_o sinhc(x) = 1 and performingthe resulting gaussian integral which yields a gamma function cancelling the gammafunction from the normalization coiistant.J(0, 0) = N2 f ths2e_23/hi = (h/d)3 (B.8)Equating J(0,0) with /2A yields d = 0.633fm’ and this in turn gives o =0.847 fm1, h = 0.615 fIn1 and g = 0.561 fm.Now, VAd(O,q)/VAd(O,O) can be found by evaluating J(0,q)/J(0,0). If the model isto generate a reasonable approximation to VAd(q’, q) then the range of this ratio shouldbe close to 0.96fm’. Using equation (13.6) it was found that,J(0, q)— e2 (g fdsse_2s2/1sinh(sq/g2) (B 9)J(0,0) — q ) f°°dss2e_23Ih(B.10)where, 1 — 1 h22 2 8 . )c7T cTi gThus, with the simple form of the deuteron wavefunction, the Ad potential is exactly agaussian with range T = 1.04 fm1. Since this result is in close quantitative agreementwith the full result one has confidence that the qualitative features of VAd(q’, q) will beFigure B.1: The sinhc function.well represeiited by the model calculation. Now let us calculate a “slice” of the potentialat fixed q’ in order to see what happens to the range of the Ad potential.J(q’, q)—-q2/a (g Jdse_232//sillh($q/g2)sinh(sq’/g2) B 12/— e 1 1 00—2s2 1h2 F 2 ( )J(q ,O) q j J dsse / sinh(sq /g)To evaluate the integral in the numerator of this expression the following identity wasused.J(q’, q)e214 sinhc(2qq’/a2)J(q’, 0) a22g= 1.56 fm1 (B.15)The function sinhc(x) is a monotonically increasing function as x increases from zero toinfinity, see figure BA. Thus, when the gaussian with range is multiplied by the sinhcfunction one expects the result to have a range somewhat larger than uT. This featureis exactly what is found in the full calculation.Appendix B. Model Calculation of VAd(q’, q) 1583C0x1sinh(A) sinh(B) =— [cosh(A-bB) — cosh(A—B)] (B.13)2With, [00e2 cosh(2ax)=F() ea2 (B.14)Joone finds,Appendix B. Model Calculation of VAd(q’, q) 159The sinhc function spoils the gaussian shape of the potential at non-zero q’ in themodel calculation. However, in the full calculation, the shape of the potential was bestrepresented by a gaussian at q’ l.Ofrn’, ( see figure 1.2 ).Since .J(q’, q) = J(q, q’), ( which follows from inspection of equation (B.6) ) one canimmediately write down the full expression for the model VAd(q’, q).VAd(q’, q) =—e 2/4e’rsinhc(2qq’/a2) (B.16)The non-separable part of this is the sinhc function which is a function of qq’ and cannotbe written as a function of q multiplied by a function of q’.Appendix CDerivation of Two Body H Pionic Decay AmplitudeIn this appendix, two details of the calculation of the two body pionic decay amplitudefor H are given. First, the effect of the Pauli principle is made clear and second anidentity for the reduced spin matrix element is proved.We wish to calculate the matrix element (‘3He Heff 1’H) where 1’He and arefully antisymmetrized wavefunctions and Heff is given by equation (2.14). The effectiveHamiltonian is the sum of three terms and can be written,Heff = H 11(1) +P2H’P +P3H’P (C.l)i=1,2,3The form of the last two terms follows from requiring that (1 H1l 1) = (2111(2)12).The fully antisymmetrized hypertriton wavefunction is given by equation(1.6). Let uswrite‘I’3He) = NAAI ‘/‘-e )for the Helium wavefunction where NA is a normalizationconstant and A = 1 + P2 + P3. Thus,(3He HeffI 3H) = ( A [H(1) +P2H’P +P3H(1)P] A ) (C.2)= /3( He W’)Al /) (C.3)where the latter equality follows from P2A = P3A = A and P = P3, P = P2. Thecreation and annihilation operators associated with H’ are a1)ta and so only thatpart of J “H ) j which the lambda particle is labelled (1) will contribute to the matrixelement. Therefore,( ‘He “effj ) = \/3( M’fP’l j,(1) ) (C.4)160Appendix C. Derivation of Two Body H Pionic Decay Amplitude 161which is equation (2.21).The operator M’ has both an s-wave and a p-wave part, the latter of which containsthe nucleon spin operator (1)• Due to the particular form used for the hypertritonwavefunction, the matrix element of the p-wave part is simply related to matrix elementof the s-wave part in the limit that the E2 multipole is zero. This is shown below.Consider first the matrix element j(1) The convention used for reduced matrixelements is,(aJMjT1bJ’M) = (JMjI J’M; KQ)(aJTKQUbJ’) (C.5)where TKQ is any tensor operator of rank K and projection Q and a, b represent thestructure of the states with total angular momentum J, J’ and projection M, M. Thedefinition of the spatial tensor operators RKQ(Iclk2)is,RJQ(A k2) = (4)Y(, ). (C.6)The wavefunctions are expanded in channels where the channels are defined by,=6(x-)y [Y )®x(1)](1). (C.7)For the hypertriton there are only two channels which have support in Theycorrespond to channels 2 and 6 of the 8-channel wavefunctions of Kamemeya et al. andhave the properties shown in table C. I.The matrix element of 11(1) is found by partial wave expanding the plane wave.(3He; nij Heff(q)I H; ) =m; kmk)( 3HeRk(0k)jk(2qy/3)) (C.8)Appendix C. Derivation of Two Body H Pionic Decay Amplitude 162Table C.i: Hypertriton Channel Specifications.C 10, ). L0, s ,$,- i0,2 0 0 0 1 1/2 06 2 0 2 1 3/2 0The Clebsch-Coiclan coefficient immediately constrains k to be 0 or 1 and the reducedmatrix element further restricts k to be even only because of the even parity of the stateswith the result that oniy the zeroth order multipole contributes.f dx dy2jo(2qy/3)( He Y)(Y (1))(C.9)To calculate the matrix element 1)fl(1) it is necessary to combine the spin operatorwith spatial tensor operators and to that end the following definitions are made.5o u, +1k/2(u + iu)1’TKQ(kl k2) = (KQ I k1rn;k2m)Rk17711 (ll)Sk2772 (C.10)ml ,?fl2The product )H(’) yields operators with well defined tensor properties thus,Rk,lk(0k)Sl?fl = krnk; 1m)°TJQ(k1) (C.11)KQThe value of A’ is constrained to be 0 or 1 siice the states have total angular momentumwhich implies that the value of k may be 0,1,2. The even parity of the states restrictsk to be even anci so in this case both the zeroth and second order multipoles contribute.The zeroth order multipole is,(3He°°TO(01)71) (3He ) (‘U’)( ‘I k) (C.12)0, 0,Appendix C. Derivation of Two Body H Pionic Decay Amplitude 163where the spatial integrals and Bessel function have been suppressed. This form shouldbe compared to,(3HeH°°TO(OO) b(’) = (P3He I) ( 1 ‘) ( (C.13)a a’where (awl a’)=c’). The reduced spill matrix element appearing in equation(C.12) is given by,(1/)= 6(la, l&)6(Aa, AaS(La, La’)(sa, Sa’) x S(Sa, 5’a Sa’, La) (C.14)where, S(sa,Sa,Sa’,La) = (i)L+sa+S Si2[($a + 1)(2Sa’ + 1)]1 1111 12 2 2 2 (C.15)I Sa Sa’ La J I Sa Sa’ a Jand so there exists the possibility that Sa Sa’ which was not allowed in the case ofthe matrix element of (1)• However, if La’ = 0 then La = 0 and since the total angularmomentum is this implies that in turn that 5a and 5a’ = and hence Sa = Sa’.If La’ = 2 then La = 2, so that 5a , S’ = and hence 5a 5a’ Further,= S(l,,2)= (_) (C.16)and so for the specific case where j) has no L = 1 component and only s = 1components,= (-)6(a, &) (C.17)The second order multipole is likely to be much smaller than the