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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 3 He. By J.G.Congleton B. A. (Physics) Oxford University, 1987. M. A. (Physics) Oxford University, 1991.  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS  We accept this thesis as conforming to the required standard  Signature(s) removed to protect privacy  THE UNIVERSITY OF BRITISH COLUMBIA  March 1992  ©  J.G.Congleton, 1992  In presenting this thesis in  partial fulfilment of the requirements for an advanced  degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department  or  by  his  or  her  representatives.  It  is  understood  that  copying  or  publication of this thesis for financial gain shall not be allowed without my written permission.  Signature(s) removed to protect privacy  (Signature)  Department of  PR Y5/CS  The University of British Columbia Vancouver, Canada  Date HIhCCH ?  DE-6 (2/88)  Abstract  The thesis is in two parts. Part I is covered in chapters 1 and 2 and concerns a simple model of the hypertriton developed by the author. The model is based on the fact that the lambda particle is loosely bound and so a lambda-inert core approach should be reasonable. The core is taken to be exactly like the free deuteron and a separable AN potential is used to construct the l)inding potential for the A particle. The model is tested in chapter 2 by calculating the ratio of two body to all pionic decay rates of the hypertriton and the result is found to agree well with experiment. Chapters 3 to 7 concern muon capture by 3 He. Using the elementary particle model it is shown that the spin observables for quasi-elastic muon capture by 3 He are much more sensitive to the nuclear pseudoscalar form factor form factor  )  (  and hence the nucleon pseudoscalar  than is the rate. Reliable and sophisticated wavefunctions for 3 He and  are 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 is  developed for the muon wavefunction overlap reduction factor leading to the result C  =  0.979. Chapter 6 details a calculation of muon capture by 3 He leading to the deuteron neutron break-up final state in the plane wave impulse approximation.  Finally, the  processes leading to muonic atom formation are considered in chapter 7 with particular reference to final hyperfine population densities and their dependencies on target and beam polarization. It is shown that if only the intra-atomic processes are included, the results for the hyperfine population densities are unreliable.  11  Table of Contents  Abstract List of Tables  vii  List of Figures  ix  Acknowledgements  xi  1  A Simple Model of the Hypertriton  1.1  Introduction  1.2  Model  1.3  Formalism  1.4  Lambda Nucleon Potential  1.5  Lambda Deuteron Potential  1.6 2  1  1.5.1  Spin Average  1.5.2  Momentum Average  1.5.3  Evaluation of the Ad Potential  1.5.4  A Separable Fit to the Ad Potential  .  .  Solution of the Schrödinger Equation  An Application of the Hypertriton Model  2.1  Introduction  2.2  Lambda Particle Decay Amplitude  2.3  Two Body Decay Rate  27  27 .  .  28 31  111  3  2.4  Total Decay Rate  35  2.5  Results  38  He: Rate and Spin Observables in the Elementary Muon Capture by 3 Particle Model and Their Sensitivity to the Pseudoscalar Form Factor. 41  4  5  3.1  Introduction  41  3.2  Definition of Elementary Particle Model and Kinematics  42  3.3  The Hadronic Current  43  3.4  Rate and Spin Observables  48  3.5  Results  52  Muon Capture by 3 He in the Impulse Approximation  57  4.1  Introduction  57  4.2  The Trinucleon Bound States  58  4.3  The Kamimura Wavefunctions  60  4.4  The Effective Harniltonian  64  4.5  Matrix Elements  69  4.6  Summary and Conclusions  72  Muon Wavefunction Overlap Reduction Factor  73  5.1  Introduction  73  5.2  Method  78  5.3  Corrections  80  5.3.1  Perturbation of the Muon Wavefunction  80  5.3.2  Relativistic Effects  83  5.:3.3  Numerical Solution of the Dirac Equation for an Extended Charge Distribution  87  iv  o.3.4 5.4 6  7  Neutrino Wavefunction  89  Summary  90  Muon Induced Break-up: the Deuteron Channel  93  6.1  Introduction  93  6.2  Kinematics  94  6.3  PWIA  97  6.4  Two Body Break Up Momentum Distribution  6.5  Muon Wavefunction Overlap Reduction Factor for Two Body Break Up  6.6  Trinucleon Structure Functions  109  6.6.1  General Considerations  109  6.6.2  On-shell Impulse Approximation  111  6.6.3  Off-shell Impulse Approximation  112  101 .  105  6.7  Matrix Element Squared  119  6.8  Results  119  Muon Depolarization and Hyperfine Populations  125  7.1  Overview  125  7.2  Introduction  125  7.3  Cascade Calculation  128  7.3.1  In the Absence of Hyperfine Coupling  128  7.3.2  With Hyperfine Coupling  138  7.4  Attempt to Include Effects of Exterilal Collisions  142  7.5  Summary and Conclusions  144  Bibliography  146  v  A Natural Units B Model Calculation  154 of VAd(q’,q)  156  C Derivation of Two Body H Pionic Decay Amplitude  160  D Derivation of Inclusive H Pionic Decay Amplitude  165  E Three Body Basis and the Permutation Operators  169  E.1  Three Body Basis  169  E.2 The Permutation Operators  172  F Expansion of a Function in the Gaussian Basis  176  G Matrix Elements  177  G.l The Norm 0.2 The Non Spin-Flip Matrix Element [i]°  0.3 The Spin-Flip Matrix Element  [6}01  0.4 The Spin-Flip Matrix Element  []21  G.5 The Form of (p,q) 0.5.1  Momentum Space Representation  0.5.2 An Expression for (p,q)  .  H Deuteron Spectrum in the PWIA  200  I  The Nucleonic Weak Current and Nucleon Form Factors  205  1.1  General liltroduction  205  1.2  The Vector Form Factors  207  1.3  The Axial Form Factors  211  vi  List of Tables  1.1  AN Separable Potential Parameters  11  1.2  Strength of the Ad Potential  20  1.:3  Ad Potential Range  21  2.1  Measurements of the Decay Branching Ratio R  28  2.2  Theoretical Calculations of R  28  2.3  Parameters for the Expansion in Gaussian Basis  34  2.4  Summary of H  37  2.5  Bubble Chamber Results for T(H)  40  3.1  Sensitivity of Observables 0 to F  55  4.1  Properties of the Wavefunctions.  62  4.2  Channel Specifications for the 8 Channel Wavefunction.  4.3  Channel Specifications for the 22 Channel Wavefunction.  63  4.4  The Reduced Matrix Elements  71  4.5  The Effective Form Factors  71  5.1  Previous Calculations of the Reduction Factor C  75  5.2  Summary of Corrections to Coarse Result  91  7.1  Transition Probabilities for the 3 He-t Atom  131  7.2  Muon Depolarization with 1  136  7.3  Muon Polarization During the Cascade  —*  He Form Factor Results 3  =  1 Populated Only  vii  .  63  137  7.4  Residual Muon Polarization Pb, for Spin Zero Nuclei  7.5  Residual Muou Polarization per Unit Target Polarization for Various Val ues of  39  142  , 1 ni  7.6  Values of the A,B,C,D Parameters with External Depolarization  144  C.1  Hypertriton Channel Specifications  162  E.l  Group Product Table for the Group G  172  viii  List of Figures  1.1  Contributions to the AN interaction in the Hypertriton due to AN coupling  1.2  The ratio  4 VAd(q’,q)/VAd(q’,O)  for various values of q’. The solid line is the  full calculation and the dashed line is the fit for the soft-core potential and Bonn C deuteron  22  1.3  Lambda part of the hypertriton wavefunction in momentum representation 26  :3.1  Feynman diagram for muon  3.2  Sensitivity of observables to F  54  3.3  Unpolarized rate versus F  54  6.1  The four momentum transfer squared q 2 as a function of deuteron momen  captllre  42  tum  96  6.2  Figurative diagram of the PWIA  96  6.3  Diagrammatic expansion of the matrix element  6.4  Two body break-up momentum distribution for 3 He. Data are taken from  100  two experiments performed at Saclay [123,124] in 1982 and 1987. The straight line is a fit to the data 6.5  103  He overlap functions E>’. The solid line is the 22-channel and the 3 dclash-dotted the 8-channel 3 He wavefunction  6.6  The muon wavefunction reduction factor for two body break-up as a func tion of  6.7  107  the deuteron momentum  The off-shell  parameter  108  6  114 ix  6.8  The matrix element squared  6.9  Deuteron spectra for various ranges of d. The solid line is the off-shell  120  impulse approximation. The dashed line is the on-shell impulse approxi mation. The data are taken from Cummings et al [121]  121  6.10 The neutron/deuterou momentum in the nd CM as a function of the LAB deuteron momentum 6.11 The neutrino energy  122  Q  as a function of LAB deuteron momentum  124  7.1  Term diagram  128  7.2  Effective cascade route  136  7.3  Rose and Mann 1 distribution  138  B.1  The sinhc function  158  1.1  Comparison of realistic continuation with simple linear extrapolation of  1.2  form factors  210  The nucleon form factors  214  x  Acknowledgements  First, I thank Dr.Harold Fearing for his supervision of this work and also his encourage ment and advice. In physics we move forward by building on what we know already. The foundations are thus of great importance and I thank Mr.C.Moss aild especially Mr.B.J.Garbutt for their sound introduction to the subject which I received at Rutherford Comprehensive School. At Wadham College I was tutored by Dr.D.T.Edmonds, Dr.G.A.Brooker and Dr.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 many good lectures on nuclear physics at the graduate level. I thank the Canadian Rhodes Foundation for their financial support during my first two 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 mother who, 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 Ron and Collette MacFarlane, Max and Sally Haugen and David and Jane Armstrong for the help and friendship. It is not easy to live in a foreign country and much adjustment has to be made. I thank my ‘native’ friends Andrew, Reena and Don for their friendship and help. I have learnt much besides physics during my stay and I thank the following for their sporting contributions. Steve (skiing), Jeff (curling), Russ and Richard (ice-hockey) xi  and Kelvin (belaying). Also thanks to Dave (how to throw a party), Simon (how to win using the Neapolitan Club), Craig (see you in Australia) and IJBC cricket club for lots of fun. A quick mention for Jack and Sylvia who organize the excellent TRIUMF social events and I should be able to get away with finishing here, having left out only a few jeop1e who I should have mentioned.  xii  Chapter 1  A Simple Model of the Hypertriton  1.1  Introduction  The hypertriton piays a very important role in hypernuclear physics due to the fact that it is the lightest hypernucleus. The A has isospin zero and so the AN interaction has no one-pion-exchange tail. The AN interaction thus does not support a deuteron like bound state [1]. There have been many studies of the hypertriton using model potentials [2]-[9], the res onating 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 obtain highly accurate solutions of the Schrödinger equation for a three particle system using either the Faddeev formalism  [171 or sophisticated variational techniques [18] using real  istic potentials. One could therefore hope to make exact calculations of the hypertriton wavefunction and binding energy but, because of lack of knowledge of the fundamental YN interaction,  (  Y=hyperon, N=nucleon  ),  these solutions would be of limited quanti  tative 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 momenta  ranging from 110 to 300 MeV. The solutions listed above [2]-[16] show what effects are likely to be important in an accurate wavefunction for the hypertriton, where such a wavefunction would be based  on a YN interaction known to a much higher degree of precision than it is today. It  1  Chapter 1. A Simple Model of the Hypertriton  2  was thought that a simple model would complement the existing descriptions of the hypertriton for two reasons. • It is easily applied to calculations of processes involving the hypertriton such as its production via strangeness exchange or associated production and also the mesonic and non-mesonic decays of the hypertriton. Hence,  • it is a reference point against which more sophisticated descriptions can be com pared.  Model  1.2  The hypertriton was modelled as a deuteron and a lambda particle in an effective Ad potential. The deuteron was taken  (  as a first approximation  )  to be the free deuteron  i.e. it was assumed that the NN configuration in the hypertriton is exactly that of the cleuteron with 3 1 partial waves unperturbed by the presence of the lambda D 1 and 3 S particle. This model was inspired by comments made by Gibson [20] in which the hy pertriton is described as a system in which “the A clings tenuously to the deuteron in almost a molecular type state”. The justification for such a model is that the A is very loosely 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 the triton  (  8.48  —  2.22  6.26 MeV) although the more pertinent comparison is to the total  binding energy of the system. The ratio of the lambda separation energy to the total binding energy is 5%. An important feature of the model is that it inherently includes the NN 3 1 tensor S interaction. Gibson and Lehman [5] have shown that “proper attention to the tensor  Chapter 1. A Simple Model of the Hypertriton  nature of the up triplet interaction is necessary  3  ... “.  This is true because the np inter  action is responsible for the bulk of the binding in the hypertriton and the binding in a deuteron like NN state is sensitive to the tensor coupling between the 3 1 and 3 S 1 partial D waves. The experimental data on the hypertriton is limited. The lambda separation energy is 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 a 7T  is produced  )  of the hypertriton is,  =  F(H F(H  He) +3 ir + all)  ‘.  —>  and the implication that  =  0.35 + 0.04  reference [25]  follows because if the spin were  (1.2)  then only the p-wave  part of the amplitude could contribute and this would give a value of R far too low irrespective of the exact form of the wavefunction. Evaluation of R will provide a test of the wavefunction and is presented in chapter 2. The only other inputs for this calculation are the scattering lengths and effective ranges for the AN  and 3 1 partial waves and four different deuteron wavefunctions S  from the Bonn potentials A,B and C [26] and the Paris Potential [27]. The AN values are taken from the Nijmegen analyses [19,28] of YN scattering data which use a one boson exchange potential approach and SU(3) flavour symmetry for the coupling constants. The scattering lengths and effective ranges should provide reliable constraints on the low energy behaviour of the AN  —  AN T-matrix.  The model being proposed includes no coupling of AN to the N channel. Although this is rather unrealistic as far as describing the full AN interaction goes, it may be a reasonable approach for use in a system where the NN components of the wavefunction are small. This statement is motivated by considering a variational approach to the  Chapter 1. A Simple Model of the Hypertriton  4  Figure 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 order to find the ground state wavefunction and these matrix elements would involve only AN AN parts of the AN interaction. That is not to say that intermediate N states would not make a contribution, but rather that the intial and final state would have to be AN. Thus, provided the single channel potential reasonably approximates the full AN  —  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 the wavefunction. Dabrowski and Fedoryñska [9] found a probability of 0.36% for the NN component in  the hypertriton using the phenomenological YN coupled channel potential  of Wycech [29]. Of course, this ignores the dispersive effect of the spectator particle, which tends to weaken the coupled contributions to the AN  —  (  figure 1.1  )  AN T-matrix and that  will have to be accepted as an approximation of the model. The three body force arising from the coupling to ZN  ,  shown on the right hand side of figure 1.icouple, will also be  neglected as will the tensor coupling in the AN 3 1 channel. The latter has been shown S not to make an appreciable contribution [15]. The wavefunction derived herein was forced to have the correct binding energy by adjusting the parameters of the Ad potential. The amount of tuning which needs to be performed will give an indication of the credibility of the model.  Chapter 1.  A Simple Model of the Hypertriton  5  An 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 deuteron wavefunction. • The resulting Ad potential was fit to a separable form and only the s-wave part was used. The other partial waves were taken to be zero. • The Schrödinger equation for the lambda l)art of the wavefunction was solved and the binding energy was forced to the experimental value. Without tuning, it would not be expected that the model would provide a good value for the binding energy of the hypertriton. As is known for the trinucleon system it is very difficult to find agreement with experiment in the value of EB and this is even more true for the hypertriton where the balance of kinetic and potential energies is so precarious that it has even been speculated that the nucleus is only bound because of the presence of a ANN three body force [3]. However, for a good description of the hypertriton, it is highly desirable for the wavefunction to possess the correct binding energy since this determines its asymptotic behaviour, small momentum  ).  (  the asymptotic limit is large distance or correspondingly  This has been demonstrated in the trinucleon system where scaling  is found for all the low energy observables i.e. no matter what potential is used the  properties of the wavefunction such as r.m.s. radius, D-state probability, etc. all show simple power law dependence on the value of the binding energy which the input potential yields [30]. The argument for scaling is that the low-energy observables depend strongly on the asymptotic part of the wavefunction and in this region where the potential is very  Chapter 1. A Simple Model of the Hypertriton  6  weak the wavefunction depends only on the binding energy. One would thus expect the same 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 various potentials, all consistent with low energy Ap scattering and binding energies of is and ip shell hypernuclei.  1.3  Formalism  Following the general definition of Jacobi coordinates for a three body system, a linear transformation (k ,k 1 ,k 2 ) 3  —  (K,  )  was made in order to separate out the centre of  mass motion. In the following equations (abc) take the values (123), (231) or (312), ka is the momentum of particle a and mc, is the mass of particle a.  A  ka+kb+kc  —  (m [(me  =  —  +  mbk)/(mb  mc)a  —  + m)  ma(b  +  )]  /(ma  If the lambda particle is labelled particle #1 then writing =  +  mb  +  m)  as  (1.3)  (),  +k 2 kA+k 3  = =  [2MN  —  2 +3 MA(k )k ]/(2MN + MA)  In the centre of mass frame (CM) this simplifies to, I  IT  =  0  =  (2—i3)  =  ‘CA  The internal momentum coordinate of the deuteron is the ‘pair’ momentum lambda momentum is the ‘spectator’ momentum  I  (i.4)  and the  Chapter 1. A Simple Model of the Hypertriton  7  The Hamiltonian governing the system is H which in the CM is, -*2  -*2  =  —+ 2m  =  Hdeuteroii  up++  2u  +  Ad  (1.5)  Hianibda  The Hamiltoniaii can be split into two parts because a static approximation has been made for the lambda-deuteron potential VAd: i.e. the influence of the lambda particle on the nucleons is neglected. The Ad potential will thus only depend on the variable  ( is  after spin dependence has been taken care of rn  =  ).  The reduced mass of the two nucleons  1 and the reduced mass of the lambda-2N system is 2.379fm  ji  . 1 3.547fm  =  The wavefunction can now be written using the fact that the hypertriton has spin one half and using the model for the nucleon part.  (7H;m) =bA(q)  x  x  [YJ ( 0 ) ®x] 2  (1,S)=(O,)(2,)  x In the above equation, spin  part shows  —  j,fl//2]  (1.6)  is the radial part of the deuteron wavefunction. The  the coupling of the spin one deuteron to a spin one half lambda particle  to give total spin part requires S  /)()  [A(TJ.  3  or  .  The s-wave part of the deuteron requires S  =  and the d-wave  =  X’Is  =  (SMs lmd;  (1.7)  X1A t flA)Xi*  This spin vector is coupled to a spherical harmonic for the angular parts of the nucleons and lambda particle which has angular momentum 1 for the  pair  and 0 for the  spectator A.  YI(i3, )  =  (irn l1m;  12rn2)  }1rn1(l3)uirn2()  (1.8)  Chapter 1. A Simple Model of the Hypertriton  8  The 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 system is 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 wavefunc tion and a deuteron wavefunction.  1.4  Lambda Nucleon Potential  The ansatz for the lambda nucleon potential is a separable potential with dimensionless form factors g(k). Only the s-wave were taken to be non-zero and the potential was taken to have two free parameters, viz., • The strength of the potential ). • The momentum space range of the potential  AAN  .  This is often referred to as the  inverse range since it corresponds to the inverse range of the potential in configu ration space. These parameters were fixed by the scattering length and effective range for s-wave AN scattering and results were taken from the findings of the Nijmegen group [19,28]. The more recent analysis [19] applies SU(3) flavour symmetry to extend the Nijmegen NN potential to YN. The potential is based on exchange of ir, m, mj’, p, , includes .J  0 contributions from the tensor  2 f, f’, A  ,  6, € 8* particles and  particles and Pomeron trajectory.  It is characterized by Gaussian form factors for the couplings which give the potential soft behaviour near the origin. This is in contrast to the earlier analysis [28] which employed potentials of the hard-core type. The results for the scattering length and effective range will be referred to as soft-core and hard-core for [19] and [28] respectively.  Chapter 1.  A Simple Model of the Hypertriton  9  Taking the AN potential as separable means that the matrix element of the potential VAN  for a transition from initial state k to final state k’ is written as the product of a  function of k and a function of k’. Here, k aiid k’ are the iiiitial and final A momenta in the CM. N  (k’VAN)  =  (1.9)  -g(’)g()  MNkA—MAkN —  110  MN+MA  Changing 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,  partial wave basis,  VAN(k’,k)  =  (Iklm)  =  _ANg(kI)g(k) 7fl()  (E) 0 G  )  (1.12)  2  The T-matrix was found by solviirg the Lipmanri-Schwiriger equation (1.13)  (1.11)  (  equation  which is easily done for a separable potential. The Green’s function here is [E  —  j where H 0 H 0 is the free Harniltoiiiair.  T(E)  V + VG (E)T(E) 0  (1.13)  Writing (k’OO T kOO) as t(Ic’, k; E) we have, t(k’,k;E)=  (1.14)  where,  JE  [g(q)] dqq f 2 Jo [E q /2AN 2 —  =  reduced mass  2.5837fm’  of AN  (1.15)  Chapter 1. A Simple Mode] of the Hypertriton  By taking the principal value of  JE,  10  equation (1.14) gives the r-matrix and the fol  lowing identification with the s-wave phase shift  0 kcot  can be made,  ask—O =  —÷  (the ON superscript indicates that E shell”  =  /2j.i 2 k  —--  a  lr/IANr(k) =  2 + —rk 2  /2t 2 k’  so  (1.16)  that the particles are “on  ).  The scattering length a and effective range r reflect the low energy properties of the interaction. A Gaussian form factor was used but results for the Yamaguchi type form factor [32] are also shown for the sake of comparison to the nucleon-nucleon potential.  g(k)  ] 2 exp[—(k/AAN)  g(k)  1 =  [1 +  ] 2 (k/AAN)  Gaussian .  Yamaguchi  (1.17)  These 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 fifths  of the strength. The spin singlet interaction is stronger than the spin triplet interaction which is the assigiment one would make from the observatioll that the Apn ground state is .J= and not The fact that the range of the hard-core AN potential comes out to be almost inde pendent of the spin is a fortunate coincidence which makes the formalism much more transparent  than in the case of the soft-core potential. The method of developing a Ad  potential from  the AN potential will be given in detail with  specific reference to  hard-core potential. The method for the soft-core potential is very similar but  the  is rather  less illuminating and only the numerical results will be given for the soft-core case.  Chapter 1. A Simple Model of the Hypertriton  11  Table 1.1: AN Separable Potential Parameters. N  AN AN NN NN NN NN AN AN AN AN  -  -  -  -  Yamaguchi (HC) Yamaguchi (HC) Yamaguchi pp Yamaguchi np Yamaguchi nn Yamaguchi Gaussian (HC) Gaussian (HC) Gaussian (SC) Gaussian (SC) -  -  0 S 1  -  I5 -  -  35 -  -  ‘S  -  -  -  15 -  -  S1  -  2 /fi  0.1106 0.1017 0.2141 0.2178 0.2164 0.2550 0.1238 0.1129 0.1312 0.0915  -  -  “AN /fm’  1.354 1.359 1.133 1.144 1.128 1.418 1.400 1.402 1.452 1.597  / fm -2.29 -1.88 -17.1 -23.75 -18.45 5.43 -2.29 -1.88 -2.78 -1.41  a  r  /  fm 3.17 3.36 2.83 2.75 2.83 1.75 3.17 3.36 2.88 3.11  Lambda Deuteron Potential  1.5 1.5.1  Spin Average  For free AN scattering, the total spiii is either 0 or 1. With an unpolarized beam and target, the population densities for each value of the total spin and total spin projection will be equal and hence scattering occurs in the singlet state with probability triplet state with probability  .  and in the  For the A in the hypertriton, the relative probabilities  are not the same as the free case as pointed out by Herudon and Tang [33] and in fact depend on the total spin S. If S then the AN interaction is  -,  (  which is the dominant part of the wavefunction  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 indicated  by 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 states  Chapter 1. A Simple Model of the Hypertriton  12  of good total spin s and spin projection s, s, s  )  for particles 1 and 2.  =  l,0)  -  (1.19)  00)T]  By taking the overlap of this vector with itself it ca be seen that we have triplet and thus, H  =  spin  spin singlet. More formally, if the spin projection operators are defined  2 + 1)/4 and ilt . 1 (—o  2 = (ui.u  + 3)/4, then  2? 1 X?lHX  =  3/4  xflx  =  1/4  (1.20)  The total 5pm of particles 1 and 3 follows the same pattern because the spin vectors are 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 of  argument. 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 three angular momenta are coupled together there are three choices of construction. Two must first be combined to a subsystem total angular momentum and then this is combined with the remaining ‘spin’. There are three choices of the pair and the three resulting  A Simple Model of the Hypertriton  Chapter 1.  13  vectors are related by a linear transformation whose coefficient is proportional to the Wigner 6-j symbol [34, 5 SM  p41]. /1  —  1\i. 1. C1.K  31V15  ji  2  =  1  S  2  1  S  sJ  (1.22)  So we have the result that the probability P that the AN spin is s, is /  5 P  =  Ill 2 2  3(2.s + 1)  1  ii  (1.23)  S sJ  Equation (1.23) yields the results given in equations (1.20) directly and show that they are independent of M. Since the ranges of the triplet and singlet AN potentials are so similar the effective spin averaged potential was taken to have range 1.401 will depend  on the spin  state S. For S eff =  For  ,  eff  ‘  (3)AN  fm  and the effective strength  0.1211 fm 2  (1.24)  =  +  =  is the triplet strength  =  . The raige was taken to be 2 0.1129fm  1 .401 fnf’ in both channels for the sake of simplification. To summarize, (k’VANk) where,  g(k)  =  —Ag(k’)g(k)  (125)  ] 2 exp[—(k/AAN)  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 quantum  numbers of the NN partial wave. The subscript on  reflects the orbital angular mo  mentum of the NN partial wave and takes on the values 0 or 2. The value of AAN was  taken to be 1.401 fnf’.  Chapter 1. A Simple Model of the Hypertriton  14  For the case of the soft-core AN potential the following forms were used, go(p)go(p’) + 0 A  (p!lVANeffp) (IyVANep)  where,  go(p)  where,  gi(p)  =  g(p)g(pf) 1 N  =  ] 2 exp[—(p/A°)  (p’) for S 1 g(p)g  =  for S  (1.26)  j 2 exp[—(p/A’)  (1.27)  It is clear that the two potentials should give similar results because their strengths at low momenta are the same. Consider the strength of the soft-core potential at q’ when S  =  it is 3/4 x 0.1312 + 1/4 x 0.0915  =  q  =  0  2 which should be compared to 0.1213fm  0.1211 fm . This is the most important effective strength because the S 2  =  state  forms the majority of the hypertriton which follows from the 95% s-state probability for the nucleons in the deuteron.  1.5.2  Momentum Average  The following ansatz was made for the Ad potential. The subscripts 2 and 3 refer to the label on the nucleons so that V 2 is the spin averaged potential between the A and N 2 and 3 is the V  spin  averaged potential between the A and N . 3 (‘  IVa)  =  f  p’d d p 3  2 + V (‘)(‘‘V d(p 3  (1.28)  This potential is a classical sum over the potentials from each nucleon and is a quan tum mechanical average over the initial and final internal deuteron momenta weighted by 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 conserv 2 should not affect the momentum of N 3 so the delta ing delta function. For example, V  Chapter 1. A Simple Model of the Hypertriton  function is 6 (k 3  2 k 3 k  — =  15  ). The inverse transformation (K, 3 k (MA/s)  0  1  (MN/s)  1  —!  (MN/s)  —1  —  and in the CM where I  =  )  (ks, k ,k 2 ) is given by, 3  “  I’ where  j  =  MA + 2MN  (1.29)  0 this simplifies to, q  2 k  =  3 k  =  (1.30)  The two body potentials are thus, ) 2 (j’’V  =  6(+  ) 3 (‘‘V  =  6(-  1 2  /  1  -- (jr’  _  ‘  +)) 2  ]g[k] 2 (\47rJ ) g[k  -  ()  I  ]g[k] 3 g[k  (1.31)  In the above, k 2 is the magiitude of the A momentum in the A-N 2 CM. The vector 2 can be written in terms of k factors  and  as shown below and this introduces kinematical  and /3 which play an important role. — I2  k  MNA—MA2 MN+MA  — —  MA-MA3 =  MN+MA/2  where, =  MN+MA  (1.32)  =  0.7285 0.5430  (1.33)  The contributions from N 2 and N 3 are the same except for a sign and so the Ad potential can 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  V’,q  16  =  (1.35)  The pertinent value of  (  either A or  ),  of the deuteron amplitude is contributing,  (  should be taken according to which part  either 1  =  0 or 1  The integral was performed by making the substitutions and performing the trivial integration over  ‘.  =  2  ). and  =  .‘ =  ’ 7 j  +  Finally, only the s-wave part of the  resulting expression was taken. It was found that with the s-wave part of the deuteron only, the maximum value of the next partial wave (1  (  s-wave,  the 1  (q’OO IVAdI qUO)  VAd(q’, q)  =  and y V+  =  c +  f .  f  =  (—)  JT  =  1+  hypertriton when  ).  [V(q’, q) + V(q’, q)] where,  ds  )g[7’ + ]g[7+  (1.36)  ]  (1.37)  Further, because of the even parity of the deuteron wavefunction  V so that,  V  VAd(q’, q)  =  V(q’,q)  =  1.5.3  2) was 1.4% of the strength of the  1 partial wave does not contribute to the  =  the only NN partial waves are 3 1 D 1 and 3 S  V(q’, q)  =  —2V(q’, q) where, ‘d(+  )g[7’+]g[+](1.38)  Evaluation of the Ad Potential  This section describes the evaluation of V(q’, q) given by equation (1.38). The deuteron wavefunction  for the sub-state with total angular momentum projection m, is bd(p)  =  1-02  The  spin part  xi  ®  (1.39)  is coupled to the orbital part }‘() to give total angular momentum  1 and projection m. The isospin part represents a neutron and proton coupled to zero  Chapter 1. A Simple Model of the Hypertriton  17  total isospin. The radial part of the deuteron was taken from the parameterization given  in [26, pp355,36 ] for the Bonn potential and was interpolated from a grid supplied by 2 Oelfke in the case of the Paris potential. The expression (1.38) should be spin averaged over projections rn according to the coupling in the hypertriton. For a “spill-up” hypertriton we have 2/3 rn  m  =  =  1 and 1/3  0. From the facts that V(q’, q) is real and only even values of l contribute, it can  be shown that the contribution due to rn  =  —i equals that due to m  =  1 and hence the  spin average peculiar to the spill 1/2 Ad states can be written as a generic spin average yielding, ‘/2)/’d(+ /2)  2J+  3 ?fl  /2)  }7+  21  }i(  /2)  7/0(  ‘/2). (1.40)  1,7llj  The expression for V(q’,q) given in equation (1.38) can thus be re-written as follows.  V(q’,q)  (,qI)*N fd3sA7ffN m 1 1(q)  =  lrO,2  where,  Th q’) N1  =  The angular dependence on  (1.41)  21 + 1  f  /2  )1)(  + /2 )g[+  ]  (1.42)  was expressed using the following expedient trick  employing tile Legendre polynomials Pk (x).  =11  (lXf(X)6(X  —l  -),  (x  -)  = 4  (1.43) k,mk  Two identities for spherical harmonics of two variables are needed to find that,  1 (s,q) N -  =  1 (s)Ni(s,q) Ym  where,  sh1(q/2)12__  Ni(s,q) 11+121  dx P 12 (x)  11.12.  x  /2 1 (s2+q2/4+sqx)  s 2 exp[-(  +2 q + 27sqx)/A]( 1.44)  Chapter 1.  A Simple IVIodel of the  18  Hypertriton  The identities are, ahlbI2  (a + b) and,  + =  ()  — —  (s)  47r(2l + 1)’ 2 i] [i + 1)!(2l  + 2 [(2ll+2l  ]  ) f 7 l  —  ()  (1.45)  r (l0j l0; 120>  (21 + 1)1(212 + 1)!  4l + 1)!  Identity (1.45) follows from partial wave expanding the x  Y & 1 , b)  1!  ! 1 l =  12!  (1.46) 1 47 )  and regarding  0 limit. Identity (1.46) follows by notiig that y (., .) must transform like  and finding the coefficient by evaluating an integral of three spherical harmonics.  Equation (1.47) holds only when l + l 2  =  l which is the case for all values entering into  the sum (1.44). The final form used for V(q’, q) is, V(q’q)  fs2A(sq!)Nj(sq)  =  (1.48)  1=0,2  which follows from the trivial integration over d.. The integrations over s and x were performed 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 Potential  The 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, of the AN potential  !\AN  ).  (  the relevant scale is the momentum space range  A separable fit to this potential is desirable for two reasons.  • The Schrödinger equation for a particle moving in a separable potential is easy to solve 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 using the same formalism as the input AN potential. The level of sophistication used for each potential is then the same. The form for the Ad potential was taken as, VAd(q  / ,  q)  Ad =  —  F(q / )F(q)  where,  F(q)  =  exp[—(q/AAd) 2 ]  (1.49)  As in the case of the AN potential, there are two parameters to find; the strength A and the momentum space range AAd. The strength was found by setting q q’  =  =  0 and  0 in equation (1.38). Thus, =  I  2 x I fdss  where, {I0) 2  (s)+  2)  (s)eff}exp[9(s/) ] 12 2  (1.50)  This is the normalization integral for the deuterori wavefunction in momentum space modulated by a form factor and constant multiplicative factor for each partial wave. The structure can be understood by studying two informative limits. 1. Suppose that the AN interaction were of infinite range in momentum space. This corresponds to a point-like interaction in configuration space. Then the modulating form factor in the integral I would be one and the integral is easily performed. I  =  0+p p 2  (1.51)  where P 0 and P 2 are the s-state and d-state probabilities in the deuteron. The strength of the Ad potential is thus twice the average  )eff  where the average is over  the AN spins found in a Ad s-wave system coupled to total angular momentum 2. Suppose that the AN interaction were insensitive to spin. In that case, ) is the same for both the s-wave and d-wave part of the deuteron wavefunction and can  Chapter 1. A Simple Model of the Hypertriton  20  Table 1.2: Strength of the Ad Potential.  r[fm (HC) 1 2 l (SC) 2 ,\Ad[frn  2 P  1%  Bonn A 0.2222 0.2234 4.38  Bonn B 0.2214 0.2226 4.99  Bonn C 0.2205 0.2216 5.61  Paris 1 0.2195 0.22i] 5.77  1  be factored out of the integral. The strength of the Ad potential is then 2 times the AN strength multiplied by I’, where 1’  =  fds  2{I  (o)  2  (2)  12  ()} exp[2 (s/A) J 2  (1.52)  This is simply the cleuteron normalization integral in momentum space modulated by a form factor. The form factor is always less than or equal to one and the rest 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 the AN interaction. How much less depends on the size of the deuteron compared to the  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 the  strength approaches 2 times the strength of the AN interaction. If however the size of the deuteron is much greater than the range of the AN interaction then the Ad interaction becomes very weak. This is a physically reasonable picture. The input was a AN potential which was weak at high momentum and this was averaged over the momenta found in the deuteron. If the two nucleons are found often to have large relative momentum then the AN relative momentum will also be large and hence the effective Ad potential will be weak. For Bonn potentials A,B and C the value of I’ was found to be 0.919, 0.916 and 0.912 respectively. For the Paris potential I’ was found to be 0.910.  