zeroth order multi-pole because of the factorj2(2qy/3). From the fact that F(q) is oniy reduced by 30%from the value of F1(0) it is deduced that the values of y for which the nuclear densityis appreciable are such that qy 1. The second order spherical Bessel functionj2(x) issmall for x 1 and so will reduce the major contributions to the matrix element significantly, ( in the case of the 3He —* 3H isovector transition the second order multipole wasAppendix C. Derivation of Two Body H Pionic Decay Amplitude 164found to be 0.2% of the zeroth order multipole). Thus, if the contribution of the secondorder multipole is igiored,(/3( 3He 11) k) (C.18)which is equation (2.23).Appendix DDerivation of Inclusive H Pionic Decay AmplitudeIn this appendix, the expression (2.42) for the inclusive ir decay rate is derived. First,the requirements of the Pauli principle are analyzed and then expressions for the exchangeintegrals are found.Using closure x,11 X, rn ) ( X, rnf 1 the sum of matrix elements between thehypertriton and general three nucleon states becomes an expectation value of the operatorHeff( ) tHff( ) for the hypertriton.(X;rnjHeg(H;rnj)2= (D.1)?n,mj,X rnThe expectation value can be simplified by using the properties of the permutation operators.( HHeff ‘I) = (b H’t/‘) + (/ Ht(P + F3) ) (D.2)The first term is easily evaluated using 11(i)t) = 1 and the normalization of the hypertriton state which implies that ( j) (‘) ) = 1.= (D.3)1rn — p(’)/q0) s — p(1)/q) (1); m) = s2 +p2(/qo)The second term of equation (D.2) can be further simplified to twice the real part ofthe expectation value of H(1P2H’ by writing the matrix element involving P3 as the165Appendix D. Derivation of Inclusive H Pionic Decay Amplitude 166complex conjugate of the adjoint operator and using P = P3.IH’tP3/‘ ) (H(1)fP3H(1))tI /‘ )* (D.4)= (k H(1)tP2H’>* (D.5)fl(1)fl(2)l(L)(2)p &(‘) ) (D.6)Assuming that oniy the zeroth order multipoles of H2 and 11(i)t are non-negligible,this permuted matrix element reduces to two terms: one invo1viig the s-wave A decaystrength s and the other involving the p-wave A decay strength. The integration over deliminates the cross terms.J ( ‘i i(1); ) 2(1)() +(D.7)where, 1)() =(‘jo(x/2)jo(y) () 2P’) (D.8)The spin and isospin parts of these reduced matrix elements are given below andthe spatial parts are expressed in terms of two exchange integrals i() and 77d(q). Thesubscript indicates which hypertriton chaimel contributes according to the value of L.The matrix elements of the permutation operator P2 are,(xy P2 ‘x’y’)=6(S, Li)S12(s,s, Sa)112(ia, i1) !:,:,X(x, y, x’, y’)(D.9)where, S12(s,s, S) = (_l)’[(2c, + 1)(2cr’ + 2 2I - Sc ‘1i2(a,a’) = (_1)ii(2j + 1)(2i + 1)] { }ZcrIy, x’, y’) = (xy (lccx)LcxML P2 X’y’(l\c’)LcxML) (D.1O)Appendix D. Derivation of Inclusive H Pionic Decay Amplitude 167Using S12(1, 1, ) —1/2 112(0,0) and S12(1, 1, ) = 1 the reduced matrix elementcan be written,= () —) (D.11)too— I I / 2 2 2 2 (0) (0) ,where, i3(q) i dxdydx dy x y x y (X)’/d (x )‘/‘A(y)bA(y)Joxjo(x/2)jo(y)X2(x, y, x’, y’) (D.12)f dx dy dx’ dy’ x2y2x/2y/22)(x)2)(xI)A(y)A(y/)xjo(x/2)jo(y)X2(x, y, x’, y’) (D.13)and, X2(x, y, x’, y’) = dtP1(t)2) — a’) (D.14)= + x’2 + 2/3xx’ta’ = x2 + x’2/4 + xx’f (D.15)To evaluate (2)() the matrix elements of 12/3 [2/32 — 1] are needed. S12 isthe total spin operator for particles labelled 1 and 2. Recoupling tile states to have goodtotal spin of particles 1 and 2,(i)( S9a1.2/3‘5 ) S3(s, s, Sa)S32(s’, s’, S)ss,X(3)(8S[2/3S2— 1]’Sa’)(3) (D.16)6(Sa, Sn’) S12(s, s,S01)S12(ss,S)[2/3s(s + 1)—1] (D.17)where tile identity S12(s, ‘, 5) = S13(s’, s, S) has been used which is proved using P = P3and tile reality of For tile hypertriton there are only two cases to consider since s == 1 and S, = or . UsingS12(1,0, 1/2) = —S12(0, 1,1/2) = \//2, S12(1, 1,3/2) = 1and including 2112(0,0) = —1 from the permutation of the isospin vector one arrives attile factor (—)[3/4 + 1/12] —5/6 for tile s-wave part and —1/3 for the d-wave part.i. e., i(2= 5()—r/d(q) (D.18)Appendix D. Derivation of Inclusive H Pionic Decay Amplitude 168Gathering all the results together,- f(X;mfHeff(H;mjH2,K= s2[1 +— d()] +p2(/qo){1 — — (D.19)which is equation 2.42 and holds when only the zeroth order multipoles in the expansionof H(2)H(1)t contribute.The integrals ?1s() and ij() were evaluated using Gauss-Legendre numerical integration. The deuteron wavefunction in configuration representation is given in Machleidt[26] and the lambda part of the wavefunction was found from the Fourier transform ofits momentum space representation thus,2°°A(y)= () j dqq2jo(qy)A(q) (D.20)= G) 2N(Q)i j qsin(qy) exp[-q2/Q]= N(QA) ()2exp[_Qy2/4]x— {cerfe(——— cerfe(- + )} (D.21)The last result is taken from Oberhettinger [44]. For values of y > 10.0 fin the followingasymptotic form was used,A(y) = N(QA) () exp[ exp[a2/Q] (D.22)Appendix EThree Body Basis and the Permutation OperatorsE.1 Three Body BasisIn a non-relativistic theory, three particles each with spin - and isospin can be represented by normalized states,23sis3i i) (E.1)where k denotes the momentum of the particle labelled i and are the spin and isospincomponents.The centre of mass motion can be split off by making the following transformation,K— k1+k23-= (E.2)where (a/3’y) is a cyclic permutation of (123). The above transformation is pertinentwhen all three particles have the same mass, aii approximation that will be made forthe trinucleon systems. The pair coordinate j5*. is the momentum of particle in theCM. The spectator coordinate is the momeitum of particle in the three body CM.Whenever the suffix on or 7is omitted it should be understood to take the value 1.i.e. (E.3)The conjugate variables to (K,7) are which are reciuired to satisfy,K.R + JLX+ k.r1 + k2.r + k3.r (E.4)169Appendix E. Three Body Basis and the Permutation Operators 170so that the plane wave states of three nucleons are invariant under the transformationwhich splits off the CM motion. By writing the equations (E.2) in matrix form andtaking the inverse of the transpose of the transformation matrix, it was found that therequired configuration space transformation is,-* 1R == r—r3=— + ). (E.5)The Pauli principle requires the trinucleon states be antisymmetric under interchangeof aiy two particle labels. States ‘) satisfying this criterion are formed by permutingpartially symmetrized states b23) which satisfy,P23H!’23) = I’23) (E.6)The operator P23 interchanges particle labels 2 and 3. The state I II’) defined by,)=(1+P2+P33 (E.7)is antisymmetric under interchange of any two particle labels. The operators P2 and P3are permutation operators and are defined in section E2 where the total antisymmetryof IIi) is also proved. The states &23) are expressed as sums of chaiine1 states a whichin the LS coupling scheme are defined by,(xy23) = (x, y)( a) (E.8)M(:a) = (E.9)where, l = orbital ailgular momentum of pair in their CM= orbital angular momentum of spectatorAppendix E. Three Body Basis and the Permutation Operators 171La = total orbital angular momentum= spin