Chapter 1.  A Simple Mode] of the Hypertriton  21  Table 1.3: Ad Potential Range. q’ fm’ / 0.0 (HC) 0.0 (SC) 0.5 (HC) 0.5 (SC) 1.0 (HC) 1.0 (SC) 1.5 (HC) 1.5 (SC)  1 AAd/fnf  Bonn A 0.94 0.98 1.01 1.04 1.15 1.19 1.33 1.38  Bonn B 0.94 0.97 1.01 1.04 1.15 1.19 1.33 1.38  Bonn C 0.94 0.97 1.01 1.04 1.15 1.19 1.33 1.38  Paris 0.94 0.97 1.01 1.04 1.15 1.19 1.33 1.38  Table 1.2 gives the values of ) found using equation (1.50) and various deuteron wavefunctions. The value of ) decreases slightly as we move from Bonn potentials A to C and this reflects the increasing d-state probability and the fact that .A  <  Also  given 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 form  for F(q), (equation (1.49)). The results are shown in table 1.3 for various values of q’. If the potential really was separable then the value of AAd found would be independent of q’ 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 shown to be reasonable by a model calculation which is given in appendix B. The fits are shown in figure 1.2. A good fit is required only where the potential is appreciable, which is in the range q  0  —  2 fm’. Further, if an intermediate value of q’ is chosen at which to perform  the fit, then the resulting potential will be too large at low momentum and too small at high  momentum  and the two effects will cancel to a certain c[egree. Let us take the  soft-core fit with Bonn C deuteron at q’ AAd  =  =  I fm’ for the sake of a definite result i.e.  1.19 fn , but remeniber that anything between 1.1 and 1.3 fin’ would also be 1  Chapter 1. A Simple Model of the Hypertriton  1.0-  22  1.0 \  0.8-  \ 1 q  0.8  00 fm’  a, x 0.6-  1 fm  \i041  CD  & 0.6  >  >  .Q4  - Q4.  >  >  0.2  0.2  0.0 0. )  0.5  1.0  1.5 2.0 q in fm—i  2.5  00 — 0.0  3.  1.0  1.0  0.8  0.8  0.5  1.5 2.0 q in fm—i  2.5  3. 0  1.5 2.0 a in fm—i  2.5  3.0  a,  b- 0.6  :  >  0.0 --  q in fm—i  1.0  Figure 1.2: The ratio Vd(q’, q)/VAd(q’, 0) for various values of q’. The solid line is the full calculation and the dashed line is the fit for the soft-core potential and Bonn C deuteron.  Chapter 1. A Simple Model of the  23  Hypertriton  reasonable. It is to be expected that the calculation of  is more reliable than the calculation  of AAd since it depends on the more gross features of the AN potential and deuteron  wavefunction, viz, the AN strength A, and also the relative size of the deuteron and AN 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 Equation  The Schrödinger equation for the lambda part of the wavefunction is, (+vAd)  (1.53)  A) = —BAI’A)  The solution of this type of equation is standard and is given here for the sake of corn pleteness. Acting on the left with the bra (q I and letting a 2 )(q) 2 2 +a (q  =  2F(q)  2BA one finds,  j2(s)F(s)  The integral appearing in the above equation is a functional of  /‘A  (1.54) and is thus a number.  Writing this functional as K the wavefunction can be written, 2dF(q)K ‘‘A(q) =  =  +a 2 q  F(q) +a q 2  (1.5)  where N is a normalization constant. The eigenvalue equation is found by demanding self-consistency using the first equality above to express K in terms of itself. 2  K  =  2[tA  / Jo  ds  2 [F(s1 L  K  (1.56)  \Vith the Gaussian form for F(s) the integral is expressed in terms of the exponentiated complementary error function cerfe(x) [35, p338,3.466(2)]. =  (  -  cerfe()  (1.57)  Chapter 1.  A Simple Model of the Hypertriton  where,  cerfe(x)  =  ex [ 2 l  —  24  erf(x)J  erf(x)  Equation (1.57) links the A separation energy  BA,  2 =  j  2 dt e  (1.58)  the strength of the Ad potential  and the range of the Ad potential AAd. Given two of these quantities it can thus he  ,\Ad  used to determine the third. In order to know the wavefunction, equation (1.55), the values of a and AAd  AAd  are needed. Since the calculation of ) is more reliable than that of  it is desirable to use  as an input parameter  and to see if the value of AAd needed  to 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 product Ad A A  AdA  Thus, the values of hyperbola “A d 1  )1d  ()  =  ()  + (2  acerfe(a)  (1.59)  AAd  and  AM  2  0.225 fm. The amount of binding depends on how much  =  which just bind the hypertriton,  (  a  —*  0  ),  lie on the  greater 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 be illustrated by taking ) to be 0.222 fm 2 and using the measured value of the separation energy to fiuid AAd. Equation (1.1) gives a  =  (6.8 + 1.3) x 2 lO f nf’. Equation (1.59)  Was solved by iterating the right hand side until it converged with the result AM  =  1.17 + 0.03 fm . 1 The first term on the right hand side of equation (1.59) is 0.225 fm and at the solution the second term is 0.035 fm. The 20% error in a yields only a 2% error in  AAd  which  illustrates the J)oint that equation (1.59) is very weakly driven by a. Thus, in order to successfully 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 find a reasonable wavefunction. The experimental binding energy was used along with the  Chapter 1  A Simple Model of the  25  Hypertriton  calculated value of ) to yield the consistent  AAd  value which is l.17fm . From the 1  results of the fit for F(q) it can be seen that this value is in reasonable agreement with the calculated values. The theoretical uncertainty in  AAd  should be taken as roughly the same as the un  certainty 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 A  is without performing a more sophisticated calculation with better knowledge of the AN potential. However, when using this wavefunction it would be prudent to vary the value of 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 to be,  A(q)  =  N(QA)  exp[q/Q]  1.17 fm (+ 1  (1.60)  -  10%)  (1.61)  (6.8 + 1.:3) x 2 lO f nf’ N(Q) =  {—  + [cerfe()(1  2 [N(l.05)]  =  0.106061fm’  [N( 1. 17)j2  =  0.10391 1 fm  2 [N(l.29)]  =  0.102188fm’.  -)  -  (1.62)  ()] }  (1.63)  (1.64)  Figure 1.3 shows the wavefunction given in equation (1.60) where the value of QA is taken to be 1 .17 fm’. The peak of the momentum distribution is very near q 0.068 fni.  =  a  =  Chapter 1. A Simple Model of the Hypertriton  26  70 6050-  Lambda part of tritonw cv ef unction  2010  -  00.  5-  —  I  I  0.2 0.4 0.6 0.8 Lambda momentum 1 q/fm  Lambda particle momentum distribution  1•  0.0  0.2 0.4 0.6 0.8 Lambda momentum 1 q/fm  1.0  Figure 1.3: Lambda part of the hypeitriton wavefunction in momentum representation.  Chapter 2  An Application of the Hypertriton Model  2.1  Introduction  As 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 x l0’° to :3.84 x 10b0 sec 1 [25, see table 4]. However, this quantity is a poor test of a model of the hypertriton since r(H) is relatively insensitive to the nuclear structure, the value being found to be near the free lambda particle lifetime  T(A) both experimentally  and theoretically. Its insensitivity will be made manifest in this chapter. A better test is the branching ratio, R of two body r  H  F(H  ,‘  —*  He) +3 ir + all)  —  0.35 + .04  decays to all r reference [25]  decays. (2.1)  The measurements of R from helium bubble chamber experiments are summarized in table 2.1. The branching ratio R. has been calculated using phenornenological H and H 3 e wave-  functions by Dalitz [22] and Leon [23] and also by Kolesnikov and Kopylov [11] who used H and 3 He wavefunctions found from variational calculations employing five different AN potentials all ‘consistent with fundameiital hypernuclear data”. The early work was hampered by poor knowledge of the lambda decay potential but despite this, Dalitz was able to conclude in 1958 that the spin of the hypertriton is one half. The results of these calculations are summarized in table 2.2. Note that the results of Dalitz and Leon should 27  Chapter 2. An Application of the Hypertriton Model  28  Table 2J: Measurements of the Decay Branching Ratio R. R 0.39 + .07 0.:36 t 0.30 + .07 0.35 + .04  [  Reference Block et al. [38] Keyes et al. [36,37] Keyes et al. [25] mean value  Table 2.2: Theoretical Calculations of R.. R 0.17 0.26 0.10 0.24 0.26 0.55 0.52 0.50 0.32 —  —  Reference Dalitz [22] Leon [23] Vi Kolesnikov V2 & Kopylov [11] V3 V4 V5  be multiplied by two due to an error in the application of the Pauli principle as explained  in Dalitz and Liu [24]. The range of values is clue to uncertainty in the lambda decay potential. Given the better limits available today Dalitz and Leon would have found R  =  0.52  and  R  0.48 respectively which corresponds to the top end of the range given  in table 2.2.  2.2  Lambda Particle Decay Amplitude  The A particle decay amplitude has the following general form  a  in  the non-relativistic  limit [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 agrees with the convention used by the Particle Data Group [41]  Chapter 2. An Application of the Hypertriton Model  where,  .  =  29  s-wave strength p-wave strength  Q  =  momentum of proton in A rest frame momentum of proton when the A is free  0 q  =  =  100.5 MeV.  This form follows from the conservation of total angular momentum, the spinless and both A and proton having  spin one  half. Both  pion  s and p wave pious are  being  allowed  because j)arity is not conserved in the weak interaction. The ratio of these two strengths can be found from the angular correlation a between the polarization of the A particle and the proton momentum. For a free A particle the decay amplitude is, (2.3) and so the decay rate for r(A  p+  an  )  ensemble of polarized A particles with density matrix p is, Tr[MpMt]  where,  p  =  (1 + PA.)  (2.4)  The ratio of the polarized to unpolarized rate is thus, p+ p+ where,  a  —) =  1 +  (2.5)  2Re(.*j3) s2 +  (2.6)  =  Applying time reversal invariance to the decay amplitude we have,  3Q  (  using  ., .  —*  ,‘  M  —  + j3.Q  (2.7)  and so .s and p must be real. However, the presence of the stroig interaction in the final state introduces extra phase shifts 6 and  13 for  the s-wave and p-wave strengths  Chapter 2. An Application of the Hypertriton Model  respectively. The quantity A  13  =  30  is measured to be 8°+4° [41] which is consistent  with results from low energy p scattering. In what follows the magnitudes of  and  will be written as s and p respectively and their relative phase will be expressed in terms of A. The angular correlation a can be written in terms of the ratio r  =  p/s and the  angle A thus, a but remains unchanged when r (2.6). Its value is a  —  =  2r cos(A) 1 +r 2  (2.8)  1 because of the s r  —*  p symmetry of expression  0.642+0.13 [41] which implies that r or r  0.368±0.010. From  the measurement of a alone it is not possible to say whether the p-wave is stronger or weaker than the s-wave. However, this can be found from the polarization of the proton 1 which is given by, P —  —  Tr[MpM] Tr[MpMt]  29  l+aPA.Q where,  /3  (210)  2spsin(A) 2  (2.11)  2 +p  (2.12) Thus, the quantity 7  15  the protons’ polarization in the direction PA when they emerge  at an angle 7r/2 radians with respect to the axis of polarization of a fully polarized ensemble of lambda particles. If .s < p then  is 7  =  ‘  =  —0.77 while if .s > p then  ‘  =  +0.77. The experimental result  0.76 + .03 [41] and it is concluded that the s-wave strength is greater than the  p-wave strength. In summary, p/s  =  0.368 + .010  2 (p/s)  =  0.1355 ± .0074  (2.13)  Chapter 2. An  Application of  the Hypertriton Model  31  The ratio R. can now be written in terms of matrix elements of the lambda particle  decay potential and kinematical factors. Ignoring final state interactions between the pion and the nuclear fragments, the effective Hamiltonian is given by, Heff()  >  =  M H ata)  (2.14)  i=l ,2,3  where,  i H ii  ‘,  M  =  s  ‘,‘)  =  exp[iJS ( 3 i  (2.15) —  i ‘)  ‘)8(i  —  —  ‘) (2.16)  The final state plane wave pion wavefunction has been incorporated into the decay amplitude operator using the operator H and the pion momentum is written  The  effective 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 the creation operator for a proton with label (i). The evaluation of the matrix elements is  detailed below in two sections. First, the H 3 e final state and then the inclusive case. 2.3  Two Body Decay Rate  The exclusive decay rate for the H 3 e final state is, F(H  He) +3  =  I  3 2w(2)  2 ( wq +  —  2 M 3He  ) 2 A  2 J H 3 e;mfHeff(H;m ( j) ,rnj  where,  Energy of pion with momentum  Wq  M3He  =  Mass of H 3 e = 2808.39 MeV  2 A  =  M3H  =  spin projection of H  —  M3He = 182.7 MeV  rnf = spin projection of H 3 e  =  q2  2 +m  (2.17) (2.18)  Chapter 2.  An Application of the HypertrI ton Model  32  For a two body final state there is a unique value of momentum for which energy con servation is satisfied and in this case the CM momentum q,  =  of the final state particles is  114.3MeV. The decay rate is thus,  F(H  He) +3  ,‘  ‘7U  +  f  x  = Wq,r I “ He) 3  He; rnf 3 I(  Heff(qH; m)  2  (2.19) The nuclear states appearing in the 3 H— H e matrix element must be antisymmetrized with respect to interchange of any two particle labels. The antisymmetrization of the hypertriton wavefunction is straightforward due to the fact that the inner product of the isospin vectors for nucleon and lambda particle is zero. Using the antisymmetrization operator A  =  1 + P 3 the fully antisymmetrized hypertriton wavefunction is, 2 + P H)  where Ii/,  )  (1 + F 2 + F )) 3  is the vector represented in equation (1.6), is antisymmetric under inter  change of particles 2 and 3 and is normalized  I ‘IH)  (2.20)  follows from A 2  =  (  ‘)  1. Correct normalization of  =  3A.  In appendix C it is shown that the matrix element simplifies to,  3 H e; rrzf IHeff(fl H; rn ) where,  v(  fP I 1 IM  l,(1)) = a1)ta5  )  )  (2.21) (2.22)  and also that the identity,  ( ‘1’3HeW11  1)  /( 3HeIIH’’ILZ/’ )  )  holds, due to the fact that the model hypertriton has only s total spin of particles 2 and 3 in the  i/’  )  =  1, L  (2.23) =  0,2 where .s is the  component and L is the total orbital angular  momentum of all three particles. The two body decay amplitude can thus be written, H(H  —  7r H 3 e)  =  F ( 1 q)  [s  +p./3qoj  (2.24)  Chapter 2. An Application of the Hypertriton Model  where  33  is the Pauli spin vector of the three particle system and 1 F ( q) is a form factor  for the H  —  3 H e ED transition matrix element.  v( 3He  F(q)  1jjk  )  (2.25)  This form factor is related to that of Dalitz [22], Leon [23] and Kolesnikov and Kopylov [11] by, F ( 1 q)  Fother(q)  (2.26)  The “other” authors’ definition of the form factor arises naturally when using a purely Derrick-Blatt S state wavefunction [42] for H 3 e which has the following form, this is the full antisymmetrizeci wavefunction  (x  (,2,rI3He;rnf) =  The hypertriton wavefunction  ‘(1)  note that  (  ). —  ‘)  ) He 1 2 . 7 7 , 23 31 (  (2.27)  has the form, XiH 2 r 2 i 3 ) 3 (r  (2.28)  [ +p/3qo]  (2.29)  and so the two body decay amplitude is (-)F(q)  where F(q) is the overlap of the S states modulated by the pion wavefunction, equations (A3,A5)]  (  see [22,  ).  The form factor 1 F ( q) was evaluated usiig existing programs for the H 3 e  ,‘  H tran 3  sition occurring in muon capture by H 3 e. To facilitate this, the hypertriton wavefunction was expanded in the same  type  of  Gaussian basis as used in the variational calculation  of the trinucleon ground states by Kameyama et al. [18]. The details of the method of this expansion are given in appendix F. Ninax  V-’A(Y)  AN(yj,j, 0) 2 exp[—(y/yjv) ]  = N=1  (2.30)  Chapter 2. An Application of the Hypertriton Model  34  Table 2.3: Parameters for the Expansion in Gaussian Basis. fm 0.05 0.10  Xrnjn/  s-wave p-wave  XIax  / fm  Yniax / fiu 100  ymjn/  20 20  —  N 1 1  —  20 —  flinax  (1)  1 x’I\’(x, 1) A  =  where,  fm 1.00  umax  20.0 18.0  (2.31)  ] 2 CXP[(X/Xn)  2 f(l + 3/2)  N(x, 1) =  (2.32)  The Gaussian ranges x , are chosen to be in geometric progression as defined by 7 Xmax, m 7 ax  etc. The relevant parameters are given in table 2.3. The parameters for  the deuteron were taken to be those suggested by Kameyama et a! [18j. The values of y , an(l Ymax follow from consideration of the lambda wavefunction in momentum 1  space. When Fourier transformed, the Gaussian basis has the same Gaussian form with momentum space ranges qN  2/yN. Nmax  equation (2.30)  (q)  2 AN(qN,0)ex j p[—(q/qN)  =  (2.33)  N=1  The lower and upper scales of the lambda wavefunction in momentum representation are  an(l QA respectively and the choices of  in order to cover the range  —*  Yiiii  and Yiiax given in table 2.3 were made  QA.  The expansion of the lambda wavefunction  was checked by computing the nor  malization of the Gaussian fit. The input wavefunction was normalized analytically and  so any deviation from one shows an inaccuracy in the fit. Nx  NN’1  The value of  ATt  [  2 Y NYN’ (YN  + YN’  ]  (2.34)  was found to deviate from one by 106 and it is concluded that  the fit was satisfactory. The high quality H 3 e wavefunction of Kameyama et al [18] was  Chapter 2.  An Application of the Hyperl;riton Model  35  used in the calculation of 1 F ( q) and led to the values given in table 2.4. The variation (q) from the value for QA 1 in F the hypertriton. The value at  1 indicates the uncertainty due to the model of 1.17 fm  Qp,  = 1.17 fnf’ is 1 F ( q,)  0.57 which should be compared  to 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 form of the decay amplitude given in equation (2.24) it was found that, F(H  2.4  3 H e)  2(1 +  q/9q] x 2 2 (q) 1 F . +p  W:/M3He)  (2.35)  Total Decay Rate  The total decay rate for all modes involving a r F(H  X)  f  2wq(2)3  x ?fl,rnj,X  The summation  is  26(wq +  I (X; rn  6MN  + Ex  —  ) 3 A  Heff() H;m)  2  (2.36)  (2.37)  is over all states of two protons and one neutron which are ener  getically accessible, their internal energy with respect to a static proton plus deuteron being written as Lx. The kinetic energy of the CM motion of the three nucleon system is 2 /q 2(3MN) and A 3 is  mA  — —  BA  177.2 MeV.  For the break-up channels Ex is positive semi-definite and Dalitz [22,24] has argued that it is reasonable to replace Lx by some mean value Ex corresponding to the peak in the pion spectrum. When that is done, the energy conserving delta function yields only one value of q, namely i at which the matrix element need be evaluated and the sum over states F(H  can he extended to a sum over all states regardless of energy conservation. X)  2(1 3 +WS M N)  X  2 (X;rnfIHeH;m i)f (2.38)  Chapter 2.  An Application of the Hypertriton Model  36  A correction must be made for the underestimate of the two body mode which occurs at significantly different Ex from Ex. This is done directly by adding and subtracting terms equal to the two body rate at the correct and incorrect values of q thus, 6F x 27r  qF(q [ 2 s ) 2 +p q /9q]  =  {s6 +  where,  (s 1 F [ 2 ) +p /9qJ 22  —  p22/qe}  =  —  (2.40)  F ( 1 q,) 2  —  When this is clone, the most appropriate value for X  )( X  —2 F(q)  [() F 3 l(q)2  =  Using closure  (2.39)  -  F)2]  (2.41)  is 96 MeV [43].  = I and the antisymmetry of the hypertriton wavefunc  tion, 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. In appendix D it is shown that, f(X;rnfjHeff(H;mi)2 7fl, flU  =  s2[1  1 + i)  —  7d@)]  +[ (/qo) 2 p 1  —  5 1s()  1  ?]d(q)].  The exchange integral (q) is due to the s-wave part of the deuteron and  rj(q)  (2.42) is due  to the cl-wave part: there are no cross terms. The exchange integrals are overlaps of the lambda part of the liypertriton wavefunction and the deuteron wavefunction as shown in appendix D. Results for 5 ij ( q) and Dalitz macic no estimate of  7jj(q)  ()  are given in table 2.4.  in the 1958 paper but later with Rayet [43] found  0.31 in agreement with Leon [23]. This larger value corresponds to the use of B,  0.25 MeV which was the accepted value at the time.  The lambda particle  momentum distribution found in the previous chapter peaked at  = /iB and will  Chapter 2.  An Application of the Hypertriton Model  Table 2.4: Summary of H q/MeV F (q) 1 0 0.8041 96 0.6263 114 0.5729 0 0.7876 96 0.6093 114 0.5559 0 0.8169 96 0.6397 114 0.5863  37  3 H e Form Factor Results.  ,‘  ?7 ( 3 q) 0.5401 0.2116  7)(q) 1.86 x iO 1.36 x iO  —  —  0.5280 0.1999  1.60 x iO 1.17 x i0  —  —  0.5494 0.2211  2.10 x i0 1.55 x iO  —  —  QA/fm’ 1.17 1.17 1.17 1.05 1.05 1.05 1.29 1.29 1.29  have a larger overlap with the proton in the final state cleuteron as BA increases since the deuteron wavefunction is larger than the lambda wavefunction in momentum space. In fact, s(q) should scale like At low momentum,  (  and hence \/ due to the following argument.  the deuteron scale is the pion mass = 0.7 fm  ),  the deuteron  wavefun ction must look like, N  where  iv  is a normalization constant and  momentum distribution is at  j  (2.43)  = /MNEB(d). The peak of the lambda  and so the overlap with the proton in the final state  deuteron is approximately equal to the the deuteron wavefunction evaluated at i. e. This scales with c since  f  2 dpp  2 (c/cd)  A(p)d(p)  @)  2  cN +  work =  0.68  c.f.  () 0.2  = 0.72  which support the hypothesis that the exchange integrals scale with \/. The final expression for the inclusive r =  (2.44)  < 1. The numerical values are,  hls(q)other  F(HX)  .  decay rate is,  q x 2r(1 + wq/3MN)  (2.45)  Chapter 2. An Application of the Hypeitriton Model  {s2[1  2.5  +  —  38  +61+ [ (/qo) 2 p 1  d()  +  — —  el}  (2.46)  Results  The final expression for R is, R  =  F1 (q) 2  (q [s2(1  +  )  -  + 2 [s q /9qg p j x k /q(1 2 ) +6) + p  -  where,  m()  -  (1 + Wq/M3He) (1 +/3MN)  k  (q) + e)] (2.47) (2.48)  The kinematical factor 1 is 1 .003 and thus exceeds one by only 0.3%. The corrections for the underestimate of the two body decay in the denominator are 6 and  €  1.7 x 102 in the case  —2.5 x iO  1.17 fnf’. The smallness of 6 shows that the effect  of the extra phase space at q , is almost exactly cancelled by the drop in the form factor 7 . 1 F  The error in this calculation was estimated by varying the value of QA as suggested in the previous chapter and combining this in quadrature with the error in the ratio (p/s) 2 from experiment. Varying QA by 10% gives a change in R of 0.016, i.e. 5%. The change in 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 particle decay rate is given by,  =  F(A p+) x  2  =  +p 2 (s )  27r(1  +wq / 0 mp)  x  .  2  (2.50)  Chapter 2.  An Application of the T-Iypertriton Model  39  The factor 3/2 arises from including the ir n mode which occurs one half as often as 0 the 7rp mode clue to the I  rule aid the fact that the lambda particle has isospin  =  zero. Since the isospin of the hypertriton is also zero the r 0 modes will occur one half as often as  the  3  modes and thus the total lifetime T(H) is given by,  T(AH) = T(A)  /qo\ q  + 2 [s ] kT p -  [s2(l  +  (q)  /q(1 2 + 8) + p  kT  (1 +w/3MN) (1 + wq /mp) 0  -  is(q)  —  5 i/s(q) -  —  —  17d(q)  + f)]  (2.51) where,  =  The structure of the hypertriton affects the value of grals  i()  and  (2.52)  .  r(H)  only via the exchange inte  which enhance the free rate by approximately 6%. The kinematical  factors also increase the rate by 6%, the overall effect being a reduction in the lifetime of 12%. With the values given in table 2.4 the total hypertriton lifetime due to mesonic decay is found to be, r(H)  =  0.88r(A)  =  2.32 x 10_lU seconds  The experimental results for r(H) are not clear.  (2.53)  The values found from bubble  chamber experiments are shown in table 2.5. Keyes et al. [25] argues that the result of Block et al. [:38] could be in error due to a misclassification of true decays at rest as decays 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 Keyes  et al. [36,37] it was found that T(H) be set on  i()  from this result are  =  (2.44) x 10_lU seconds. The limits that can  ()  =  0.1 + 0.2 which provide no real test of the  hypertriton wavefunction. In summary, the ratio R of two body to all pionic decay rates is found to agree with bThis 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 weighted average could then be performed assuming that the variable y is normally distributed.  Chapter  2.  An Application of the  Hypertriton  Model  40  Table 2.5: Bubble Chamber Results for T(H).  10 r(H) sec /10  Reference  0J9 0 ••95 0.15  Block et al. [38]  2 • 64t0.84 --0.52  Keyes et al. [36,37] Keyes et al. [36,37]  2 • 28  0.46  0.62 2 . 46 +4  Keyes et a!. [25]  2.20+1.02 —0.53  Keyes et al. [25]  2.64+092  Keyes et a!. [25]  0.54  experiment which lends confidence to the wavefunction.  Although the lifetime is also  found to agree with experiment, this is a poor test of the wavefunction.  Chapter 3  Muon Capture by 3 He: Rate and Spin Observables in the Elementary Particle Model and Their Sensitivity to the Pseudoscalar Form Factor.  3.1  Introduction  The Elementary Particle Model (EPM) was first used to calculate the muon capture rate by 3 He in 1965 [45].. It was reviewed with emphasis on the the value of the pseudoscalar form factor in 1968 [46] and compared to an impulse approximation result by Phillips et al 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 3 H and 3 He nuclei are described. They are treated as elementary particles with respect to the number of degrees of freedom used in the wavefunction, while their structure reveals itself through nonelementary couplings and q 2 dependent form factors. It is important to realize that these are nuclear form factors and so represent the full structure of the nucleus including contributions from the nucleons and meson exchange. Hwang [49] has applied the EPM to spin observables and noted that the triton asym metry is sensitive to the pseudoscalar form factor F. Just how sensitive the asymmetry is and also how sensitive other spin observables are to F was investigated using accurate values for the trinucleon form factors. An analysis of the theoretical uncertainties in the spin observables was also made.  41  The Elementary Particle Model  42  Figure 3.1: Feynman diagram for muon capture.  (k’)  3 H e(k)  3.2  Definition of Elementary Particle Model and Kinematics  In the EPM, the triton and helion are assumed to be members of an isospin doublet. This can be written formally as, He) H 3 j  =  )  3N:m,k)®1/2, 1/2)  (3.1)  :3N:mJ) ® 1/2,  (3.2)  —  1/2)  where k is the trinucleon centre of mass momentum and rn the latter vector is in the two dimensional isospin space I are the same for 1 3 1e and H 3,  (  =  -,  spin projection.  The  13 ) and the space-spin vectors  for the same k and mi). Given that both the spin and  isospin of the trinucleon bound states are one half, five degrees of freedom are needed to describe the state, three continuous labels (k), the spin projection m, and the isospin projection 13. The muon capture amplitude is found by evaluating the Feynman diagram shown in  The Elementary Particle Model  43  figure :3.1, writing free spinors for H 3 e and H 3. This diagram defines the kinematic 4vectors  i/v.  k and k’ whose contravariant components are written  muon momentum ii  =  }.  The  and the helion momentum k are set to zero. The neutrino momentum  is 10:3.22 MeV and the four momentum transfer squared is —0.954m. These  figures are found using four momentum conservation and taking the initial state energy to be the mass of the muonic atom Mat . 0111 Matom M3He  =  M3He +  —  11 keV = 2914.039 MeV  2808.392 MeV [51]  (3.3)  105.658 MeV [41]  3.3  The Hadronic Current  In the absence of second class currents, the hadronic current has the following form, ‘hadroic  (k’)  a 7 [FV  + FMi + FA7 5] u(k) 7 5 + F  (34)  which defines the form factors Fv, FM, FA and F. In the above, qa = —  — 7137a)/2 75  3 M  =  75  =  3 2 l O 7 j  =  2808.66 MeV  ufu =  1  In equation (3.4), M 3 is an arbitrary parameter with dimensions of energy. It is intro  duced to make all the form factors dimensionless and its value is taken to be the average trinucleon mass for the sake of convenience. In accordance with ecluations (3.1) and (3.2),  the same space-spin function has been written for H 3 e and H 3. The spinor u(k) obeys the Dirac equation for a free particle,  (  —  M ) 3 u(k) = 0.  (3.5)  The Element arv Particle Model  44  The M 3 denominators in equation (3.4) have no physical significance but the M 3 appear ing in equation (3.5) should represent the trinucleon bound state degenerate mass. Of course, H 3 e and 3 H 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 Vector Current Hypothesis (IVC) [50], one can relate the vector form factors to the electromag netic form factors of H 3 e and H. Specifically,  where  i  ) 2 Fv(q  =  ) 2 2F(q  ) 2 FM(q  =  HeFIe(q2)  —  ) 2 F’(q  (3.6)  ) 2 TrF’r(q  (37)  is the anomalous magnetic moment and F 2 are electromagnetic form factors , iF 1  analogous to F and FM in equation (3.4). F 1 and F 2 have the value one at  2 = q  0. The  values of the anomalous magnetic moments are, He  Tr  =  —8.:3689  (3.8)  +7.9173.  (3.9)  These values follow from the measured magnetic moments of anl  1t’  2.978960(1) n.m.[5l] and the relationships between He 11  =  Tr  where, and m , is the proton mass. 1 and FM= —16.286. At the  At q 2  3’  (2 +  KHe)/31  (1 +  I 3 Tr)/  =  M / 3 m  =  pertinent q , 2  1-1e  t  =  and  —2.127624(1) n.m. tc  which are,  n.m.  (3.10)  n.m.  (3.11)  =  2.99344  0, equations (3.6) and (3.7) give F  (3.12) =  1.000  world averaged values are taken from electron  scattering experiments, which yields [52], ) 2 Fv(—0.954 in Ii  0.8:34 ± 0.011  FM(—0.954 m)  —13.969 + 0.052.  (3.1:3)  The Elementary Particle Model  The value of FA at q 2  45  0 is measured by the 3 H beta decay half-life FA(O)  —g(O.96l  + 0.003)  where gA is the nucleon axial form factor at q 2  [53]-[561.  =  +1.212 + 0.004  =  0, having the value —1.261 + 0.004  (3.14)  [41]. The q 2 dependence of FA is based on impulse apl)roximation expressions for the trinucleon 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 and reduced matrix elements between the 3 H and H 3 e states. The form factors are given by [58, equations :3.1-3.3]: ) 2 Fv(q  gv(q ) 2 [1j°  ) 2 FM(q  ) + 2 :3[gv(q  ) 2 FA(q  gA(q ) 2 [j  ) 2 Fp(q  where [1]0,  dependent  [j_,  ( see  =  []+  (3.15) gM(q ) 2 ][]  gv(q ) 2 [1]°  —  32 g(q2)[j ]11  (3.16) (3.17)  5 ) 2 9gp(q ] [ + [j21  —  63g(q2)[]21  (3.18)  and [iP] ’ are the reduced matrix elements and are q 1 2  [58, equations 3.7,3.8 and appendix G] for their definition  ).  The nucleon  form factors gv, gri, gA and gp are defined by an equation for the nucleonic charged weak current analogous to equation (3.4) with M 3 replaced by MN, the average mass of the nucleon. It should be noted that it is more usual to write the mass of the 2MN in the denominator of the  gp  muon  and not  term: that convention will not be used here.  The q 2 dependence of FA is thus given by, U (  ‘Aq  2\  FA(0)  )  -  2\  I [1—( 2 gAq I L° q gA(0) []-(0) .  —  The usual argument [45,46] is that the q 2 dependence of [6]  (3 19 can be found from the q 2  dependence of FM. This will be true if the last two terms in equation (3.16) are negligible.  The Elementary Particle Model  At q 2  46  —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 trinucleon wavefunctions and so the expression is dominated by the first term.  can appeal to a smooth and small change in  [öJ  (  Alternatively, one  over this q 2 region. At q 2  =  0, FA  is about equal to (—)g so that [6](0) is about —1. The value of gv + g at q 2 —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 q 2 dependence [59) so we have,  ) 2 [](q  [5]-(0)  —  -  ) 2 gv(O) Fj(q ) FM(0) 2 gv(q  p320  Writing dipole form factors for gv and g and substituting the above in equation (3.19) gives, ) 2 FA(q  —  FA(0) FA(—0.954m)  where,  M  =  M  =  (1 (1  —  —  /Mv FM(q q 2 ) ) 2 /M 2 q ) 2 FM(0)  1.050 + 0.004  0.7 10 CeV 2  3 21 (“ (3.22)  [59)  1.08 + .04 GeV 2 see appendix I  A better method is to use the q 1 + FM which is the same as the q 2 dependence of F 2 dependence of  []  in the limit that [iP]” is zero. In that case, ) 2 F(q FA(O)  —  —  FA(—0.954nI)  (1 (1  —  —  /Mv 2 q ) 2 (Fv(q + F(q ) ) 2 /M 2 q ) 2 Fv(0) + FM(0)  1.052 + 0.004  3 23 (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 them against the ratio Fp/Fp’  .“  Fp’  “  is the value of F obtained from the simplest form of the  partially conserved axial current hypothesis (PCAC) [60] as shown below.  The Elementary Particle Mode]  47  The PCAC links the divergence of the axial current A(x) to the  pion  field ,(x):  the relationship being, ãA(x)  rn 1 a ( 3 1, x) ,  where a , is the pion decay constant and a 7 , 1  (3.25)  0.9436 + 0.0011 [41]. The aim is to link F’A  =  and F to the pion trinucleon coupling constant 1 F, defined by, ( H 3 Jj(x) H 3 e)  =  —  r 7 m  (q 1 F ) 2 ,7 (k’) s u(k) e  (3.26)  To do this one other ingredient is needed, namely the Klein-Gordon equation.  [aa. + m1,] (x) The field operators j(x) and (x) operate initial 4-momentum  k and final  matrix element can be  on  4-momentum  =  j(x)  (3.27)  elements of Fock space supporting only  k”, so that  &L  acting on  (x)  replaced by iq. This allows the replacement of j(x)  (:3.26) by (—q ) (x). At this point, the PCAC relation, 2 7, 2 + rn  (  in  in  such a  equation  equation (3.25)), is  used to replace the pion field, which leads to the axial current aHcI finally to the axial form factors. H 3 (:  j(x)I H 3 e)  =  2 + (-q  rn7,2)  (Hl(x)jHe)  7, m (—q + ) 2 HA(x)I d( H 3 e) a,, rn 7, =  A  +  7, + (—q ) 2 m  2 q 2 3 4M  2 rn 7, —a =  2 7, (rn  Evaluating equation (:3.31) at q 2  =  —  ) 2 q  2M ( 3 FA +  42  Fp) (k’) u(k)e 5  (q 7 F ) 2 ,  (3.28)  (3.29) (3.30) (3.31)  0 yields the Goldberger-Treirnan relation. FA(0)  (0) 7 F 7, —a ,  (3.32)  The Elementary Particle Mode]  48  By using this and rearranging equation (3.31) in favour of F, the following relation ship between F, FA and F 1. can be derived. ) 2 Fp(q with, €  q 2)  FA(q ) 2 [i  = 2 2_q 1 m .  —  -  2 m  (3.33)  1. 1 F ) 2 (q (0) /F . )/FA(0) 2 FA(q  1  ) 2 (-q  €(q ) +2 ]  (3 34  The above procedure is due to Primakoff [61]. If it is assumed that the q 2 dependence of 1 is the same as the q F 2 dependence of FA, dominated by the  pion  pole. The quantity F’ is defined as, F I (q 2 ) =  and will be referred to  vanishes. The q 2 dependence of F is then  €  as  the  “  PCAC  4M 2 3 2 FA(q , 2 —q 2 7 rn  value  “  of  (3.3o)  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). They then fit is  €  €  over the range —m  < q 2 <  2 using equation (3.33). The result 1. rn  —0.05  2. They evaluate FA using equation (3.17) and 1 F. with an expression of similar form and fit them to dipole expressions over the range —m These fits can he used to evaluate result at q 2 —0.954rn is 3.4 At  €  €  <  q  <  . 2 1. rn  directly from equation (3.34) and the  —0.06  Rate and Spin Observables  2 =—0.954n, the weak interaction is well apl)rOxirnated by a point-like interaction q  so that evaluation of the Feynman diagram in figure 3.1 yields,  M  = /2  Vd  J° hadroiiic  (3.36)  The Elementary Particle Model  49  where the leptomc current is, jiePtOfliC  CF  =  (1’){7(1  —  ‘ ) 5 }u(t)  (3.37)  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 form thus, M  V (1  —  iL){C + Gi+ GA.}  (3.38)  where the two dimensional spinors have been dropped and, ) 3 (w+M  N’  11  Fv (i  3 w+M  3 w+M —  CA  =  w =  FA  (Fv + FM + FA  —  (1’  +w  —  ) 3 M  3 w+M  F) 3 2M  0.854 + 0.011  0.6027 + 0.00098  ‘  3 w+M  /(,,2  )  F 4 3 2M  —  (Fv + FM)  1.29:3:3 + 0.0041  ) 2 3 +M  lepton Pauli spin matrix trinucleon Pauli spin matrix The numerical values for G ,Cp and CA are at. q = —O.954m and the value for 2  Op  only  holds when F takes on the PCAC value, Fe’. N’ is the normalization constant for the H spinor and w is close to the energy of the triton. Despite the appearance of equation 3 (3.38), it is not a non-relativistic reduction. If a 2x2 matrix, M, is defined by, M  (1  {Gv + Cp + GA.}  —  then the rate is proportional to T  2,  T  (3.39)  where 2  Tr [pj MtM]  (3.40)  The Elementary Particle Mode]  50  where p is the initial state density matrix and MtM is given by, MM  Go2 + ç2  where,  gL  =  +  +  2 2 + 3GA G 2+O ri  ‘-‘  A  ri 2  2  ri  ‘—‘P  ‘-  fri  ohr-i  =  (3.41)  +  2 v H-  ri  “—‘V’--’P  —  ‘‘—-‘  ri  A ‘-‘ i  n-i  ‘  ‘—‘A)  on  —  ‘—‘A  2  2GA(Gv+Gp—GA)  7 =  —20p(Gv +  GA)  The rate is given by, F=  V  2  C 2 N’  I)  2  ,2  (_  )f  IT  (3.42)  2  where q(;) is the spatial atomic wavefuiiction and C is a correction factor which takes into 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 for the four lowest lying atomic states. These are all associated with the iS state and the population densities are written as N(f,f), where angular momentum and  f  f,  on the quantization axis  .  