of pairSa total spin= total angular momentumMJQ = total angular momentum projection= isospin of pair‘a = total isospinMia = total isospin projection (E.1O)The symbol ® indicates the coupling of La to Sc. to form total angular momentumJ0, with projection The function Y and the vectors X and are defined below.yML() (LMi lc.mi; 1()Y() (E.11)n mxs:r5(1) = (ScyM8 sm5; mi)(sc.m3Irn2;m3) yrnlyrnsyrn3 (E.12)7775m2rn3i’v’ 2rni)(ic.mj rn2; m3) 771772173 (E.13)m2rn3Note that for the spin and isospin vectors, particles 2 and 3 are first coupled to someI and then the third particle is coupled according to s 0 and not 0 sThe requirement that‘23) be antisymmetric under interchange of particle labels 2and 3 is satisfied by including only channels for which (_1)15 = —1. Further, therequirement of positive parity for the 3He and 3H states is satisfied by demanding that1a + ‘a = even.Appendix E. Three Body Basis and the Permutation Operators 172E.2 The Permutation OperatorsThe permutation operators P2 and P3 are defined by their action on three ordered objectsthus,P2(.o*) (o*.) P3(.o*) = (*.o). (E.14)Along with the unit operator defined below these cyclic permutations form an abeliangroup G according to the group product table E.2.1(.o*)=(.o*) (E.15)The operators P2 and P3 can be written in terms of the non-cyclic permutation P23and P12 thus,P3 = P1223 (E.16)where interchanges the objects in positions i and j. This identification is used to provethat the matrix elements of P2 and P3 between the partially antisymmetrized channelstates ) are equal as shown below.(cP3c’) = (cjP12P23c’) (E.17)= (cP23cV) (E.18)= (—1)(—1)(P23P12’ (E.19)= (cP2c’) (E.20)1 P231 1 2’3P23 1•P3 1 2Table E.1: Group Product Table for the Group G.Appendix E. Three Body Basis and the Permutation Operators 173where the trivial property (P)2 = 1 is used as well as the fact that a) is antisymmetricunder interchange of particle labels 2 and 3.The total antisyminetry of the vector‘), which was introduced in equation (E.7), isproved as follows. There are three choices of particle labels which may be interchanged.They correspond to the actions of P12,P23 and P31. Consider first P23.= (1--P2+P3)3 P2323)= +23) (E.21)P23’P) = (P23 +P2312 +P2312)R& (E.22)= (P23 +P1223 +P2312)I (E.23)= (—)(l+P+P)Ii23 (E.24)—HP) (E.25)The action of P12 on ‘I’) is found similarly. To find the action of P31 on‘) it isuseful to make the identification P31 = P233. Using the group product table it is foundthat,(1+P2+P)=(1+P (E.26)so we have,P31i) = P23I’) = —‘I’) (E.27)This concludes the proof that Ii) is antisymmetric under interchange of any twoparticle labels.The permutation operators are not hermitian and their hermitian adjoints P and P( along with the identity element 1 ) form a group G’ isomorphic with C according tothe following transformation.1 1p2 — p (E.28)P3 p2tAppendix E. Three Body Basis and the Permutation Operators 174The above identification follows trivially after finding the irreducible representationof C. Since C is abelian, Schur’s first lemma implies that the irreducible representationsare one dimensional. Defining a basis by,(. o *) 1 (o *.) e24 (* • o) (E.29)the irreducible representations are,1 = 1 P2 = e23 P3 = (E.30)The tilde indicates that the quantity is a representation and not an operator. Theirreducible representation of P is given by,= (t_ (e2/3)* = e_i23 == P3 (E.31)and the correspondence given in equation (E.28) follows from the equality of the irreducible representations of P and P3 etc.The equality P P3 can be verified using the definition of the hermitian adjoint andcalculating a matrix element between two three-particle states which are simple tensorproducts of three orthonormal single particle states 0,1 and 2. The definition of theherniitian adjoint is,(BPA) = (AP2B)* V A), B). (E.32)Consider the state ) with particle labelled • in the state 0, particle o in the stateI and particle * in the state 2.= 0)• ® 1 )() 0 2)() 012) (E.33)The bracketed suffix to the ket labels the particle and the number inside the ketindicates the state.Appendix E. Three Body Basis and the Permutation Operators 175Using the normalization of the single particle states ( 012012 ) 1. Taking thepermutation operators to act on the particle labels, we have(201P012) 1(012P201) = 1* = 1 (E.34)P2t20l)= 012)i4=p3.Appendix FExpansion of a Function in the Gaussian BasisWe wish to express a smooth function f(x), ( which tends to zero as x —* oc at least asfast as i/x ), as a sum of Gaussian functions of range x. where the x, can be chosen tobe any values thus,flmaxf(x) = Aflx1exp[—( /x7)2J. (F.1)fl= 1The value of 1 should be chosen by inspection of the x—÷ 0 limit of the function f(x).Multiplying each side of equation (F.1) by exp[—(x/x7j2]and integrating one has,nia__j dx f(x) exp[—(x/x721 = A7 j dx x1 exp[—(x/x7n)2] (F.2)n= 11 1 1where, —— =—i-- + — (F.3)X XThis can be written as a matrix equation thus,XA F = A = (X)F (F.4), 1 1 1+1where, (- ), = (f )fl/rn = f(l + r)(xm7i)(A)7, = A7,(F)= j dx f(x) exp[-(x/x772] (F.5)and the problem of finding the expansion coefficients A has been reduced to tile mversionof a real symmetric matrix of dimension umax and also tile evaluation of umax integralsinvolving the function f(x). Tile IMSL library routine DLASF was used for matrixinversiOn ill tile application of the above method to the hypertriton wavefunction.176Appendix GMatrix ElementsTo calculate the rate for muon capture by 3He to the 3H final state, three reduced matrixelements need to be evaluated. They are,[11° = (3H Ijo(vr)3He)[jO1= (3HW (0.1)[]21= (3HU Ij2(v ®]1 He)/v.The reduced matrix elements are reduced in the spin-angular co-ordinates but notin the isospin space because the wavefunctions which were used were not exact isospinpartners. The are the spatial co-ordinates of nucleon j in the three body CM and arethus functions of the Jacobi co-ordinates £ and j and not R. Equation (B.5) definesB, iE and by,D 1 1 1 r10 1 —l . (0.2)By matrix inversion we have,1 l0 Ir2 = 1 — (G.3)1 1 1 -*73 hiand since x2 =—ii, y = x3 = —r2, y3 = —1+2), the followingexpressions for (2, Y2) and (, y) in terms of (, ) = (, ) are easily derived.177Appendix G. Matrix Elements 178= —2X—Y3— 1—Y2 =X3 =__ (04)The operator in the []21 reduced matrix element has the spherical harmonic Y2()coupled to the spin operator 3j to give total angular momentum 1. The precise definitionis,{Y2(i) 0 = (1M 2rni; 1mg)Y2m)S (0.5)fl2q7flwhere, S= +(a+iu). (0.6)Before deriving the form of the matrix elements, an expression for the norm of thewavefunctions will be expounded. The matrix elements are then obtained by minormodifications of the expressions for the norm.G.1 The NormThe full wavefunction ‘4’ ) is the sum of a Faddeev component ‘/) and its permutations.(G.7)The norm is,(0.8)Appendix G. Matrix Elements 179It is possible to simplify this expression using P = P3, P P2 and (1 + P2 + P3)2 =:3(1 + P2 + P3) but this will not be done in preparation for the matrix element calculations.In that case an operator sits between the permutations and does not commute withP2,P3. The expression (G.8) is partially simplified however by using the properties ofthe permutation operators.= (b1 +4P3 +2PP3+2PP3Rb) (G.9)This simplification will not quite be possible in the case of the matrix elements butthe expression (G.9) is general enough to explain all the techniques needed. There arethree parts; DIRECT, PERMUTED and DOUBLY-PERMUTED.DIRECT = (G.1O)PERMUTED = 4(iP3Ib) (G.1l)DOUBLY-PERMUTED = 2IPP+I/’) (G.12)A useful check on the calculation of the doubly-permuted parts was to verify numeri cally the following identities.KPP3) = () (G.13)(PP3) = (P3) (G.14)DIRECTThe Faddeev component is expanded into channels as described in chapter 4, equations(4.4-4.8).M 1() = x,y) [Y()®x1(i ] I(1) (G.15)> Cy)( (i a)La,sS; ) jc) (G.16)Appendix C. Matrix Elements 180The notation (l\)La, sSa; M) will be a useful shorthand for the spin-angularpart of the channel wavefunction. The spatial part of the channel wavefunction (x,y)is expanded as a sum over gaussian basis states (x)(y).(x,y) = AN(x)(y) (G.17)nNq(x) NiaX1exp(—x2/ ) qj(y) = NNy exp(—y2/y) (G.18)21+3/2 2 2 2 2 2= [() F(l + 3/2)] NN = [() f( + 3/2)]Expanding both bra and Ret,( I) = ( (la)L, sS; (l1A)L,sa’S’; Ica’ n]Vn’]V’f dx dy x2y(x),(x)(y),(y) (0.20)The spin-angular and isospin part of this expression yield a factor 6(,‘) so thedirect part of the norm only receives contributions when the channel in the bra matchesthe chaimel in the Ret.la+(/ /‘) AANf ( 2xxflt 2ynyn’ (0.21)nNn’N’ X + X71 y +The total number of contributions is 405,000 for the 8-channel wavefunction and545,000 for the 22-channel wavefunction.PERMUTEDThe permuted part is the overlap of the Faddeev component /‘) with its permutationP3 ‘/‘). The spatial representation of the permutation is,(IP3)= (X2, y2) [@ y2) 0X1(3)] ‘(3) (0.22)where the property P = P2 has been used. The spin and isospill vectors are symmetrizedwith respect to interchange of particle labels 1 and 2 as indicated by the label (3).Appendix C. Matrix Elements 181The spin vector X’(3) is expanded in terms of the basis X(1) as shown below.= S(s, , S) XMs(l) (0.23)SrSwhere, Si3(s, s,S) = XMt (1 )X(3) (0.24)11= (—1)[(2s + 1)(2s + 1)} j 2 (G.25)1 Sc So J2The isospin vector is expanded similarly and having the same SU(2) algebra the resultfor‘13 follows immediately.‘i’(3) =I13(i,i) ‘(1) (G.26)1 1where, 13(i, i) = (—flt{(2i + 1)(2i + 1)] { 1 (0.27)11.12 2 )Tile spherical harmonic is also expanded. In equation (0.28) the sum is over all l,and 11 = 0 to l, ‘\i = 0 to A. where 12 is set to l — 11 and \2 is set to— ‘\2./xh11yl22 N LMYj(2, i/2) = R(l1Ai)( Y (,l;) (0.28)\ X21J ,‘11 + 12 =+ ‘2Iwhere, R(l111)=(—1 )Ai (2l + 1 )(2 + 1) [ (21)! (2A)! 2(2l )!(212)! (2 )!(22)!]lI 12 lyI h1+2()’\1(i0 110; o)(o 120; 20) { A1 A2 A (0.29)(2) L JAppen clix C. Matrix Elements 182This expansion is found using some standard recoupling theory ( see [34] ) and also thefollowing two identities.__l1b2 4 (21 + 1)’ 2-Yi( b) [(2l + i)!(212 (G.30)2) = y@) [(2li+i2i+ 1)]2(laO I hO; iO) (G.31)Identity 1) follows from partial wave expanding = eieib and regarding thex —* 0 limit. Identity 2) follows by noting that y[(&, ) must transform like ‘()and finding the coefficient by evaluating an integral of three spherical harmonics.The channel wavefunction evaluated at the permuted arguments x2, Y2 has a simpleform and this feature exemplifies the beauty of the Kamimura wavefunctions.= AN(x2)(y (G.32)nNa 1 12 2—i-- 2NX2 exp[—(x + y + x.y )/x7j](Y2) = Nniy exp[—(x2+ y2 — )/y)] (G.33)Notice that tile unpleasant xy dependence cancels with the xmQya dependenceof yM(2,Y2) as shown in equation (G.28). This is a consequence of the basis functionsq’4(x) having sensible x —* 0 behaviour. There is an angular dependence on andcoming from (x2)q(y which is partial wave expanded as shown below.exp(2c) = ik(2cxy)(_1)k[2k+ 1]4y() (0.34)where, ik(Z) = jk(iz)/(i)k 2c =— + (G.35)x YNExpressing tile permuted Faddeev component as a sum over the dummy index a’rather than a, tile PERMUTED part of the norm is,4( k IP3i) = 4 ANAjNIN?1Nflh1iNNANNIA, Xaa’ nIVn’IV’Appendix 0. Matrix Elements 183I x2+ +11 1y2 12+\2 k [x (_i_ + 31Jdxdy2(i+i+9)2(i+i+ N1(XI expYN jj Jk I L \xl + l = lal+ A =Ra’l 1 ‘)\al)S13(5, so’, Sa’)I13(i, ia’) X21 1Y(_1)k[2k + 1]( (l\a)La, saSa; 4irY@, (la’i)La’, sSa’; )( i i). (G.36)The spin-angular matrix element which needs to be calculated is thus,(_l)k[2k + 1]( (1 a)La,saSa; (la’’)Lai,sSa’; ) = (G.37)I(kllaa’La) X 6(s, Sa)(Sa, Sa’)6(La, La’) (G.38)where, I(klalaALa) = (_l)k+L(2k+ 1)[(21a + 1)(2a + 1)] Xla l(laO kO; iO)(AO kO; O) { ,.(G.39)L JThe isospin part yields a simple i). The final task is to evaluate a double integralH with the following form which follows from equation (0.36) after a scale change ofvariables.H1(k, m, ri; c)= f dx dy x2+2rny2+2fl+ke_se_Yik(2cxy) m, n = 0, 1,2...(G.40)The integral is performed by expanding the bessel function [147, p348].1 (cxy)21’ik(2cxy) — /7r (cxy,___________(0.41)— 2 F(k + ) (k +In the above (k +,is the Pochammer symbol defined by,F(a +p) (0.42)F(a)Appendix U. Matrix Elements 184Performing the gaussian integrals yields F functions and so,F(m+k+)F(n+k+) k2Fi(m+ k + ,n+ k+ ;k+ ;c2)11(k, rn, n; c) = 8 F(k+) Cir(2rn + 2k + 1)!!(2n + 2k + 1)!! ck(2k + 1)!! 4+,n+n+k (1 — 2)rn+m+k+2F(—m, —n; k + ; c) (043)The last equality follows by using F(n + ) = /ir(2n + 1)!!/2’ and also a lineartransformation for the hypergeometric function [147, p143] which yields a finite (Jacobi)polynomial.The final expression for the permuted part is,4(/P3H/)) = 4 (La,Lai)6(Sa,Sa’) XxatN’G’ (kl ) )H1 (k, n, n; cp) 2m+k+3,3,n+ (G44)kl + l = la’A + ,\,‘a a’where, (klA) = I(klaAala’Aa’L lRa’1lIA/l ) xa) 21 Jlot‘‘a313(Sa, ba’, Sa’)113(ia, ia’) (0.45)Cp2m = la+l+Aik Cp (cp<1)2y2n= ‘a + 12 + — k (0.46)1 1 1 9= x2+4x2+16y21 1 1 1= YN x, 4y,1 1 3----+—- (0.47)= x, 4YN’Appendix G. Matrix Elements 185The coefficients G’, S13 and ‘13 were calculated separately using routines for ClebschGordan, Wigner 6-j and Wigner 9-j symbols written by JGC.The total number of contributions is 6,885,000 for the 8-chaimel wavefunction and29,985,000 for the 22-channel wavefunction.DOUBLY-PERMUTEDIn principle, the calculation of the doubly-permuted parts is no more complicated thanthe calculation of the singly permuted parts. This is because of tile gaussian natureof tile basis functions (x) and qSj(y). The unpleasaiit dependence arising from=1 — — and Y2 = — in both bra and ket is combined simply and partialwave expanded as in equation (G.41).exp(2c1)exp(2c2x.y) exp(2(c1 +c2)) exp(2c) (G.48)Tile filIal result for tile doubly-permuted parts of the norm can be written downdirectly.2(l/PP3+ PP3) = 2(La,Lci)6(Sa,Sci) XANANlN?1aNflh1iNNNNlXi XnArn’ f\J’C33 dDp) 2m+k+3 2n+k+3anp IDP (.)+ G)(kllilA)H2(k,rn,n;c23, dDp)11 + 12 =1 + 2 =1! 1! 