f  is the projection of the grand total  The quantum number  f  equals 0(1)  may take the values 0(-1,0,i) giving four states in total.  p  2 N(f,f,) ) (f,fj If,f  (3.43)  f,fz The density matrix is taken to be diagonal in the 5 I -Ie  f,f  space rather than the  ,  space because j,f are good quantum numbers. This follows from the fact that the  width of the IS state,  (  due almost entirely to the muon lifetime  hyperfine splitting. F(1S)  =  3.0 x 10° eV  ),  is much less than the  The Elementary Particle Model  51  E(hyperfine)  0.2 eV  (3.44)  The values taken by 2 N(f,f ) , is the subject of chapter 7. Using,  Tr [ 2 f,f ) ( .f,fz (A =  +  +  +  +  A + B[2f(f +1) -3] +  [f(f +1)  +  +  x  2 [ + F.jf  -11+ [-f(f +1) + 3f j 2  -  (3.45)  one finds, T  0o2  2  [N(1,i) + N(l,0) + N(1,-i) + N(0,0)]  —(a’ +  ) cos0 [N(1,1)  —  N(1,-1)]  [N(1,1) + N(1,0) + N(1,-1)  +  + 6’ [N(1,0)  —  (3.46) —  3N(0,0)]  N(0,0) + cos 9 (N(1,1) + N(1,-1) 2  —  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 clearly seen. T  2  02(1  + APPi(cos0) + 2 AtPtP ( cos0) + AP)  (3.48)  where P, P and P express the deviation of the hyperfine populations from the statistical values and A, A and A are the “Analyzing Powers”. The factors 1 P and P 2 are the  Legendre polynomials. P  =  N(i,i)  =  N(l,l) + N(1,—1)  =  N(1,l) + N(1,0) + N(1,—1)  —  N(1,—1) —  2N(1,0) —  3N(0,0)  =  1  —  4N(0,0)  (3.49)  The Elementary Particle Mode]  Using  52  the Madison convention [64] to define polarization of the  f  = 1 state, the vec  tor polarization Pz is 4P/(3+P) and the tensor polarization p is 4P/(3+P). The analyzing powers are given by, A  =  0 —(+/3 2 )/G  =  2 0 26/3G  =  . 2 (‘ + 6/3)/G  (3.50)  There are thus three spin observables. The fact that there are three corresponds to the four states in which the muonic ion can  find  itself prior to capture. These four states  give four rates which can be measured. For a statistical population one arrives at an isotropic rate given by G . For iiou-statistical populations, the rate is anisotropic the 2 0 angular 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 the difference between the  f  0 and  f  = 1 total rate.  rf-1 -  stat  Ff=o Fstat  =  1  —  3A  (3.51)  The unpolarized rate was found by setting all of the N(f,f) to 1/4. 0 F  = G 2  2 C I (O) Vud 12 N’  2 ,2  2 a  (i  (3.52)  —  3.5  Results  The value for the unpolarized rate, with F set at its PCAC value, was found to be 0 = 1497 + 11 F  s  which should be compared to the experimental results of Auerbach  The E]ementaiy Particle Model  53  et al [65] and Clay et al [66].  0 F  11505 +46 s_i  [65] (3.53)  =  1465 + 67  [66]  The value found herein agrees with the weighted average of the above results, which is Fo Fo  =  1492+:38 srn’. Phillips [47] finds F 0 =1425 s , and Frazier and Kim [46] find 1 1449 s, the difference between the two being solely due to the values of Fv, FM  and FA used. The enhancement of the present calculation with respect to the result of Phillips, is 5.1 %. The corretion factor is 1.4% larger and the form factors give an enhancement of :3.7%. The values of the spin observables, with F set to the PCAC value are, A  =  0.5243 + 0.0057  (3.54)  1 A  =  —0.37933 + 0.00064  (3.55)  A  =  —0.0959 + 0.0060  .  (3.56)  It is pertinent to ask which of the four observables A, A , A, Fo is the most sensitive 1 to the value of F. The sensitivity to F is shown in figure 3.2 where the observables are drawn scaled to their PCAC values i.e. the curve labelled A is actually A divided by its value at F  =  F’ etc. It can be seen that A is more sensitive to F than A 1 which  is more sensitive than A which is more sensitive than Fo. Quantitatively, the pertinent figure of merit for an observable A is the slope of log(A) with respect to log(Fp). This is given for F , A, A 0 1 and A in table 3.1. When “measuring” F it is necessary to combine an experimental error with a the oretical error. This should be done in the following manner. Let error in the theoretical prediction of F 0 and  be the fractional  be the fractional error in the measured  value of F . The fractional error in a quantity A is the error divided by the value of A. 0  The Elementary Particle Model  2.0  1.5  1.0  0.5  54  -  -  -  -  0.0 0.0  0.5  1.0  1.5  2.0  Figure 3.2: Sensitivity of observables to F.  1;  0.0  0.5  1.0  1.5  Figure 3.3: Unpolarized rate versus F.  2.0 p  The Elementary Particle Model  55  Table 3.1: Sensitivity of Observables 0 to Fp.  0  Fo A A A  Define a quantity  (F)  %  dF  F,  0.11 0.38 0.75 0.89  by,  /  ((F))2 =  (  \ 2  (P)  F dFp F  /  JJ -b (  (F) FF dFp F  2 ‘\  J I  (3.57)  This 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 F and would equal the absolute error in F 0 divided by the slope of F . This is equal to 0 the fractional error divided by the slope of the log of F . When both an experimental 0 error and a theoretical error contribute, the probability density function for F is the convolution of two normal distributions,  (  and has variance equal to o+o. Thus,  variances o and u  ),  which is itself normal  is the total error in the value of Fp resulting  from the measurement of Fo. Evaluating this expression one finds, (  2 (F) Fp)  —  2 N 25 (°•°  (0.0072  o.ii) +o11} =  0.24  (3.58)  By evaluating C 2 for different values of F it is easy to find what value reproduces the 0 experimental result for F . The dependence of F 0 0 on F is shown in figure :3.3. If all the assumptions made in the EPM are correct then one has the following “measurement” of F. =  1.03 + 0.24  (3.59)  The Elementary Partide Model  56  What precision is needed in a measurement of A, A or A in order to better the above measurement of Fp? By evaluating expressions analogous to equation (3.57) it was found that, (Av))2  =  (8))2+(1)2  =  ((A)2  (3.60)  (0M017)2  (At))2  (3.61) =  ())2  ())2  +  (03)2  (3.62) which lead to the following requirements for the fractional errors in the measurements of A, A and A. (Av)  <0.24 if qj) <0.09  (3.63)  <0.24 if  j) <0.18  (3.64)  <0.24 if  j)  (3.65)  Also, it can be seen that the minimum and the minimum  <o  is 0.061, the minimum  is 0.0023  is 0.071.  These results indicate that all the spin observables A, A and A offer a better determination 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 the microscopic calculation detailed in the next chapter.  Chapter 4  Muon Capture by H 3 e in the Impulse Approximation  4.1  Introduction  The elementary particle model (EPM) describes quasi-elastic in a  compact  a  muon capture by H 3 e  and useful way. However, it is unsatisfactory in that it is a convenient  parameterization of the H 3 e  —  H transition rather than providing a complete theoretical  understanding. A more fundamental approach is at the ‘microscopic’ level which means using nucleons and mesons to describe the iiitial and flial nuclear states and the muon capture interaction. The microscopic approach is applicable to muon capture by other nuclei and also muon induced break-up channels in a uiifiecl way, whereas the EPM introduces a new set of form factors for each nucleus and final state and thus has limited predictive 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 currents  are included. Other currents clue to the virtual mesons also present in an interacting system 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 and the effective muon capture Hamiltonian. Highly sophisticated and reliable wavefunctions for H 3 e and H were used aid are described in detail in section 3. The muon capture  Hamiltonian was derived using the on-shell IA which is expected to be a reasonable aThe 3 He H channel is termed quasi-elastic rather than elastic because the transformation of a 3 muon into a neutrino involves a release of energy.  57  Chapter 4. Muon Capture by 3 He in the Impulse  58  Approximation  first approximation for the nuclear current. This calculation is intended to eliminate any uncertainty clue to the description of the nuclear states so that ai unambiguous conclusion regarding the quality of the IA can be made. An early calculation by Peterson [106] used a simple phenornenological based form for the wavefunction and found that the rate fell short of the experimental value by approximately 13% in the IA. Phillips [47] has also calculated the total rate in the IA using wavefunctions for H 3 e and 3 H based on separable NN partial wave potentials.  The result fell short of the  experimental value by approximately 10% and it is unclear whether this discrepancy was due to the wavefunctions or the IA. The calculation by iKlieb and Rood [58] employed a wavefunction which is a solution of the three body Schrödinger equation with the Reid soft core potential constrained to act only in the  I0  and S D channels. These authors found the rate to be 1268 3 1  seconds which is 15% short of the experimental value. Again, the wavefunction used is deficient in that it does not represent a complete solution of the three body problem and also does not reproduce the correct binding  4.2  energy.  The Trinucleon Bound States  The triton 3 ( H ) and helion 3 ( H e) form a good isospin doublet and have spin one half and positive parity. To find wavefunctions for these nuclei it is necessary to solve the three body Sclirödinger equation. In solving the three body Schrödinger ecluation, it is usual to separate it into the Fadcleev equations [17] as shown below. If Vc. is the potential acting between particles  and -y, where 3 (c/ y ) is a cyclic  permutation of (123), then the Schrödinger equation for the wavefunction 3 (T+ + 2 — E)) V V+ =0 V  Ji  is, (4.1)  Chapter 4. Muon Capture by H 3 e in the Impulse Approximation  59  where T is the kinetic energy of the three particles. The Faddeev equations result from writing the full wavefunction as the sum of three Faddeev components ,L’ , 1  /‘2  and  The Fadcleev components are found by solving the following coupled equations.  ) 1 E)I  =  -E) 2 (T+V )  =  (T+V  -  )) ( 2 -V ) 3 ) 1 + I ) 2 ( 1 —V ) j +  2 (T ) ) 1 ( 3 ) =+l E +V ) V b ) For three identical particles the Faddeev components mutations of /‘  )  ‘2)  (4.2) and  ‘) are simply per  as can be seen from the equations (4.2). The total wavefunction ‘P is  thus, =  (  3 + 2 +P ) ) P  (4.3)  where P ,P 2 3 are permutaton operators as defined in appendix E. The solutions of the Facldeev equations for the trinucleon bound state  have attracted much attention over the  last twenty years [67]-[93] but the most important property of the resulting wavefunctions is the phenomenon of scaling as exposed by Friar, Gibson, Chen and Payne [:30]. These authors clearly show that various low energy observables of the trinucleon bound state all depend on the binding energy of the solution obtained according to a simple power law irrespective of the details of the NN potential used in the Faddeev equations. In order for wavefunctions to provide good descriptions of 3 He and 3 H then, it is a necessary not sufficient  )  (  but  condition that they have the correct binding energy.  Most NN realistic potentials underbind the triton by ‘--1 MeV. This is not a had  disagreement 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 body force enhances the binding energy. The amount of enhancement depends sensitively on the irNN form factor cut off A. It is thus possible to tune A in order to reproduce the experimental binding energy for H 3. This procedure is perhaps the most natural way  Chapter 4.  Muon Capture by H 3 e in the Impulse Approximation  60  to ensure that a :3N bound state wavefunction satisfies the condition of possessing the correct binding energy.  4.3  The Kamirnura Wavefunctions  The wavefunctions for the helion and triton used in this calculation were developed by H.Kameyama, M.Karnimura and Y.Fukushima and their properties are described in detail in [18]. These wavefunctions represent accurate solutions of the Schrödinger equation for the 3N system, include the Coulomb repulsion in H 3 e exactly  (  i.e. non-l)erturbatively  )  and use a tuned three-body force to make the small adjustment necessary to reproduce the experimental binding energies. The Argonne 1 V 4 NN potential [69] was used as well as the Tucson-Melbourne 2r exchange model [70,71] of the three body force. The Argonne V 1 4 potential gives an excellent fit to the deuteron properties and NN scattering phase  shifts up to the pion production threshold. The wavefunctions are found from the variational principle. This method has the advantage over regular solutions of the Faddeev equations that the potential need not be 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 variational approach leads to fast convergence of the binding energy with respect to the number of channels in the Faddeev component, a statement which will he explained below. Previous variational calculations [94]-[100J were not as precise as the Fadcleev solutions but this deficiency 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 channel is defined by the quantum numbers l, A, L, s, S,  ) where each  and the Russell-Saunders  Chapter 4. Muon Capture by H 3 e in the Impulse Approximation  61  (LS) coupling scheme has been used.  ()  =  Ik)  =  i  (x, y)  In equation (4.4), particles 2 and 3 i and have orbital angular  orbital angular  and total  spin  isospin  defined precisely  in  .  (  (, ) the ‘pair’  momentum 1 a  )  ] 1 x  ®  Mj  1  (4.5)  have been coupled to spin sa,  in their CM. Particle 1  (  SOSJflfl  the ‘spectator’  )  has  with respect to the CM of the pair and is coupled to the  momentum  pail to give total  (4.4)  S, total orbital angular  momentum  The spherical harmonic )‘,  L, total angular momentum  vector X and isospin vector  spin  ij  are  appendix E (see equations (E.8)-(E.13)). The chaiinel wavefunctions  i/(x, y) are written as a sum of Gaussian basis states whose ranges are  in  geometric  rogressioIL Thrnax  (x,y)  ,]V,nsx  ANq(x)q(y)  =  (4.6)  n,N=l  q(x)  exp(— / x) x 12 N?IlGx =  p max  (  n—i max—  ql(y)  =  NNy (  )  2 exp(— / y) y (4.7) N—I  tNmax—i  =  jInax  Ymin  (4.8)  The distribution of ranges is dense at small distances which allows the description of short range correlations. At large distances the basis functions add coherently so that with 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  j  [(y)] 2 dyy  The non-linear variational parameters  (fliiiax,  =  1  =  1  Xiji Xmax)  (4.9) and (IVrnax, Ymin, ymax) can  be chosen independently for each channel and thell the expansion coefficients AN and  Chapter 4.  Muon Capture I)y 3 He in the Impulse Approximation  62  Table 4.1: Properties of the Wavefunctions. 3 H (22) I H(8) 8.43 8.34 8.48 90.61 90.70 0.16 0.14 9.23 9.16 49.07 48.64 —57.50 —56.98  EB EB (exp.) P(L=0) % P(L=1) % P(L=2) % K.E. MeV V MeV  3 H e(22) I He(8) 7.76 7.67 7.72 90.64 90.73 0.16 0.14 9.20 9.14 48.21 47.78 —55.97 —55.45  binding energy are found from the Rayleigh-Ritz variational principle which yields a generalized eigenvalue equation for the AN.  b  It has been demonstrated [18] that these wavefunctions are as accurate as the Fad cleev solutions and further that the binding energy of the solution converges with respect to the number of channels faster [101, see figure 5]. This convergence is especially im portant when a three-body force is included since then the Faddeev solutions show poor convergence because of the strong dependence of the three body force on the angular orientation of the three particles. Two wavefunctions were used comprising of an 8 channel or 22 channel expansion  of the Faddeev component. Some properties of these wavefunctions are given in table 4.1. The label H 3 (8) indicates that the properties listed pertain to the 8 channel triton wavefunction. 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 of  the three nucleons. The parameters defining these wavefunctions are given in tables 4.2 and 4.3. bThe values of  the basis coefficients AN  were provided by M.Kamirniira.  Muon Capture I)y 3 He in the Impulse Approximation  Chapter 4.  63  Table 4.2: Channel Specifications for the 8 Channel Wavefunction. Channel 1 2 3 4 5 6 7 8  — —  l, 0 0 2 2 2 2 0  ,\. 0 0 2 2 2 0 2  2L  L 0 0 0 1 1 2 2 2  s  0 1 1 1 1 1 1 1  Sa 1/2 1/2 1/2 1/2 :3/2 3/2 3/2 3/2  a 1 0 0 0 0 0 0 0  umax  j 1 X  X11-lax  15 15 15 15 15 15 15 15  fin 0.05 0.05 0.10 0.10 0.10 0.10 0.10 0.10  frn 15.0 15.0 15.0 15.0 15.0 15.0 15.0 15.0  Nmax  Yniin  15 15 15 15 15 15 15 15  frn 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3  Yma frn 9.0 9.0 9.0 9.0 9.0 9.0 9.0 9.0  Table 4.:3: Channel Specifications for the 22 Channel Wavefunction. Channel  —  5  6 7 8 9 10 ii 12 1:3 14 15 16 17 18 19 20 21 22  —  —  —  Ci  1 2 3 4  —  0 0 2 1 1 2 :3 2 2 1 1 1 2 :3 :3 2 0 2 1 :3 1 3  ,\ 0 0 2 1 1 2 3 2 2 1 1 1 2 :3 3 0 2 2 1 1 3 3  La 0 0 0 0 0 0 0 1 1  1 1 1 1 1 1 2 2 2 2 2 2 2  5 a 0 1 1 1 0 0 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1  Sa 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 3/2 1/2 3/2 1/2 1/2 1/2 :3/2 3/2 3/2 3/2 3/2 :3/2 3/2 3/2  1 0 0 1 0 1 1 0 0 1 1 0 1 1 1 0 0 0 1 1 1 1  umax  15 15 15 10 10 10 10 15 15 10 10 10 10 10 10 15 15 15 10 10 10 10  Xmjfl  XnIax  fin 0.05 0.05 0.10 0.10 0.10 0.10 0.10 0.10 0.05 0.05 0.10 0.10 0.10 0.10 0.10 0.10 0.05 0.05 0.10 0.10 0.10 0.10  fin 1.5.0 15.0 10.0 10.0 10.0 10.0 15.0 15.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 15.0 15.0 15.0 10.0 10.0 10.0 10.0  Nmax  Ymin  15 15 15 10 10 10 10 15 15 10 10 10 10 10 10 15 15 15 10 10 10 10  fin 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3  Ymax fm 9.0 9.0 9.0 6.0 6.0 6.0 6.0 9.0 9.0 6.0 6.0 6.0 6.0 6.0 6.0 9.0 9.0 9.0 6.0 6.0 6.0 6.0  Chapter 4. Muon Capture by 3 He in the Impulse Approximation  4.4  64  The Effective Hamiltonian  The effective Hamiltonian Hoff for muon capture was first derived by Luyten, Rood and Tolhoek [103]. The basic scheme is to sum the contributions of each proton iu the nucleus according to a non-relativistic reduction of the 1 r+p  —*  v, + n amplitude. For a proton  of 4-momentum k’ and neutron of 4-momentum k this amplitude is, M(p + p  v + n)  (4.10)  = VudJaJadroic  where, jlePtornc  =  Ti(V){7a(i  =  (k’) [gv7a +  and,  —  (4.11)  7 ) 5 }U([t)  5 + gp75 +g  2 gMia  u(k)tu(k)  =  1  and,  qa  2  ]  u(k)  =  (4.12) (4.13)  —  The hadronic current is discussed in appendix I where the values of the nucleon form factors 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 initial proton and final neutron respectively and obey the free Dirac equation. The current in equation (4.12) is that of a free proton  —>  free neutron transition. There are two problems  which arise when this current is applied to bound nucleons. First, the current of a bound proton that of a free proton  —+  bound neutron transition is not the same as  free neutron transition. In general there are extra terms and the  form factors gain a k’ , P dependence as well as the more familiar q 2 2 dependence [104]. Second, relativistic wavefunctions for H 3 e and 3 H are not available and so even if one did know the current for a bound p(k)  —>  n(Ic”) transition there would still have to be  some 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 extra  terms and also the k , k’ 2 2 dependence of the form factors.  