111 + 2 =) + ) =where, G I(ki R(l11i )R’(l 1Y) x10 _)0 1cy -‘‘Appendix C. Matrix Elements 186xSIi3(, ia)Ji3(,i1) (0.50)= > I(kii1AL)R(l1Al )R(l iJ1) xLi23- a tl \ C1XS12(i, i)Ii3(i,i1) (0.51)DP/32m =l++l+)—k c33 273DP/3DF2n 2+,\l)—k C23= (0.52)2731 1 1 9 94x2 + 2 + 2 + 16y,4x,‘6YN1 1 1 1 1= 224242xnl1 1 1 3 3—---+-—-+——= X72 x1 4YN 4yr’1 1 1 3 372 = (0.53)The cIefiiition and value of S12(s, s, S) are,S12(s, aj S) xSMt(l) XSOM(2 (0.54)Sar( 1)$ /— 13S,Sa,Sa). (0.55)The symbol R(l1Ai)arises from the expansion of yM(3,)and is equal to(—1) 1+12 R (11 A1 i).The total number of contributions is 35,336,250 for the 8-channel wavefunction and190,931,250 for the 22-channel wavefunction. The evaluation of the norm for the 22-channel wavefunction required 20 hours of CPU time on a VAXstation 3100-M76/SPX.Appendix C. Matrix Elements 187G.2 The Non Spin-Flip Matrix Element [lj°There are two extra features in the matrix element [i]° as compared to the norm. Theyare the isospin lowering operator 1 and the zeroth order spherical Bessel function jo(vr).Before tackling these features a simplifying identity for the matrix elemeilts involving thesum of three operators is proved. The sum of three operators 0 0 + 02 + 03 can bewritten,O O1 +P2O1+P3OP (G.56)where the identification of 02 = P20P has been made to ensure that (1 IOiI1) =2 I02 2). By noting that,(l+P+3)(O1(l+=3(l+P1 (0.57)we see that the matrix element of 01 + 02 + 02 between symmetrized states is threetimes the matrix element of 0 between symmetrized states. Thus,[110 =(3H3Ijo(iiri)e). (G.58)The matrix element of three times the isospin lowering operator is,23J22= i)(3— 4a) (G.59)which necessitates only a simple modification of the expressions for the various parts ofthe norni.The bessel function jo(vr) = jo(2iy/3) modifies the double integrals over x and y asshown below. The direct part integral is straightforward, the permuted part integral isnot.a) Direct Part.In this case, the integral f dy y2+2e 2/YNl becomes f dy y2+2e_Y/vNljO(2I,y/3)Appendix G. Matrix Elements 188where YNN’ is defined by,2 = + (0.60)YNN’ YN i/N’The latter iitegral is performed by expanding the Bessel function using equation(0.41) and using Kummer’s formula [147, p298j to reduce the confluent hypergeometricseries F1 to a finite polynomial.J dy y2+2e_YN1jo(2vy/3)— exp (—p2)1F(—A; ; p2) (0 61)f° dy2+>e’ NN’where, p = 1JYNNI/:3 (G.62)a) Permuted Part.In this case the integral which needs to be performed is H2.H2(k, n, n; c, d)= j dx dy x2+2flky2+2ke_Xe_Yi( cxy)j0( d (0.63)where d and c are real and the magnitude of c is less than one. The fact that c < 1 forC Cp, c33, c33 can be shown by expanding 4_2_2—y and noting that only positiveterms remain.Expanding both k and j0 using equation (G.34) yields a form for H2 containing ageneralized hypergeometric fullction of two variablesr(2m + 2k + 1)!!(2n + 2k + 1)!!H2(k,m,n;c,d)= (2k+1)!! 4+m+n+kc1(n + k + ; m + k + ; k + , ; c2, —d2) (0.64)where, i(a;;7i,2;x,y)()pq()pxPy(0.65)p,q=O ()p(72)q p. q.The fuiction Ji1 is the confluent form of the Appell series F2 [1481,[147, p192] andhas been discussed briefly by Humbert [149]. The complicating aspect of ‘IIi is the mixedPochammer symbol (a)p+q. The following theorem was developed in order to reduce ‘I’to a finite polynomial. The proof is given in detail since the theorem is a novel result inthe field of generalized hypergeometric series.Appendix C. Matrix Elements 189Theoremm () x+ m; i, 72; , x, y) = (1 — iF@ + 7); i —p=o (y 1 — XI(0.66)where, rn=0,1,2... Ix< 1, Re(71) >0, mC (0.67)p!(m— p)!and, 1F(a;7;x) = (0.68)q=O (7)q ¶7!ProofNeed,i) (—rn) = (—1) rn!(rn—p)ii) 1(1_x)a = x< 1iii) ( +p + q) — (c)p(c+p+r)g (0.69)7’— (c)p(c7+p)qiv) 2Fj(a,;7; 1)—Re(7 —— ) > 0v) ()p( + p)q = (a)pqThe identities i) ii) iii) and v) are easily proved using the definition of the Pochammersymbol, equation (0.42). The identity iv) is proved in Wang and Guo [147, pl561. Letus work on the right hand side (R.H.S.) of equation (0.66) by expanding the confluenthypergeometric function and expressing 7C in terms of a Pochammer symbol.00 /R.H.S. =(—x)° ( + p)q yq 1 (0.70)p=Oq=O (7) p! (y q! (1Now expand l/(1 — x)+P+ using ii).m ((—m) p+r+p + q) + p)qyR.H.S. = (—1 (0.71)pO q,rO (71)p p!r!rNow use iii) to express ( + p + q).00 ()p+r( + p + r)q yR.H.S. =p=op,r=o (71)p(72)q p!r!q!(0.72)Appendix G. Matrix Elements 190Consider the term with q fixed i.e. write,1 yqR.H.S. = Tq (0.73)q=O (72)qq!mthen, Tq = (—m)P@)P+r@ + p +r)q p+rA3 (a)3p=Or=O (71)p p!r! s=O(0.74)=2F1(—m, —s; 7i; 1).where, A3 (_1)P(Th1)Ps!(‘Yi)p p!(s —p). 0 (‘ii)(0.75)Applying iv) and checking that Re(71 + m + s) > 0 we have,2F1(—rn, —s;7; 1) = (71 + Tn)3 (0.76)and so replacing the dummy index s by p and using v),(a)p+q(71 + 7n) x 1qR.H.S. = (G.77)p,q=O (71)p(72)q p! q!= ‘I1i(a;7 +m;7,-y2;x, ). QED (0.78)The integral H2 can therefore be reduced to a finite polynomial.ir(2m + 2k + 1)!!(2n + 2k + 1)!!H2(k,rn,n;c,d) x (G.79)(2k + 1)!! 24+rn+n+k3’/ —d2C (m+ I____ ____m(1 _c2)7 2 (k+) _C2) iFi(n+k++p;; 1 _2)lr(27n + 2k + 1)!!(2n + 2k + 1)!!x (0.80)(2k + 1)!! 4+m+n+k3’ / 2 p d21 m + k + r) ( cck exp[—d2/(l — c2)j mCp_______ ______(1 — c2)7 0 (k + \1 — C2)1F(—n — k— 1 — 2)where Kummer’s formula has again been used.The final expressions for [110 are,[1}° = (‘3HW(1 + P2 +P3)O1(1 + F2 + P3)3He) (0.81)Appen clix C. Matrix Elements 191= (3HWO 3He) +2 [K OP3W3He) +(3HeO3)]+2( 3HW10l13 + 14OlPI/)He) (G.82)DIRECT(l) + PERMUTED(l) + DOUBLY-PERMUTED( (G.83)DIRECT(l) = (3HHOl3He) Oi = 3Ijjo(iiri) (G.84)= (3—4i)x ( 2x7x’ )1a+ ( 2AN(3H)A,N,(e) + , + 2yn,!nI’/n’IV’. 2exp(—p2)Fi ;, ) (G.85)PERMUTED(l) 2 [(‘3H OlP3He) + (3HeOP3W’3H)] (G.86)= 2 6(Lc, La’)(Sc, Scx’)Si3( ,a’,Sc)I3(ic,i’)(3 — 4ic) Xca’[A(H)AN,(3He) + AN(He)AN,(3H)]N711 N7111,NNNN1, XnNnl Al’2m+k+3 2n+k+3G(kl)H2(k,7n,n;c,dp)cp (G.87)k7/11 + 12 =)4 + = Ac,’DOUBLYPERMUTED(l) = 2 [( P301U/’3He) + (3H POlP3iRb3He)]= 28(Lc,,Lc,’)6(Sc,,Sa’) Xc,cNc,1, NNIA, XnNn’N’Appendix G. Matrix Elements 192F 33(1)(kllllil)H2k,rn,n;c,dDP) 1 2m+k+3 2n+k+DP /3DP (0.88)[+G)(kllll)H(k,rn,n;cdDp)j‘1 + 12 — 1+ .A2 =‘ + 12 = lcxlI+ ) = ‘“cr’Iwhere, d = v/3p dDp = v/3Dp 3 (0.89)G(kl1Aj) = > I(klcycjaiArLa) R(l lcy)’) xa at•513(s, s,S)I13(i, ii)(3 — 4i) (0.90)33(kl1Al)= I(klc\cy1crJc’Lcr) R’2(li\ilcx\ “P’l’ )‘ lci)t.cyi) X) 2 1 1Ta a TatS13(s, s, S)S13(s, xS13(i, i)I13(i,it)(3 — 4i) (0.91)G’’ (kl11l) J(klct\cslcti,\aiLa) R(l1\i\ XC) 21 123(i)i0 T,Si2(s,s,S)S13i,S)SI2(i, i)Ii3(i,it)(3 — 4i) (0.92)There are exactly tile same number of contributions to each part of [110 as there areto tile corresponding part of tile norm.G.3 The Spin-Flip Matrix Element []0IThe presence of the spin operator in the matrix element [6]0t1 has two effects. Firstly, itchanges the value of each channel-channel contribution. Secondly, it increases the numberAppendix C. Matrix Elements 193of channel-channel combinations which can contribute since the S, = S’ selection ruleno longer holds. The fundamental result that is needed is the reduced matrix element ofi between the LS coupled channel states.((la, )La, sS; i (lc, c’Sa’; )= (la, la’)6(c, a’)6(L, L’)(c, ) (0.93)where, S(8aLaSSa’) = ( l)L+So+S_SI2{($+ 1)(2Sa’ + 1)] X1 1 (G.94)I 8a S’ J L S Sa’ L JThe convention that has been used for reduced matrix elements is that of Brink andSatchler [34] i.e.(J’M’T1J ) = (J’M’ JM; KQ)(J’WTjJ) (G.95)where T1 is a tensor operator of rank K and projection Q. (j-) is the reducedmatrix element of the rank one spin operator between two spin states and equals \/.The expressions for the various parts of [5]01 are given below.[]Oi DIRECT’)’ + PERMUTED + DOUBLY-PERMUTED (0.96)DIRECT’ (/)3HO1 IhHe) Oi = Ijo(iii1) (G.97)Xcyc’(3 — 4ia)S(ScL,Sc,S,’) XAN(H)A:N,(He)(2x7xi!) ( 2YYi) 2 x,iNn’N’ ••+- —I— y,2 2exp(—p )iFi(—;r;p) (0.98)Appendix G. Matrix Elements 194(0,1) —PERMUTED() — 2 [( 0lP3Wb3He) + (1i3HeOP33H)] (G.99)= xFA 3I nN H)A:N,(3He) + N71,NN NN’, XnNn’N’G’(klçA’jH2(k,m,ri;c,dp) 2rn+k+3 2n+k+3/3p (0.100)k1 + 12 = lfA + A =DOUBLY-PERMUTED°1 2 [(?/)3HIIPOlP3Wl/’3He) + (h/’3HII140P3W1/’3He)]= 26(L,L,1xAN(H)A:N,(He)N?ll NNNN’,f\J’ N’G’ (ki AllAc)H2(k,7n,n;c33,dDp) 1 27n+k+3 2n+k+333(e) 1DP IDP (G.101)k 23(a)G’ (kllA)H,rn, n; C23, dDp) j11 + 12 la+ A2 _l’l + l2 = lc’A + A =where,G’ (klcAc) = I(klaAala’A’Lcx) R’(lcAcic’Acr’) X()1cr a15a’ Sc,r)S(aLaS,ySoi) xI13(i,i)( — 4i) (0.102)C,33()(k1IAI1IAI) = I(kii \L) R(liAii )R’(lcAciaii) X1a Ctl01:5c0,Appendix 0. Matrix Elements 195Si3(S,a,Sa)S13(S,5a’,Sa’)S(SLaSaSa’) X13(’i, a)Ii3(,a’)(3 — 4i) (G.103)G(kl1,\l, ) = I(kl la’Ac,’Lc,) Xla)kalai\aiS12(s, s, Sy)S3(5, so’, Sa’)S(LaSaSa’) X2 12(Z z,)I3(z, zi)(3 — 4z) (G.104)G.4 The Spin-Flip Matrix Element []21For this matrix element we need the reduced matrix element,((laa)La;saSa; [Y2(y) 0 (la’&)La’;sa’Sa’;= 6(la, la’)’(Sa,Sa’)S(SaL La SaSa’)RYl z\a)iLaL&) (0.105)where,S2(aLaLa’Sc$ar) (i)2[3(2La + 1)(2La’ + 1)(2Sa + 1)(2Scx’ + 1)] XLa La’ 2{ } S Sa’ 1Sa Sa’ 1RY2(laaa’LaLa’) = ( l)1a+LQ[(2 + 1)(2Aa’ + 1)] XI ,\ 2 1(20 )a0; )‘a’O) . (0.107)I La’ La la JThe spatial integral differs due the second order spherical Bessel functions whichreplaces the zeroth order spherical Bessel function appearing in [1)0 and [8]01.For the DIRECT part, the y part of the integral is,j dy2’exp(—y2/y,)j2 2vy/3) = (0.108)/ir(2\ +3)”y++Aal p2exp(—p)iF(—Aaai + 1; ;p2)Appendix G. Matrix Elements 196a +where, Ai = (G.109)2The spatial integral for the permuted parts is H3 with rn = 0, 1,2... and n 1,2,3...‘00H3(k, rn, n; c, d) J dx dy x2+2ky2+2ke_Xi( cxy)j( d (0.110)0— ir(2rn + 2k + 1)!!(2n + 2k + 3)!!— 15(2k + 1)!! 3+L+n+k x (0.111)ckd2 exp[—d2/(1— c21 m (n+ k+ ( c2)P — k—p+ 1;; 1 _2)(1 — c2)fl+k+ (k +The final expressions for [121 are,[]21 DIRECT’ + PERMUTED + DOUBLY-PERMUTED (0.112)DIRECT2’1— (3HWO1U7!3He) = j2(v i)v[Y2()0 (0.113)(77)6(lc, 6(j7,i771)(3 — 4i77) X•$2(SaLc,La’Sc,S771)RY1 \ Acx’LL ,l)X2x,x,’ANA:N,NNNN/, 2 ) x (0.114)nNn’ N’+ 3)!! 2YNN’ 30 2 p exp(—p)iF1(—)77+ 1; ; p2)PERMUTED21 2 [K /)3H OP3U’3He) + K3HeO3W3H)] (0.115)= 2 [AN(H)AN,(He) + AN(He)AN,(H)] x7777’X2’,n+k+3G’ (klA)H3k,m,n;c,dp)cp ‘‘‘ (0.116)3(,.21k11 + 2 177’+ =Appendix C. Matrix Elements 197DOUBLYPERMUTED2’1— 2 [( b3H POIP!hHe) + 3HPOlPb3He)](u) —= 2 AN(H)A:N,(He) xcc’ nI’Jn’I\J’NfllQN?’1,NNNN’, x (0.117)33(2l)(kllllAi)H m,n; C33 dDp) 1 2m+k+3 2n+k+3I aff, /3DPk[+ 2321(kl11)H,rn, n; c23, dDP)]11 + 12 lc+-2l’l + l = lyl,v1 + =where,03(2l)(Lli\i) = I(klc lc\L) R’(llc’)RY2(1a’) cx’LcLc’) XIQI513(Sa, ‘,Sa1).92(SaLLiSaSal) X— 4i) (G.118)rac’ (kl1)l) = I(klalcLc) R(liilc )R(lAccy’) XLi3321TQ )\(1QlRY2(l’\c’LaL’) xS1(s, s, S)S3(s, so’, XSIi(, a)Ii(i,c1t)(3 — 4i) (0.119),‘ (1cl\il)) = I(1la la#\La) R(liilc a)R’(l ii) XTc ,\RY2(lc’AAa’LaL’) X.912(S, 5cy, Sa)S13(5, Sc’,Sa’)S2(sLoLcr’So a’) )<SI(i, i)Ii3(i, ii)(3 — 4i) (0.120)Appendix G. Matrix Elements 198G.5 The Form of (p, q)In the calculation of the two body break-up momentum distribution p2(d) the followingchannel decomposition of the full wavefuiiction in momentum space representation isrequired.M(pqIW3He;MJ) = (p,q) [Yh()ØX1(l ] (1) (G.121)In this section the momentum space representation of the Faddeev component is foundand then an explicit form for I1a(p, q) in terms of the Faddeev components is presented.G.5.1 Momentum Space RepresentationGiven the relationship of the basis kets,(Iz7) = exp(—i)exp(—i (0.122)and the expansion of the Faddeev component‘/‘ in channel states,Mj ii= (x,y) [Y@)®x!(1)] (1) (0.123).Mj ii(7) = (p,q) [Y()øx1(i ] (1) (0.124)we have the fo1lowiig transformation formula for the channel wavefunctioni(p,q) (_1)1 () jdxdyx2y2ji(px)j(qy)i(x,y) (0.125)With the sum of gaussians form for y) the momentum space representationb(p, q) has a very similar form to 7/’(x, y).q) = (—1) p1qexp(—p2/ )exp(—q2/q) (0.126)nNwhere, p7, = 2/x qN = 2/YN and, (0.127)(2 3/2 2 2 (2 +3/2 2 2= ) F(l + 3/2) = f( + 3/2)(G.128)Appendix G. Matrix Elements 199G.5.2 An Expression for (p, q)The function Jia(p, q) has a direct and a permuted part.= (pq(lak)L;saS; (1 + P2 +P3),b) (G.129)= L’g(p,q)+(p,q) (G.130)The function /‘(p,q) equals the function (p,q) given in equation (G.126). Thepermuted part will eventually be taken in an inner product with a wavefunction anti-symmetric under interchange of particle labels 2 and 3 and so the identity “P2 weaklyequal to F3” can be made. The term “weakly equal to” is applicable when two operatorsare equal only in a restricted sub-space, in this case the restriction being to the sub-spacespanned by vectors antisymmetric under iiterchange of particles 2 and 3._____I(p,q) 2(—l) 2 xcr1 n]V()21 G’(kl) ph1+1ql2+2 exp(—p/N)exp(—y2/)ik(pq/7) (G.131)k1! II I+ —)I’ + ‘21 1 1 1 3 1where,—j-- = — + -- -— = — — + -i-- (G.132)nN 4P q 7nN 4Pn N32 62+2 (G.133)Pn NAppendix HDeuteron Spectrum in the PWIAIn this appendix, equation (6.17) is derived and an expression for p(q) is found. Thestarting point is Fermi’s golden rule and the safe approximation that the weak interactionis pointlike at the values of q2 encountered is made so that the matrix element can bewritten in a current-current form. Normalizing the states to one per unit volume, thedifferential rate is given by,dF= CF2V1d2CLW’3(2) (H.1)The factor 2 underneath CF comes from the definitioii of the weak coupling constant andthe factor 47r underneath the atomic wavefunction evaluated at zero separation comesfrom the use of the s-wave Bohr orbital i.e. (O)= p (i= 0). La is the leptoncurrent-current tensor.L = (;s7(1—75);s)si ,sv[iv + — (.v)g +i6v}/(mv) (H.2)W is the hadronic current-current tensor combined with some of the phase space factors.= (i Jt f) J (2)3 (236 + d - P - q)( fJI) (11.3)The currents J and are the sum of three single particle currents from each of thenucleons in 3He.