Chapter 4. Muon Capture by 3 He in the Impulse  65  Approximation  With regard to the second problem let us consider what choices could be made for ko.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 nucleons in  3 H e 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 two  particles and thus, =  MN  —  Eu( H 3 e)/3.  (4.15)  Yet another view is to notice that the muon capture process is a low q 2 process and 3 H e is asymptotically like a bound deuteron-proton system so that the energy of the struck proton should be, MN  —  EB( H 3 e) + EB(d).  (4.16)  There is certainly a certain uncertainty as to what to take for k°. Of that we can be quite 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 is exactly that of a free proton  —*  free neutron transition. This approach is standard and  since the emphasis in this work is on the nuclear structure, it is desirable not to cloud the issue with an unusual nucleon current. It is necessary to expand the hadronic current in powers of k/MN, k’/MN so that the integrals appearing in the matrix elements of the effective Hamiltonian can be evaluated analytically. An alternative approach is to evaluate the integrals numerically using mo mentum space wavefunctions but that approach has not been pursued. After expansion to some order in k/MN, k’/MN the identification Ic’  =  Ic  —  17 was made. Finally, the  resulting current was contracted with the lepton current and simplified. dHaviig made  this choice k’° follows  from energy  conservation at  the vertex  Chapter 4. Muon Capture by 3 He in the Impulse Approximation  Using this procedure,  it  is  important to  66  have some idea of the various sizes of terms  so that a consistent expansion can be made i.e. an expansion which includes all terms contributing at a given level. A useful expansion parameter is neutrino energy  ii  m,L/MN  =  0.11. The  in any muon capture reaction is constrained to be between zero and  the mass of the muon so that quantities like i//MN can safely be taken to be first order in the expansion parameter. The matrix element of k can be identified with that of i/3 as will be explained in the  and so taking k/MN to be first order in the expansion  next section  parameter is a very safe identification. A very crude estimate of 2 /(k M)fo1lowing from this is 2 v / 9M  0.1% and this value is in accordance with the size of the 2 k / M  contributions found by Friar [105]. The contributions of these terms will be neglected. In conclusion, by keeping terms second order in  il/MN  and first order in k/MN an effective  Hamiltonian was found which should be good to about 1%. One further point concerns the size of the terms involving the pseudoscalar form 2 2 m factor gp. In appendix lit is shown that g which implies that —-gp g. It is therefore necessary to keep extra terms which involve gp because of its large size and this is affected by assigning gp order minus two and the other form factors order zero.  The result for q= k’  —  expanded to order two is shown below where the identification  hadronic  k has been used. ‘2  hac1ronic  =  gv  1 —  —  ()IVIN  4M q°  +  .k  2MN 01 f 1 J r  —  v  k+k’ 2MN  +  +  2 Q1,1  2 AT/1  iu.(’x) 4M.  +  8M  —  iX  2MN  -  8M)  2MN  + k) -i  tji  “\ —  —  (  2MN  ,2 —  (  8M  —  3k’ 2 8M}  (7 ix(+’) + 2MN 2MN 2MN  q°  iX  +gM  +  2 ATK ‘ILVIN  A . 9 (k  ‘2  3/c2 —  iu.(i’ x  ‘tIVIN  N 1 °‘‘  1  +  (4.17)  Chapter 4.  Muon Capture by F 3 Ie  the Impulse  iii  \  2  L(  -  4M  8M  o / , Et L2MN  c\  q  2 P  3k 2 8M  —  +  8M) + 4M  -  8MN)  —  +  (  ‘.)]  “1  /  e5tI  —  67  Approximation  2 M N  —  8M  —  3k’ 2 8M)j  +  (4.18)  The leptouic current in two component form is,  k’  leptonic and  k  =  and q°  i7  —  1 (l  =  -(1  m,,  =  —  1  Jieptonic Setting  =  hadronic currents and  v  —  —  I, ± 3 L)L)  in equations  including  (4.19)  (4.17), (4.18), contracting the  the neutrino wavefunction the following  effective Hamiltonian was found. Heff  CF dMIJexp(_ii7 1 V 1 .ij)  =  I  —(1  .  (4.20)  iL){GN + Gi +  [G1 +  +  + (1)  +  -.L  -  GLNa (v)+  MN  +  GvLN11(Ji)  2 J)  = iN  gv (i + ,  iN ETA  2MN  g —  (  rl(1)  r [  ‘gv+gM  v 2 8M) ii  =  =  “  \  8M)  —  I  = —  2MN  —  (i  (i).  GNv(z’.Jj)  —  —  2 V M 4  q°  2MN)  (v  +  /  + Gi x  2 rnv  163 +gP 41 gA(1+  (4.21)  —  (4.22)  —  2 P  / 1  3,,2  —  iiq° 2 +M)  J}  2 4 +M  8M)]  (4.23) (4.24) (4.25) (4.26)  —  11  (4.27)  Chapter 4. Muon Capture by H 3 e in the Impulse  GLvN  =  ii  q°  4MN  2 V 1N  =  gA  (4.28)  q°  V  (1)  68  Approximation  —  (4.29)  gM, N  =  2 gp G  The Pauli  =  4 gv  (4.30)  2 +M  Spin operator 3 j acts  (4.31)  on the  nucleon and the effective couplings  have the superfix N to indicate that they are nucleon couplings which distinguishes them from the effective couplings appearing in the EPM formalism.  ‘T  is the isospin  lowering  operator which has non-zero matrix element oniy for an initial state proton and a final state 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. The terms of order two are absent of course in that result. A comparison was also made with the result of Friar [105] who used a Foldy-Wouthuysen reduction to obtain an effective Hamiltonian including “corrections of order 2 (1/MN) ” . The differences between that result and the oiie found here are of two types. First, extra terms have been included for gp which was a result of the realization that gp has order minus twod. Second, there are differences which are like iiq°/M  =  3 x iO. This quantity is of the same order of  magnitude as the error one would make by identifying q° with k’°  —  k°. It is stressed  that this identification has not been made at any point in the derivation of the effective Hamiltonian where q°  =  1 m  —  ii  was used.  the more conventional definition of gp it has order minus one. Of course, the conclusions regarding which terms to keep involving gp are independent of the choice of convention. dWjtI  Chapter 4. Muon Capture by He in the Impulse  4.5  69  Approximation  Matrix Elements  The effective Hamiltonian has terms of order (k/iW)° and (k/M)L The matrix element  of the latter terms can be simplified by making the approximation that the H 3 e and H wavefunctions form an exact isospin doublet. i.e.  I H 3 e)  =  3) H  =  )II=,I3=—U  (4.32)  In that case the matrix element of Ic can be written in terms of the matrix element of I by making an integration by parts following the identification Ic 1 i=2’/3.  =  —ivy  and also  -  (H ZIj 3 exp(— H e) i)  H 1j exp(—i)j 3 He)  =  (4.33)  This procednre is due to Peterson [106]. The quality of this approximation was tested by evaluating the matrix element of the isospin lowering operator 1  =  1j + 1j + 1  between H 3 e and H 3. If H 3 e and 3 H formed an exact isospin doublet then this matrix element would eqnal one. In fact it was fomid that, HIH 3 ( H e)  =  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 have total angular momentum  (H  I exp(—i.i) 3 He ). Since H 3 e and  the total angular momentum of any operator which  has a non-zero matrix element can be only 0 or 1. A further selection rule is provided by the parity of intial and final states which implies that operators with negative parity have zero matrix element. Expanding the plane wave in the matrix element the following simplification was arrived at. 3( H 3 ;  ?flj  IH 3 exp(— e;in W.i)j )  =  [1]°6(rnf,rn)  (4.35)  Chapter 4. Muon Capture by H 3 e in the Impulse Approximation  [110  where, For the :3( H:  operator  Spin  1 m  i jo(vr1) { He) 3  H 3 (  =  70  (4.36)  there are two multipoles which contribute.  3 exp(— H e; i) rn)  =  xt {[]o  —  3J))1[121} (4.37)  []01  where,  =  [5]21  ( H 3 11  He )/ ITjo() 3  (H  Ij2(vr)  (4.38)  [}) ® jj  3 H e )/  (4.39)  The matrix element between H 3 e and 3 H has been written in terms of the Pauli Spin  vector  for the trinucleon states, two-dimensional Pauli spinors X 7 and X for  the final and intial states and also the reduced matrix element of the vector operator between two spin one half states which equals  By using this form and writing  in the non spin-flip matrix elemeit the total matrix element of the  (mf, 7n)  effective Hamiltonian between the nuclear states can be written:— (RH;  where,  M1A  =  mj 11 eff  3 H e; rn)  Vud (1  —  =  XtMIAX  3’) {G +  (4.40) (;IAL}  (4.41)  which should be compared to the expression for the matrix element .41 in the elementary particle model (EPM) given by equation (3.38). The effective form factors G, G, G are given by,  =  =  {c; +  + 3M +  =  -  3MN (G +  3MN  G)] GvvN  [6]  []0  -  GN)]  -  3M  GLN]  [6121  (4.42)  Chapter 4. Muon Capture by H 3 e in the Impulse Approximation  where,  [8J°”  []+ [6]—  =  +  71  \/2[6J21  [ë]°’  (443)  1[8j21  (4.44)  These effective form factors may be directly compared to the effective form factors G, Op and GA arising in the EPM. The calculation of the reduced matrix elements [1]°, [6j°” and smallness of  [6]2  [6j21  is detailed in appeidix 0. The results are shown in table 4.4. The  is a result of the low three momentum transfer  ii  in the process.  Table 4.4: The Reduced Matrix Elements.  8 channel 22 channel  [110  [6]01  []21  0.851 0.853  —0.808 —0.809  0.0015 0.0015  Table 4.5 shows the value of G, G and G which follow from the values of the nucleoii form factors given in appendix I and the matrix elements given in table 4.4. The first row gives the effective form factors from the EPM along with their experimental Un certaiuty. The second and third rows give their values in the IA along with the deviation from the EPM value both with and without relativistic corrections.  Table 4.5: The Effective Form Factors.  EPM IA O(/MN)1 IA O(k/MN)°  0.85 + 0.01 0.836 —2% 0.866 +1%  0.603 ± 0.001 0.551 —9% 0.506 —16%  GA 1.293 + 0.004 1.205 —7% 1.202 —7%  Using the values for G, G and G given in table 4.5 (i.e. those including k/MN corrections  )  and the expressions developed in chapter 3 for the following results for the  total rate and analyzing powers in the IA were found. =  1325 seconds’  Chapter 4.  Muon Capture by 3 He in the Impulse Approximation  A A 4.6  =  0.55  =  —0.372  72  —0.077  Summary and Conclusions  A calculation of quasi-elastic muon capture by H 3 e has been performed using sophisti cated and reliable wavefunctions for the nuclear states. Care was taken to include all necessary terms with the pseucloscalar form factor gp in the effective Hamiltonian where the 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 ana lyzing 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 the EPM whereas Gp and GA fall short of their EPM values by about 10%. and G are proportional to  [6101  Since G  and G is proportional to [110 this can be expressed  as a lack of magnetic strength in the IA. A possible remedy is the inclusion of meson exchange currents (MEC) which are likely to enhance the magnetic strength in the muon capture hamiltonian. This is because MEC contribute at low order to isovector magnetic  nioments as has been shown by many authors [107j-[11 1].  Chapter 5  Muon Wavefunction Overlap Reduction Factor  5.1  Introduction  At the values of q 2 relevant to muon capture, the weak interaction is very close to being pointlike. The initial muon and proton must be coincident with the final muon neutrino and neutron as is made manifest in the matrix elements for the process, M  I exp(—i)(r)M 3He)  (3H  (5.1)  where the s-wave muon wavefunction ,(r) and neutrino wavefunction exp(—i.P) are evaluated at the initial/final proton/neutron coordinate amplitude for capture  M  7(1  .  M() is the muon capture  by the particle labelled (i) and has the form. —  UL.V)  (Gv +  + GA.) + smaller terms.  (5.2)  The standard procedure in evaluating the matrix element in equation (5.1) is to remove the muon wavefunction and replace it by an average value  ()  where the average  is over the nuclear transition density appearing in the matrix element M. At this point no approximation has been made. The value of  will be of the order of a fermi because  that is the size of the trinucleon systems. Since 1 , ( r) varies slowly at small r,  is  written in terms of the value of the Bohr muon wavefunction p(r) evaluated at r  =  0  thus, M  2  C (0) where,  2  I exp(—i)Mj  (3H  (0) 73  2  3 4(Zmred)  3He)  2  (53) (5.4)  Chapter 5. Muon  and  red 11  Overlap Reduction  Wavefunction  74  Factor  is the reduced mass of the muon-helion system. The reduction factor C expresses  the reduction in  p,(r)  as r increases from zero. For small nuclei such as H 3 e C is close  to 1. The evaluation of C is the subject of this chapter. 2  Consider the evaluation of M  ignoring the  r  dependence of the muon wavefunc  tion. The rate receives contributions from the spin-flip and  non spin-flip  reduced matrix  elements giving,  J  spins  = 2 MJ  rI  ri2 ‘-(1)  i wnere,  2 + G)[6] G[1] 2 2  2 + (GA 2GA  —  2 Gp)  [1]  =  (H  Ijo(vr)3He)  [6]  =  ( H 3 f  0 J(i) ( vrj)3He). j  and,  (5.5)  (5.6)  (5.7)  In general, the r-dependence of the reduced matrix element [1] will not be the same as that of [6] so that C is a combination of a spin-flip correction factor C() and a non spin-flip correction factor 1 C( ) . =  C  2 ) 1 C(flG [1] + )G[6j 2 G)[1] +  (5.8)  The exact definitions of C( ) and C(a) are, 1 =  where  jdrr2p()(r,zI)  takes on the values 1 and u and the density  (5.9)  is the pertinent density arising  in the reduced matrix elements. [1]  dr  rp(l)(ri,  ii)  x constant  (5.10)  dr  VP() (ri  ii)  x constant  (5.11)  =  [6] =  1  =  f I  drr p 2 (*)(r,  ii)  (5.12)  Chapter 5.  Muon  ‘vVavefunction Overlap Reduction Factor  75  The constants appearing in equations (5.10,5.11) are determined using the normal ization condition, equation (5.12). The densities p)(r, ii) are thus trinucleon isovector densities, not charge densities, modulated by a factor jo(I’r) from the zeroth multipole of the neutrino wavefunction, the higher multipoles making iegligib1e contribution. The isovector density is related to the point proton charge densities of H 3 e and 3 H thus, He Pisovector(r) = 2p3 ()  where both  p H 3 e  H 3 p (r)  v)  and  are found by making the choice i  p()(r, ii)  =  are given below for the sake of clarity. They  1 in equations (5.10,5.11) and using the relation  2y/3 where y is the usual Jacobi spectator co-ordiiate p(l)(rl, ii)  P(g)(71,11)  (5.13)  3 are normalized to one. and H  Explicit forms for 1 p( ) (r,  =  —  (  see appendix E  ).  x8 3 fd dy 1 ’ 2(3 H; in )I’j(2iíy/3)( j He; in) 3 =  ()3  xd 3 y( H; m )Jjo(2vy/3)( 3 3 He; in) fd 8/2) xd 3 y 3 H; in I) IL u’)jo(2vy/3)( £I 1 fd He; in) 3 (3 ()3 mjg) ( 2iiy Jl)ul /3)( )j j x3 3 y( d H ;0 He;m) 3 fd  (5.14)  (5.15)  In the above equations in is the spin projection of the trinucleon and can either take the value + or  —.  Table 5.1: Previous Calculations of the Reduction Factor C. C 0.965 0.9704 0.965  Reference Kim and Primakoff [112] Peterson [106] Donnelly aid Walecka [113,114]  The results of previous calculations are given in table 5.1. Kim and Primakoff used a square well nuclear density with radius fixed to reproduce the r.m.s. electric charge radius of H 3 e. Their calculation is criticized for several reasons.  Chapter 5. Muon Wavefunction Overlap Reduction Factor  76  1. The nuclear density should be an isovector density and not an electric charge density the latter being the sum of isoscalar and isovector densities. 2. The charge deisity measured in electron scattering experiments includes the finite extent 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 element and so it is the magnetic radius rather than the charge radius which is the more pertinent parameter. 4. The effect of  the neutrino wavefunction is ignored.  5. The density used is uiwealistic for the H 3 eH system which has a long tail in 3 configuration space. 6. The pertinent parameter for the nuclear density is not the second moment (r ) but 2 rather the first moment (r). The last point follows from the fact that the nuclear density is much smaller than the muon 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)  so that the value of  =  1  —  , +2 1 v/a r / 2a +...  (5.16)  is given approximately by,  1- (r)/a.  (5.17)  aThe spatial distribution of the nucleons has already been included via the nucleon form factors appearing 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  77  Since (r) < a, the higher order terms in the expansion (5.16) will give oniy a small correction to C so it is important for a model nuclear density to reproduce the first moment of the real nuclear density. The density taken by Kim and Primakoff was not realistic 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. The charge distribution of 3 He measured by McCarthy, Sick and Whitney [115] has a root mean 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 square  radius of 1.88 fm has a first moment of 1.63 fm. The calculation of Peterson used a realistic H 3 e charge density to find the non-Bohr muon wavefunction numerically but it is not stated whether the overlap of this wavefunc tion was taken with an isovector density or a 3 He density. However, taking the overlap of an unperturbed Bohr orl)ital with the H 3 e charge density given in the paper gives C  =  0.9749. This result suggests that the overlap was indeed taken with the H 3 e charge  density since the perturbed muon wavefunction will be diminished near r  =  0 and this  leads to a slightly smaller value of C. The effect of the neutrino wavefunction was not included. 