= I(j)j(j) (H.4)200Appendix H. Deuteron Spectrum in the PWIA 201The fact that a proton in the initial state must become a neutron in the final state istaken into account with the isospin lowering operator for particle (i), Ia).Using plane waves for the final state, it will he shown that the hadronic tensor Wcan be written in terms of an analogous nucleonic tensor W thus,W= () fd3W2(II) (H.5)where the nucleonic tensor is defined by,W = (p,spjfl,Sn) f(2)36fl —p — q)(n,S1jp,sp). (11.6)S Spand p2 is the two body break-up momentum distribution.The spin projections of the struck proton and ejected neutron have been written asand s, respectively. Equation (H.5) has an appealing form. It states that the totalnuclear tensor is that of a single proton averaged over the range of proton momenta foundin the nucleus weighted by the probability of finding a proton-deuteron with the relevantrelative momentum. The proof of this equation follows.The plane wave form for the final state is,(l+P2+P3) dJj ) — d (where, ( n; s, = (263(* — q (H.8)(J7d;Jd)(l) (p)[(j3) ØX1j. (11.9)1=0,2 2The permutation operators which affect antisymmetrization of the final state are writtenP2 and P3 and is the neutron momentum in the neutron-deuteron CM,= (— -d).The correct normalization of the final state,i.e. ( .1’ f) (27r)36i* — *)6(si s1j6(J, Jd) (H.10)Appendix H. Deuteron Spectrum in the PWIA 202follows upon the assumption of incoherence. This assumption makes the approximationthat the knocked out neutron has negligible overlap with a bound nucleon.To calculate the matrix element of the current J a representation of the one bodycurrent in the three body Hilbert space is needed. To this end the decoupled spin/isospinbasis {, y} is introduced. The set of discrete quantum numbers € describes the spectatorparticle and the set -y describes the pair.— {s1,i1} = in3,i23, rn} (H.11)where .s and i are the spin and isospin projections of particle 1, and i23 are thetotal spin and isospin of particles 2 and 3 and rn3 and rn are the total spin and isospinprojections of particles 2 and 3. The one-body current has the following representationin the the three body Hilbert space.•a--I I I c -I 3 - -*1 3 — -.1 /(p, q,€, p , q , € , ) = , q ) (p — p )6 (q — q — 2/3Q)(7,7) (H.12)The current imparts three momentum Q to nucleon 1 and does not change the momentaof the other two nucleons. The spin and isospin of nucleons 2 and 3 are not changedwhich explains the‘,“) factor. The factor j,(7, q) is the current for the transitionof a nucleon with momentum spin projection .s, and isospin projection z, to a nucleonwith momentum 7’ = + Q, spin projection s and isospin projection i.Using the antisymmetry of the initial state and the properties of the permutationoperators it is found that,(fJi)= (‘)( ns1;dJd J) + J(2) + J 3He) (H.13)The assumption of incoherence is applied again this time to the overlap of the neutronwith the :3He wavefunction. i.e. (j)(flhlJJ3He) = 0. This corresponds to neglecting theAppendix H. Deuteron Spectrum in the PWIA 203direct cleuteron knock out process.(f IJ i) = (1)( d; J Ijl)I 3He) (H.14)The current thus decouples into a proton —f neutron one-body current and a deuteronhelion overlap.(fii) jd3p(2:3(+-+ ) ISp ja(q) (H.15)where,= (iJ 1m; un8)I 0,271l1= {l,ms,0,0}W3He;Ji) (H.16)The hadronic tensor is now easily evaluated and found to be,W = fd3pP()*Q)JJi811 Sp8xj83(p,q)j8p,q)6(EP + Q° -E1)/(2)3 (H.17)Performing the sum over J and Jd on dsP()Id3P()* yields a quantity which iszero unless = and is otherwise independent of s. The sum over s, in equation(H.17) can thus be moved to cover only the one-body currents. Defining the two-bodybreak-UI) momentum distribution p2(p) in the standard way,P2(P) JiJis() 12 (11.18)‘d 8Pwe have,W= () fd3pp2(p)W (H.19)Appendix H. Deuteron Spectrum in the PWIA 204where the nucleonic current-current tensor is given by equation (H.6). Extracting theenergy conserving delta function from W/ to define the matrix element squared by,() JLW =M2 (2 )36(Efl - - Q°)the expression in equation (H.1) reads,dF 7Jp2dp(E + Q° — E1)p(p) M2(p,Q). (H.2flThis is ow written in terms of d, x and Q(d, x) using,d3ii 1 i1ii= Q J—=-] dx] Q2dQ47r 2—1 0p=dE = M3Re_(d2+md2)E11= (Q2 + d2 — 2dQx + mi2)= rn,— Q. (H.22)The integral over Q is eliminated using the following identity for the delta function,6(E + Q° - E) =8([Q()). (H.23)Performing the trivial integration over Q, equation (6.17) is arrived at.=7p2(d)d f, dx (Q-dx) M 2 (d, x) (H.24)Appendix IThe Nucleonic Weak Current and Nucleon Form Factors1.1 General IntroductionThe charge changing weak interaction current of quarks and leptons has a pure vectorminus axial vector (V-A) form. For example, the muon to muon-neutrino current is,(iii, IJwK) = ü(v)(1 —5)u() (1.1)where the coupling constant GF/\/2 has been omitted. The nucleonic current does nothave the same simple form because nucleons are composite objects and thus their totalcurrent is a matrix element of the sum over currents from the constituent particles. Thenucleonic current can, however, be conveiiently pararneterized in terms of form factorsby writing it in terms of independent quantities with known Lorentz properties,•a,i3 aaq____gv7-- gM 1a—LIVIN(n J< p) = u(n) a u(p). (1.2)(gA + + gp2where, qa = a pa= 4-rnomeiitum of neutronpa= 4-momentum of proton—75= j70l23=(1.3)205Appen clix I, The Nuc]eonic Weak Current and Nucleon Form Factors 206gv = vector form factorg = weak magnetic form factorgs = induced scalar form factorg = axial form factorg induced tensor form factorgp = pseudoscalar form factor (1.4)In equation (1.2) q is the 4-momentum transfer to the nucleon and the form factorsare Lorentz scalars. In general, the th are functions of the scalars q2,p and n2 butwhen the nucleons are on-shell ( i.e. p2 = = M ) this simplifies to the familiar q2depence.The current splits into a vector and an axial vector part according to the propertiesunder improper Lorentz transformations ( i.e. those transformations containing an inversion of the space axes or “parity” transformation ). The vector part has parity +1 andis given by,(‘ Jw,v) = (n) [gv7a + gMi2 + gs2u(p). (1.5)The axial vector part has parity —1 and is given by,(“AL WIK,AI) = ü(n) [gA7a75 + + gTi2 5] u(p) (1.6)The fact that both vector and axial-vector currents contribute implies parity violation,a feature peculiar to the weak interaction.The current may be further categorized into first and second class parts using a classification due to Weinberg [150]. The current is said to be first (second) class if undera G-parity x parity transformation it is even (odd). The G-parity transformation is arotation of 1800 in weak isospin space combined with the charge conjugation operationAppendix I. The Nucleonic Weak Current and Nucleon Form Factors 207C, G=Cexp(i7r12). The second class currents correspond to the scalar and tensor formfactors and standard practice is to set these form factors to zero. At low q2 their contribution to the current is small and so this may be a reasonable approximation. Anotherapproach is to hypothesize that the weak current is invariant under GP, where P is theparity transformation. In that case only one sign of GP can contribute to the currentand first class currents are kept in favour of secoid class currents by the properties ofneutron beta decay.To have complete knowledge of the current then, it suffices to know the values of theform factors gv,gM,gA and gp.1.2 The Vector Form FactorsThe isotriplet vector current hypothesis [151] provides a simple link from the vector partof the weak current to the electromagnetic current of the nucleons. This hypothesisstates that the isovector part of the electromagnetic current and the vector parts of thecharge raising (n —* p) and charge lowering (p —* n) weak current form an isotriplet ofcurrents corresponding to the components 13, 1+ and 1 respectively, where 1+ I + ‘2.This hypothesis ( as it was in 1958 ) follows from the standard model of electroweakinteractions given that the proton and neutron form a good weak isospin doublet.Writing the isovector part of the proton electromagnetic curreit as,(1’JMp) + Fiu2]u(p) (1.7)and the neutron current as,(JEM) = u(n’) [F7 + Fiu2 ] u(n) (1.8)with Dirac and Pauli isovector form factors F1 and F2, the following identification canAppendix I. The Nucleonic Weak Current and Nucleon Form Factors 208be made using the isotriplet vector current hypothesis and the Wigner-Eckart theorem.— p n— 1 1g = — (1.9)The form factors F are normalized to one at q2 = 0 and 1c’(i) is the proton (neutron)anomalous magnetic moment.= 1.793= —1.913 reference [41] (1.10)The values of the F can be found for space-like q2 from electron scattering. In therange of q2 relevant to muon capture (—m < q2 < m) gv and g are very close tobeing linear in the variable q2 and a convenient parameterization is in terms of radii TVand 7M•gv(q2) = gv(0)[1 + rq2/6]gM(q2) gM(0)[1 + i’q2/6] (1.11)Equation (1.9) implies gv(O) = 1 and gM(0) = 3.706. Höhler has made a thoroughanalysis of nucleon form factors for space-like q2 and provides the following values of TVand TM, see [152, Table 5,fit 8.2].= 0.576fm= 0.771 fm2 (1.12)The linearity of the form factors in the region —m < q2 < 0 was tested by using thedipole form for the Sach’s form factors GE and GM from Kirk et al [59] to find gi andg. The Sach’s form factors are related to the Dirac and Pauli from factors as shownAppendix I. The Nucleonic Weak Current and Nucleon Form Factors 209below.F= (142)(GE42GM)F2 = 1— A1I12 (GM — GE) (1.13)\ ‘±lVIJFitting a straight line so that it passed through the q2 0 and q2 = —m points, themaximum deviation from linearity was found to be 0.01% for gv and 0.02% for g. Thelinear form is thus completely adequate.For the3He(,u,iij reaction q2 is fixed at —0.954m = —0.273fm2.Equation (1.11)gives the following values of gv and gj at this q2.gv(—0.954m) = 0.9737gM(—0.954mj = 3.576 (1.14)For the 3He(, i’d)n reaction, q2 varies from —0.84m to n. It is impossible tomeasure the electromagnetic form factors of the nucleons for 0 < q2 < 4M using electronscattering or electron positron annihilation and so some educated guess must be madein this region. The values of gv and g were found by analytic continuation of theexpressions in equation (1.11) i.e. the straight line is simply extrapolated into the 0 <q2 < m region.It is not clear a priori that this procedure is valid due to the presence of a cut in theF for q2 > 4ni 7m. The procedure was tested by comparing the values obtained tothose from a fit to both space-like and time-like form factors incorporating the correctanalytic properties by Houston and Kennedy [153]. The fit used used poles from thep(T70) and p(l2SO) isovector mesons placed on the second Riemann sheet as well as afourth order polynomial to simulate both more distant poles and also the crossed singlenucleon exchange cut on the second sheet. The fit allows a realistic analytic continuationAppendix I. The Nucleonic Weak Current and Nucleon Form FactorsCC’1.041.02•1.000.980.96—1.0210Figure 1.1: Comparison of realistic continuation with simple linear extrapolation of formfactors.0.0q2/m2Appendix I. The Nucleonic Weak Current and Nucleon Form Factors 211into the 0 < q2 < m region and is compared to a linear extrapolation in figure 1.1.The figure shows that the use of equation (1.11) to continue the form factors into the0 < q2 < m region is perfectly adequate the maximum deviation being 0.3%.1.3 The Axial Form FactorsThe particle data group [41] give the value of the axial form factor at zero q2.g(O) = —1.261 + 0.004 (1.15)This value is found from the correlation between the electron momentum and neutron spinmeasured in neutron beta decay. The q2 depeudeice of g for spacelike q2 is measuredby observing the ip nt reaction for various neutrino energies. The analysis of theexperiments assumes a dipole form for g and fits the parameter MA.2 — gA(0) 116gA(q )- (1_q2/M)2 (.Taking a weighted average of various results [154]-[162] the best value for M is given by,M = 1.08 + 0.04 GeV2. (1.17)The expression in equation (1.16) was used to analytically continue g into the timelike q2 region.The equivalent linear form for gA(q2) valid in the range of q2 found in muon captureis,gA(q2) = gA(0)[1 + rq2/6] (1.18)with r 0.433 fm2 which can he compared to the radii of the vector form factors givenin equations (1.12).Appendix I. The Nucleonic Weak Current and Nucleon Form Factors 212The value of the pseudoscalar form factor gp is not well known experimentally. Inprinciple, its value can be found from the muon capture rate by hydrogen but the experiment is difficult to perform, both final state particles being electrically neutral, and alsothere arising problems of interpretation due to p ——p molecule formation.As a starting point the value of gp will be taken from theoretical considerations. Oneof the aims in this research is to develop the theoretical understanding of muon captureby 3He to such a stage that the only uncertainty ii the theory is the value of gp. In thatcase experiment could set its value.The theoretical value of gp arises out of the partially conserved axial current hypothesis (PCAC) as well as knowledge of g and gNN, the pion-nucleon form factor. Theapplication of PCAC in the following form is due to Primakoff [163] and the analogousderivation of F, the nuclear pseudoscalar form factor for the 3He —* 3H transition, isdetailed in the latter half of section 3.3.1.4M2gp(q2) gA(q2)(2 q2) [1 + E(q2)]where, (q2)= () [ — (1.19)—q g(q )/gA(0)The value of€ can be estimated using a monopole form for gN(q2)as suggested byThomas and Holinde [164].gNN(q2)= A = 800MeV (1.20)For small q2, € is given by,€ ( — = —0.006 (1.21)The smallness of this value is due to the similar q2 dependence of g and NN forsmall q2.Appendix 1. The Nucleonic Weak Current and Nucleon Form Factors 213A useful quantity to consider is gp which is the combination of kinematical factorswhich apl)ears with gp in the matrix element squared for the rp ,‘ un reaction. Usingequation (1.19) we have,2gp(q2)= gA(q2)(i2/q -1) (1.22)For q2 in the range —m < q2 <m the magnitude of gp is in the range °•4gA tol.3gA and so the following rule of thumb can be employed which will prove useful in theexpansion of the effective muon capture Hamiltonian.q24M2 gp g (1.23)NThe form factors given by equations (1.11) (1.16) and (1.19) are plotted in figure 1.2.Appendix I. The Nucleonic Weak Current and Nucleon Form Factors 2143.853.803.753.703.653.603.55)—t29-——1.0I I.0 —0.5 0.0 0.52 2—q /m—100 I—200—300—400—500—0.5 0.5 1.0 — .02 2—q /mFigure 1.2: The nucleon form—0.5 0.0 0.5 1.0—q2/m,L2factors.

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