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 H 3 e charge density to find a relativistic muon wavefunction and took the overlap of the muon probability density with the pertinent nuclear transition density i.e. exactly that density arising in their matrix elements. However, their work can be criticized on the grounds that the nuclear  transition density is based on a (i.s) shell model configuration of the trinucleon bound state. Apart from lacking any two particle correlations, the model neglects the d-state of the bound state which represents 9% in probability. Our knowledge of the trinucleon bound states has increased greatly over the last 15  Chapter 5. Muon Wavefunction Overlap Reduction Factor  78  years and this allows a better calculation of the correction factor C. The method that was used to calculate C is given in the next section.  5.2  Method  An accurate trinucleon isovector density for the case v  =  0 was found from the H 3 e and  H wavefunctions of Kameyama et al [18]. This density yields a coarse result for C which 3 was fine tuned by applying the following corrections. • Perturbation of p(r) due  to  non-pointlike nature of the H 3 e 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 of the neutrino wavefunction which is equivalent to setting the magnitude of the three momentum transfer to zero. The corrections were calculated using a model transition density which allowed systematic study of the corrections. The model density which is supposed to represent the  p(*)(r,O)  was taken to be,  p n 7 (7)  =  exp(—r/a).  (5.18)  The value of a was set by exact calculations of the first moment of the isovector transition density at zero momentum transfer.  Since there are two matrix elements  which arise in muon capture, there will be two values of a. They correspond to the non-spin  flip amplitude [1] and the spin-flip amplitude [5]. Using the antisymmetry of  the nuclear states to simplify the matrix elements, the values of a( ) and 1  a() are given  Chapter 5. Muon Wavefunction Overlap Reduction Factor  79  by, (r)  fdrr p 3 (r)  =  (3Hjj 3 A3)U H (HIHe)  :3a(1)  1.661 fm (5.19)  3a(a) where the particle label  (j)  —  —  He) ( H 3 WJ 3  1.536  ITE  on the operators can be 1,2 or 3 and the matrix elements are  reduced in spin space but not in isospin space. Since the Bohr muonic wavefunction is of the same exponential form as the model density, there arises a dimensionless scale parameter ‘s’ which is the ratio of the size of the nuclear density to the size of the muonic orbit. =  a( ) 4 /a  (5.20)  From equations (5.19) we have, 4.17 x i0  S(i)  and,  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.  =  The term  692  f  dr r p(r) 2  =  1  0  —  3 + 6s 2  —  3 +... lOs  (5.22)  contributes at the 1 x iO level. Since the model density has a reason  ably realistic 1 sha ) 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 level of accuracy will be 2 x iO at the very worst. Second, all the corrections to the coarse result depend on the nuclear density at small r i.e. r If the model density reproduced all the moments  (rn),  the size of the nucleus ii  1—2 fm.  0, 1,2 ... then it would equal  Chapter 5. Muon Wavefunction Overlap Reduction Factor  80  the real density. Although the model density does not do this, the important moments  are for small values of n since we hope to reproduce the small r behaviour. Fitting the model density to the smallest non-trivial moment not only gives an accurate coarse value for C, but also ensures that it is realistic in the iml)ortant region. The coarse results for C( ) and 0 1 C( at zero momentum transfer, using a Bohr muon wavefunction and realistic trinucleon isovector densities to set the values of  (i)  and (8 u)  are,  5.3  ) 1 C(  0.9753  (5.23)  C(j)  0.9772.  (5.24)  Corrections  5.3.1  Perturbation of the Muon Wavefunction  The effect of the perturbation of the the muon wavefunction should lessen the value of C. This is because the potential from an extended charge distribution is less attractive than that from a point charge. The muon is thus more likely to be found further away  from the nucleus which corresponds to a greater amplitude for large r. By normalization constraints the perturbed wavefunction is smaller than the unperturbed wavefunction at small r and this implies a smaller value of C,  (  remembering that the correction factor  C is defined with respect to the value of the unperturbed wavefunction at  7’  = 0  ).  The  3 H e charge density is taken as, 7 H 3 p . ) e( =  with ac  =  exp(—r/a)  (5.25)  0.543 fm fixed by the r.m.s. charge radius of 3 11e which is 1.88 fm [77, table  II]. The scale factor for the charge density is thus,  s  =  , 1 a  =  4.09 x i0  (5.26)  Chapter 5. Muon Wavefunction Overlap Reduction Factor  which is close to both  S(i)  and  S).  81  The potential for such a charge distribution is found  by solving Poisson’s equation analytically. 2 V  where,  (5.27)  =  V  =  electric potential  p  =  charge density  =  electric permittivity of free space  =  (Ze)p(r) (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-point  like nature of the nucleus and ca be treated as a perturbation. The perturbing poten tial is then, AH  (Zn) exp(—r/ac)  (  +  (5.30)  and the perturl)ecl wavefunction is expressed as, i)  =  (r)°(i) +  ) 7 a(nlm)i(r)Y (nim)  (100)  (5.31)  Chapter 5. Muon Wavefunction Overlap Reduction Factor  where,  82  (nlrnIAH 100)  a(nlm)  (5.32)  = —  Ejm  =  unperturbed energy eigenvalue for state I nirn) \2 L) “red (‘7 —  —  2 2n  (‘7 — —  (•  2 2an  —  33)  Since H is not a function of i, only the S states are mixed. The general form of the s-wave Bohr orbitals is [116], no  N(n)  exp[—r/(na)], F 1 (1  —  )), 1 n; 2; 2r/(na  N(n) = (4/n ) 3  (5.34)  (a)2  where F 1 (; 1; z) is the confluent hypergeometric function. By evaluating the matrix element in  equation  (5.32) it was found that the coefficients a(n)  a(nOO) are second  order in the scale parameter s.  N(n) 8c(ii/2)  )  The change in  535  is thus,  =  a(n)N(n)  J  dr rp(,)(r, ii) exp[—r/(na)JiFi (I  0  —  n; 2; 2r/(na))  a(n)iV(n)[1 + O(s)].  (5.36)  n1  Ec u 1 ation (5.36) follows from the normalization condition, equation (5.12), after ex pansion of tile  muon  wavefunction  r. Thus, to second order  ill  and the confluent hypergeometric function for small  ,  722  = —8s  = —8s = —0.0001  (5.37)  n=2  The sum over n  was performed  by expanding into partial fractions and observing the  cancellation of all but two of the terms. The effect of the non-pointlike charge distribution on the muon wavefunction thus affects C at the 2 x i0 level and corrects  to  =  1_3s+62  =  1  —  3s  —  2.s2  (5.38)  Chapter 5. Muon Wavefunction Overlap Reduction Factor  where use of the proximity of .s to  (i)  and  83  has been made.  The above calculation was repeated with a charge density of slightly different form 2 exp(—r/a r ) as used by Peterson [106]. In that case the perturbing potential is, 2 = (Z)exp(-r/a ) 2  with a 2 corrects  [1+  (  + 18)]  (5.39)  +  0.34:3 fin which is fixed by the r.m.s. charge radius of H 3 e. The perturbation  by —20a/a  —0.0001 which is the same as for the exponential charge  clensi ty.  5.3.2  Relativistic Effects  The 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 of  the calculation, may or may not change C appreciably. The solution of the Dirac equation for a particle moving in a Coulomb 1/v potential will be used to calculate the overlap with the nuclear transition density. The form of the wavefunction is [116], g (r) Y ( ) X  cp(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-i spinor with projection rn. Standard practice [117,118,114] is to redefine (7 thus, C’  —  1  dr  1 H 3 e Pc \ I  )t( 7 ( ) (O)2  15 41  where p’(r) is the charge density of the initial H 3 e state. However, the overlap of the nuclear 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 be  Chapter 5. Muon Wavelunction Overlap Reduction Factor  84  found by considering the construction of the muon capture effective Hamiltoiiian which is the contraction of a leptonic current with the non-relativistic reduction of a hadronic current. Xx M XLXN where,  •‘pto1l  =  =  Jj  u(ii)[y(l  —  (5.42)  (5.43)  7 ) 5 ]U(t)  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, 7(1  leptoIl  =  XL  Jepton  =  xL {(1  —  XL  .i)  —  LL)}  (5.45)  XL.  The matrix element of the resulting effective Hamiltonian is evaluated thus, i exp(—i  where  )y(r)M  (5.46)  3He)  contains the spinor structure and nucleon form factors etc. This is equivalent  to 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 already present in the leptonic current. i.e. take  =  eptofl  L)NR}  XL {(i  XL {(i  —  L  XL  L)YNR}  XL.  (5.47)  Replacing the product spinor by the relativistic spinor, i.e.  i  0  )  (  —f =  1 g(r)°() ’ X  i \\ if(r) [}‘ c XL]im  i  I  (5.48)  Chapter 5.  Muon Wavefunction Overlap Reduction Factor  85  the lepton current becomes,  ‘ePton  Jiepton  x {(i  -  L.,)[g(r)y()  4 {(i  .i)(_)[g(r)Yo()  where the angular dependence on one half.  if(r)Yi()]} 0 XL  —  —  if(r)Y)]}  ® XL  (549)  is understood to couple to total angular momentum  By comparing the al)Ove with equations (5.47) it is seen that the effect of  relativity can be taken into account by making the substitution, NR()y ( 0 )  [g(r)Yo()  0 XL  —  if(r)Y ( 1 )] 0  XL.  (5.50)  This leads to the following relativistic definition of C. CREL  j,  2  —  where, =  Cf =  j j  (0)2  +  2 (ã)  (5.51)  dr r p(r)jo(vr)g(r) 2 dr r (r)ji(iir)f(r) 7 p 2  (5.52)  The density appearing in the definition of C has the factor 1 j ( vr) rather than the jo(vr) because the small component of the muon Dirac spinor has orbital angular  mentum I  .  mo  Conservation of total angular momentum along with a parity selection rule  imply that the zerot.h order multipole of the neutrino wavefunction makes no contribu tion to Cj and the first non-trivial multipole is of order one. The functions g and  f  are  [116], g(r) =  () C  =  where,  71  (r2 ‘ 1)) g(r)  (5.53)  (Z)2]  =  [1  =  [1 + (Za/7l)2j_  —  exp(-r/a)(2r/a’  =  7i•  (5.54)  Chapter 5. Muon Wavefunction Overlap Reduction Factor  A useful small parameter is 6  2 (Z)  =  86  2.13 x iO. Expressing everything in terms  of 6 we have. i/4)  2  g(r)  2 exp(_r /a)(2r/a  (-)(6/4)g(r)  f(r)  (5.55)  1—6/2 I Consider the size of  c  —  6/2.  (5.56)  compared to that of  g 0  First replace 1 (j z’r) by j (vr) in the 0  clefiuiition of C . This will give an upper limit for Gf since the factor ji(vr) will reduce 1 1 considerably. G )2  <  (2  (1_— \1 +€J  6/4  =  5 x 10  (5.57)  Thus, 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 a point-like iucleus given by equation (5.53) and neglecting terms smaller than or equal to 6/4 it was found that. C  (2s)(1  —  3s + 6s . ) 2  (5.58)  The factor (2.s) comes from the logarithmic divergence of g(r) at small r. This enhances  C and the effect is of the order —6 ln(2s) 8 (2s)  =  1  —  =  +1.0 x 10 since,  61n(2s) + [61n(2)j /2 2  —  ...  (5.59)  However, it is not clear that equation (5.58) is the correct result due to the following argument. The enhancement of C is due to the  1/rS/2  enhancement of the relativistic  wavefunction g(r) over the non-relativistic Bohr wavefunction (v) =  ()  2 e xp(r/a)  Chapter 5.  Muon Wavefunction Overlap Reduction Factor  ()  87  2  exp(—r/a) x  2 / 6 (2r/a)  (1  —  0.;346 +  0(62))  (5.60)  In arriving at equation (5.60) use was made of the gamma function expansion,  F(:3 where  i/i  —  8)  =  F(3)(1  —  6’(3) +...)  (5.61)  is the digamma function having the value,  =  —‘y + 3/2  Is the singularity at r  =  0.92, 0,  (  =  Euler’s constant  =  0.57722.  and hence enhancement at small r  ),  (5.62)  peculiar to the  relativistic nature of the Dirac equation or to the singular nature of the Coulomb potential 1/r? i.e. when a more realistic non-singular potential is used in the Dirac equation, is  there 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 Dirac equation for the electric potential given in equation (5.29). The calculation is detailed in the next section. 5.3.3  Numerical Solution of the Dirac Equation for an Extended Charge Distribution  The 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 binding energy. The exponential decay of the functions g and  f was factored out  by defining new  Chapter 5.  Muon Wavefunction Overlap Reduction Factor  88  functions G and F thus, g(r)  =  exp(—r/a L 1 )G(r)  f(r)  =  exp(—r/a ) 1 F(r)  (5.64)  and the equations that were actually solved are, + E + eV(r)jF(r) + C(r)/a  =  =  [mred  E  -  7  -  eV(r)]G(r) + F(r)( 1 -  7  (5.65)  with boundary conditions, G(r)  1 r’  —*  F(r)  (—)r’’  as r  ()2  —* 00  as r  00.  (5.66)  Fourth order Runge-Kutta extrapolation b was used to solve these equation starting at r  =  20 fm and using variable step size down to r  0.02 fm. The method was checked by  solving for the Coulomb J)otefltial 1/r and comparing the result to the analytic solution. Agreement was found at the i0 9 level for G and the i0 level for F at the final r  =  0.O2fm point. When solving equations (5.65) with the non-singular potential the energy of the so  lution was taken to be, E/mreci  which is the Coulombic solution  (ti)  1 = E  8s 2 + (Zo’)  (5.67)  plus an estimate of the change in binding energy  found from first order perturbation of the non-relativistic solution. AE  =  ([HI)  =  4  j  dr r 2  (5.68) exp(—2r/aL)  fl7red(ZQ)8.Sc[1 +  0()]  exp(—r/ac)  (  +  (5.69) (5.70)  bSee Numerzcal Recipes by W.H.Press, B.P.Flannery, S.A.Teukolsky and W.T.Vetterling page 550  an explanation of this technique.  for  Chapter 5. Muon Wavefunction Overlap Reduction Factor  B  89  taking the overlap of the solution for G(v) with the model density pm(r) the  correction factor C() was found to be, REL -‘perturbed  0..  —  0  5. 1  which 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 neglected  at the i0 level. 5.3.4  Neutrino Wavefunction  The nuclear transition density includes the factor  jo(iir)  stemming from the neutrino  wavefunction. The presence of this factor gives a value of C which depends on the value  of the three momentum transfer v. In this section, the result for C is calculated as a function of  i-’.  Using the model density the dependence of C on v is,  =  f  exp(— ( r)jo(iir dr r 27 r/a)p )(1 + v . ) a 2  (5.72)  ) factor comes from the renormalization of the total density p( a 2 The (1 + ii )(r, v) 4 according to equation (5.12). After expansion of the muon wavefunction for small r the integral is easily performed and it was found that, 3(1 2/.3) (1 6s +2 (1 + 712) (1 + —  —  =  where,  1  =  —  2) 7/2)2  vu.  (5.73)  The apparently paradoxical result that the correction factor becomes larger than one for large v is understood by noting that the effect of the decreasing  jo(vr)  becomes negative for ir  muon wavefunction in this region  <  zir  <  27r so that  is to reduce in magnitude  the negative contribution to the matrix element thus increasing its overall value.  Chapter  5. Muon Wavefunction Overlap Reduction Factor  For the H 3 e =  —*  90  H transition the neutrino momentum i is 103.22 MeV which gives  0.29 for a( ) and 1  =  0.27 for a(). The correction to the term of order  s2  corrects C  by 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)  =  0.9778  C()(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, this correction will be taken into account by modifying the coefficients of .s thus, (1)  where,  =  1  =  1  5.4  —  3st(l) + U(s ) 2 3st() H-  Q(2)  =  t(i)  t()  —  (575)  0.897  =  =  0.911.  (5.76)  Summary  Including relativity, neutrino wavefunction and muon wavefunction perturbation the ex pression for C(*)  i5,  C()  (1  —  6s(&)t() + 5%))  (5.77)  Inserting the numerical values,  ) 1 C( C()  =  0.9776  0.9790.  (5.78)  Chapter  5. Muon Wavefunction Overlap Reduction Factor  91  Table 5.2: Summary of Corrections to Coarse Result. Effect neutrino wavefunction muon perturbation relativity  +0.0020 —0.0002 0.0000  In the impulse approximation, the matrix elements [112 and [8]2 contribute to the total capture rate in the ratio 1792:8208. The elementary particle model however suggests that the ratio of spin scalar to spin vector contributions to the rate is 1602:8398. The  pertinent combinations of C() and C() are then,  eff  =  0.1792C(I) + 0.8208C()  =  0.1602(7(1) + 0.8398C()  =  0.9787 0.9788  (5.79)  and the latter result will be taken for the sake of definiteness. The sizes of the various effects are given in table 5.2. In comparing this result to the other calculations the relevant ciuantity is the deviation of C from one,  ( since this is what is calculated ), even though  it is the absolute value of C  which affects the muon capture rate. The deviations from one of the 1)revious calculations are 0.035 [112], 0.0:30 [106] and 0.035 [113,114] which should be compared to 0.021. The present result is thus significantly different from previous calculations. In heavier nuclei where the deviation of C from one is large, the theory used here may yield a significant change in the absolute value of C itself. The difference in the value of C obtained in the present calculation compared with previous work, is due mainly to the fact that previous calculations have overestimated the first moment of the pertinent density. There are two important points which led to the small value of (r) used in the present work.  Chapter 5.  Muon Wavefunction  • It is the flip  (  spin-flip  Overlap Reduction Factor  92  density which makes the dommant contribution to Cs’. The spin-  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 real density available to experiment. Thus, in previous calculations of C which based the nuclear density on measured charge densities, the smearing of the charge due to the structure of the nucleons was implicitly included and this led to a density whose first moment was too large. Further, the effects of the neutrino wavefunction had not been included in previous calculations. While it is the isovector densities which are relevant for the H 3 e  —  H transition, 3  other transitions will involve different densities. For example in calculating the inclusive rate in the closure approximation, the relevant density is the electric charge density of the initial state and this should be taken with the muon probability density and not the muon amplitude. To conclude, the muon wavefunction overlap reduction factor is found to he 0.9788 which is larger than previous calculations.  Chapter 6  Muon Induced Break-up: the Deuteron Channel  6.1  Introduction  As well as the quasi-elastic reaction  He +3  —  ii,  H, there are two break-up channels -b 3  which are open to muon capture by H 3 e. They are, j+ H 3 e  ,  v + p+n+n  (6.1)  and are referred to as the cleuteron and proton channels respectively, the label being given 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 ap proximation using the Arnado model for the final states with s-wave NN interaction only and a phenomenological H 3 e wavefunction. Those authors found, f(dllV)tiieor  =  414  F(Pllh1l1)theor  =  209  s_I  (6.2)  Until recently the only experimental results available were for the total break-up rate and total  neutron  rate.  .+ 1 F(dnii  pnnzI,)x ) 1 .  F(dni, + 2pnnviL)exp.  =  660 + 160 s  =  665 1200 + 170 s 93  reference [119] reference [65] reference [120]  (6.3)  Muon Induced Break-up: the Deuteron Channel  Chapter 6.  94  While 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 energy spectra of  1)rOtOflS  and deuterons with high momentum [121]. There are eight data points  for the proton channel which span a range of proton momentum from 180 MeV to 320 MeV. There are only three data points for the deuteron channel. One is at deuteron momentum 275 MeV and the other two are at deuteron momentum :325 MeV. The integrated deuteron and proton rates cannot be deduced from these measurements since not enough of the spectrum is covered. The calculation described in this chapter is of the deuteron spectrum in the plane wave impulse approximation (PWIA). The formalism used is similar to that used in electron induced l)reak-up reactions and the present work  represents  the  first step  in  raising the level of sophistication of muon induced break-up calculations to that of the electron case.  6.2  Kinematics  To kinematically define the final state requires two variables. This result is arrived at as follows. There are three particles each with three independent momentum components which gives nine unknowns. • The 3 particles must be in a plane. Set one coml)Onent of each momentum to zero 6 unknowns.  • There  are  equation  now 2 momentum conservation equations and I energy conservation  3 unknowns.  • There is a trivial rotation in the plane  2 unknowns.  Chapter 6. Muon Induced Break-up: the Deuteron Channel  95  The following list defines the kinematical variables appearing in the formalism.  pa  =  helion 4-momentum = [M3He, 0]  =  muon 4-momentum = [m,,0]  =  neutron 4-momentum = [Es, 17]  =  deuteron 4-momentum = [Ed, d]  =  neutrino 4-momentum =  =  4-momentum transfer =  =  4-momentum of struck proton = [E,y7]  17]  [ii,  = [Qo,  —  Q] (6.4)  The two kinematical variables that will be used to define the final state are the magnitude  of the deuteron momentum d and the cosine of the angle between the deuteron and the three momentum transfer x = d.Q. Using energy and momentum conservation we have, Matom  =  11 + Ed + E  Q (6.5)  where,  Ed  (j2  +  inj2)  (2  +  2) 1 m 1  (6.6)  Equations (6.5) lead to the following expressions for the ma.gmtude of the three mo mentum transfer  Q  and neutron energy E , 1  QQI,x)  =  in terms of  (1  —  —  /3(d)  =  (d)  =  (6.7)  d.x//3)  =  where,  d and x.  Maton, [1-  (1 —  —  1/3)]  Ed +miñ]  (6.8) (6.9) (6.10)  Chapter 6.  IVIuon Induced Break-up:  the  Deuteron Channel  96  Figure 6.1: The four momentum transfer squared q 2 as a function of deuteron momentum. 1.0  0.5  0.0  —0.5  —1.0 0  100  200  300 (MeV) 400  Figure 6.2: Figurative diagram of the PWIA.  LI  w±  Chapter 6.  Muon Induced Break-up: the Deuteron Channel  97  The deuteron momentum is constrained to be less than :356.35 MeV, the upper limit corresponding to the situation where deuteron and neutron are emitted back to back and the neutrino has very small momentum.  When the cleuteron and neutron have  zero relative momentum the neutrino has its maximum energy of 97.18 MeV. The four momentum transfer squared q 2 is given by, 2 q  and ranges from —O.84m 2 to 1  =  . 2 +m  m ( 2 1 1  —  j 1 2Q/m  Its value as a function  (6.11) of  d is shown in figure 6.1.  The range of values of q 2 for each value of d corresponds to the variation of x from —1 to + 1. Unlike the quasi-elastic chaimel or any kinematical region available to electron scattering, q 2 may become time-like, (i.e. q > 0 2  ).  This allows the interesting possibility,  as in radiative muon capture, of finding observables sensitive to the value of gp which according to PCAC should rapidly increase in value as q 2 approaches +m . 2 6.3  PWIA  The formalism of PWIA is a great. simplification of the full problem l)ecause of the following three approximations. 1. The strong interaction between the neutron and cleuteron in the final state is ig nored.  The final state wavefunction is then a product of a plane wave for the  neutron and the free deuteron wavefunction.  2. The weak current is taken to be that from the nucleons only i.e. meson exchange currents are neglected. :3. Only the direct nucleon knock out amplitude is included. The direct deuteron knock out amplitude is ignored.  Chapter 6. Muon Induced Break-up: the Deuteron hanne1  98  A figurative diagram which describes the PWIA is shown in figure 6.2. To put the PWIA in perspective it is useful to make a diagrammatic expansion of the matrix element. Let us represent the fully antisymmetrized iiitial H 3 e state by the symbol, 1  i)  =  He  2  (6.12)  3  where the numbers label the nucleons. The final state must also he fully antisymmetric and using the permutation operators P 2 and P 3 can be written,  1) where  f )1  =  2 + 1 (1 + P )f)( 3 P )  is the Faddeev component and is antisymmetric under interchange of  particle labels 2 and :3. The symbolic  representation of  I =  the Fadcleev  component is,  1  1  (i)(f  (6.13)  +  H  (6.14)  where the first symbol represents the disconnected part of the wavefunction waves  )  (  plane  and the second the connected part.  In general, the current has one-body two-body and three-body parts where the n-body current depends simultaneously on the coordinates of n particles. These currents are represented symbolically by a wavy line attached to one, two or three nucleons according  to the nature of the current. Using the permutation symmetry of the current and also the final and initial wave functions, the matrix element can be simplified to,  M  =  (fIJI)  =  :3()(  fIJIi).  (6.15) (6.16)  Chapter 6. Muon Induced Break-up: the Deuteron Channel  99  This matrix element is represented in figure 6.3. The PWIA corresponds to including oniy the top left diagram (la) in evaluation of the matrix element and thus represents oniy the very first stage of a complete calculation. Diagrams lb and ic represent direct deuteron knock out and diagrams 2a-2c represent the 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 the two-body current and diagrams 5d and 5e represent the disconnected and connected amplitudes for the three-body current. It is shown in appendix H that, including only diagram la, the deuteron spectrum takes the following form. = 2 d 7 ( d) p  j  dx  M  2  (d, x)  (6.17)  In equation (6.17) 2 p ( d) is the two body break-up momentum distribution for 3 He and equals 3 times the probability of finding a deuteron of momentum d and proton with equal and opposite momentum when observing a 3 He nucleus at rest. a The factor y is, 2 G  F 0  2 (o)I2. 2 C Vud  (6.18)  is the Fermi coistant and Vd is a Cabbibo-Kobayashi-Maskawa matrix element  [62][63] linking the up and down quarks. C 2 is the muon wavefunction overlap reduction factor 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 performed  analytically. In the final calculation this integral was performed using Gauss-Legeidre numerical integration including the x dependence of M  2  but the deviation from the  aThe factor three arises because of the normalization of the bound state to one. The probability a proton in 3 observing of He is thus 2/3 and this must be multiplied by 3 to find the total number of factor of three arises naturally when properly antisymmetrized states are used and is not This protons. inserted by hand.  Chapter 6. Muon Induced Break-up: the Deuteron Ohannel  a  b  100  C  1 +  +  +  +  0 H  +  d  e  2  3  4  5  H H°(  Figure 6.3: Diagrammatic expansion of the matrix element.  Chapter 6. Muon Induced Break-up: the Deuteron Channel  101  approximate analytical result was found to be 5% at the most. The analytical result is, (d)/3 M p d 7 2 k  2  (d, 0).  (6.19)  where k is a kinematical factor close to one.  k  ic(1 + 2 /3,9 d )  —  -  ) / 2 (1-d  1  6 20  . -  As the deuteron momentum ranges over zero to its maximum value, 0.09 to zero, k varies from one to 1.14 and /3  —  m,  —  tc varies from  m varies from —5.5 MeV to —-39.1  MeV.  6.4  Two Body Break Up Momentum Distribution  The two body break-up momentum distribution 2 p ( d) is a theoretical construct central to the PWIA formalism. It is defined exactly by equations (H.18) and (H.16) but can be thought of simply as the square of the overlap of a proton of momentum —d and deuteron of momentum d with  a  3 H e nucleus.  p ( 2 d)  He) 12 3 H ( proton(—d), deuteron(d) l  (6.21)  A calculation of p (d) was performed using the H 2 3 e wavefunction of Kameyama et al [18] and the deuteron  the same  wavefunction from  Gaussian basis as  (2.31) and also appendix F  p ( 2 d)  the Bonn Potential C which was expressed in  the H 3 e wavefunction,  ). =  (  see chapter 2, section 3, equation  The final expression used for p (d) was, 2 {E(°)(d) + 2E(2)(d)}  (6.22)  Chapter 6. Muon Induced Break-up: the Deuteron Channel  102  2  6(la, l)(Aa,  (ia, )6(Sa, 1)  a,1=O,2  where,  E(d)  =  la  La  1  Sa  x[6(2Sa + l)(2La + 1)]  1 X  f dp  2(l)  (P)’He(P  .  (6.23)  1  2  2  dl)  The above expression follows from equations (11.16) and (11.18) in appendix H and can be shown to equal that of Meier-Hajduk et al [122] by making the following transformation from  jj to LS coupling. =  ((ic a)La,(saa;  X  S L,  la [(2ja + l)(2ta + 1)(2Sa + 1)(2La + 1)]  ‘\a  a Ja  La Sa  (6.24)  ka  The form of I1(p, d) is given in section 5 of appendix C. The results of the calcula tion for 8-channel and 22-channel H 3 e wavefunctions are shown in figure 6.4. Also shown are results of electron scattering coincidence measurements H 3 e(e, e’p)d which have been used to infer the value of 2 p ( d) in the context of PWIA theory. It should be noted that the ‘experimental’ values for 2 p ( d) rely entirely on the validity of PWIA and thus are in error due to the presence of final state interactions (FSI) and the contributions of meson exchange currents (MEC). The calculations of Laget [125] indicate that the inferred value of 2 p ( d) is expected to be up to 50% too small because of FSI and  10% too large be  cause of MEC in the kinematical range of the experiments. These numbers indicate the size of the uncertainty involved in inferring p (d) from electron scattering experiments 2 rather than being precise corrections that could be applied. Bearing this is mind, the experimentally inferred 2 p ( d) agrees reasonably with the theoretical calculation. The  Chapter 6. Muon Induced Break-up: the Deuteron Channel  103  10 1982 data  0  •  102  ‘1987 data  a  fit to data 8 channel 3 He  22 channel H 3 e  1 .  ND  10°  > 0  S.  C 101  -o (N  Q  102 0 100 200 300 Deuteron momentum d [MeVI Figure 6.4: Two body break-up momentum distribution for 3 He. Data are taken from two experiments performed at Saclay [123,124] in 1982 and 1987. The straight line is a fit to the data.  Chapter 6. Muon Induced Break-up: the Deuteron Channel  104  enhancement of the theoretical value over the fit to the data ranges from one, i.e. no enhancement at the zero deuteron momentum point d  0 to 2.4 at d  =  :350 MeV. The  form of the fit was, p ( 2 d)  exp[—ad + b  d in MeV  25  —  (6.25)  and a least squares analysis led to the following values for the free parameters a, b and c. (1  =  3.40 x 2 l0 M eV  b  =  7.5  c  =  :30 MeV  (6.26)  The theoretical results for 8 and 22 channel H 3 e wavefunctions deviate from each other for d  >  200 MeV. This is due to the fact that the higher partial waves are absent  in the Faddeev component of the 8 channel wavefunction. Although these higher partial waves are present in the fully autisymmetrized 8-channel wavefunction, they are poorly constrained by a Schrodinger eciuation with a Fadcleev component containing only the lower partial waves. The 22-channel wavefunction thus represents a better solution of the nuclear three body harniltonian for large momentum. The theoretical value of 2 p ( d) is well reproduced by the form, pl1(d) =  ’d + b t1 exp[—a t’ 1 —  where,  cthl  d + 25  =  3.23 x l0_2 MeV  =  7.9 37 MeV.  (6.27)  (6.28)  Chapter 6. Muon Induced Break-up: the Deuteron Channel  6.5  105  Muon Wavefunction Overlap Reduction Factor for Two Body Break Up  The muon wavefunction overlap reduction factor for two body break-up C 2 is not the same 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 the H 3 e(t, vd)n process is not the same as in the 3 He(t, 1 v case. The neutroll produced in the former case flies H 3 ) away from the recoiling deuteron and, in plane wave approximation, is equally likely to be found in any fixed volume of space. In the quasi-elastic process however, the neutron is captured by the strong nuclear forces of the other two nucleons and a triton is formed which confines the neutron to be within a fermi or so of the centre of mass of the H nucleus. Thus, C 2 can be expected to be less than C since the nuclear density has larger spatial extent for the deuteron channel. The matrix element which appears in the PWIA is, M  ( ns; dJcj  =  j(l)I()  exp(—i  3 ) 1 H (rj) e; J).  (6.29)  The final state has the following representation in configuration space using the decoupled spin/isospin basis {€,  (  }  introduced in appendix H.  dJ(j 7y)  =  eXp(_iri*.y) x  (1Jd  irni; 17fls(X)1(X)  x6(.s, s )(i€, 1 1 —)6(s237, 1)(i , O) 237 (mj, 0) 6  (6.30)  where i is the neutron momentum in the nd CM. 9 =  Combining the  (i7—  1—  d)  (6.31)  7 dependence of the fluial state with that of the neutrino wavefunction  gives an overall exponential modulating factor which was found to be a function of d only after the momentum conservation condition given in equation (6.5) was used. 1 . 7 exp[i(-i i  -  j*)]  =  exp[i(  -  +  =  exp[icJ  (6.32)  Chapter 6. Muon Induced Break-up: the Deuteron Channel  106  Following a calculation similar to that for 2 p ( d) and averaging the matrix element squared over all deuteron angles it was found that, 1  dd  JsflJa  f  = 4 2 M 11 S ) 1  2  f  x A—0,2  dyy2E(A)(y)L(y)jA(dy)  .  (6.33)  This implies the following definition of C . 2  2 +2C 1 0 C 1 2 =  where,  f  ‘(A)  C(A)  =  12  (0)  1212  (2)  dy 2 y E (y)j(dy)  (6.35)  E(A)(y)j(dy) exp(—2y/3a) 2 f dy y  2  °° 2A (y)JA(dy) 10 dyy  (6.36)  The functions E(A)(y) are given by equation (6.23) with the variable q replaced by and the dummy variable p replaced by x. They are equal to the 3 clH e overlap functions  u(y) which are used to determine the 3 He asymptotic normalization constants Cs and C,  (  see [18, equation 4.7, figure  b 71  and [126]  In calculating E(x)(y) the deuteron  ).  wavefunction 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 their first and second moments (r) and (r ). 2 (r)  —  —  f dr r3E(A)(r)  ( 2 7 )(A)  E(>’)(r) 2 j dr r  —  f dr r4E(A)(r) E(”)(r) 2 J° dr r  (6 37)  The first moments (r)(°) and (r)( ) were found to be 3.42 fm and 3.14 fm respectively. 2 The value of (J(2)/I(o))2 was found to he 7 x iO and given the proximity of (r)( ) to 2 (r)(o) the safe approximation C 2  C(o) was made.  bTIe author is grateful to M.Kamirnura who provided numerical values of the dHe overlap functions 3 for the sake of corriparison to this calculation.  Chapter 6. Muon Induced Break-up: the Deuteron Channel  107  0.35  0.30 0.25 -  0.20 0.15 0.10 0.05 0.00  —0.05 3 r=2y/3 (fm) Figure 6.5: 3 cl 1 -1e overlap functions E(>). The solid line is the 22-channel and the dash-clotted the 8-channel 3 He wavefunction. A fit to E(o)(i) was made so that the integrations in equation (6.36) could he performed analytically. 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 r 2 dr. The free parameters d, e,  f  which gave the best fit are, e  =  1.295  f  =  0.316  g  =  0.542  (6.39)  Chapter 6. Muon Induced Break-up: the Deuteron Channel  108  Figure 6.6: The muon wavefunction reduction factor for two body break-up as a function of the deuteron momentum.  100  200  300  Deuteron momentum d [MeV]  0  A single exponential was found to be inadequate for values of the deuteron momenta above 1.0 fm 1 clue to the influence of the jo(3dr/2) factor. The correction factor C 2 was evahated by finding the first and second moments of the distribution p(r) defined by, p(r)  r u 2 (r)jo(3dr/2)  =  (6.40)  and then using, (()(2))2  C ( 2 d)= The expressions for (r) and =  (r)  (7.2)  (6.41)  are,  /g {(i+)41 5 [] + 24f 2e + 6f /g [] 4  6e  (642)  [(1,)2]  -  (.2)  24e  [(14]  2e where.  i =  [()2]  2 (3da/2)  The value of 02 ranges from 0.95 at d 6.6.  + 6 120f / g  =  +4 6,f / g  {1+2]  [] =  0 to 1.02 at d  2 (3dc/2)  =  (6.44)  356 MeV as shown in figure  Chapter 6. Muon Induced Break-up: the Deuteron Channel  6.6  109  Trinucleon Structure Functions  6.6.1  General Considerations  The hadronic tensor W describes fully the structure of the H 3 e  —÷  dn transition with  respect to its charge changing weak interaction.  (ilJf)J (3  2  -  P  -  q)(fjJi)  (6.45)  A general expression for the tensor can be written in terms of five structure func tions as shown below where the hypothesis of time reversal invariance has been made to eliminate a contribution like i(Pq W  qP).  —  We(g) +  M3He  8 qP  +  2 M 3HeMN  P + (P 1 ‘ I  +He  2  +Wie  M3HeMN  (6.46)  The numbering of the structure functions follows that of Llewellyn-Smith [127] and factors 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 = W  =  ()  (p,sp{j  ()  2 fd3p ( II)W p  n,s ) 1 J  Sn ,Sp  (2  ) ( 6 3 fl  (6.47)  —p —q)(n,s’jp,sp).  (6.48) The nucleonic tensor lends itself to the same type of expansion in terms of structure cn4 functions as WHe  =  —Wg + +W  + 1 +2 W M 2 (pq + q9)  (6.49)  Chapter 6. Muon Induced Break-up: the Deuteron Channel  110  Let us define nucleonic structure functions T to be the W without some of the kinematical factors thus, W  =  T  ()  6(E  -  -  . 3 Q°)/(2ir)  (6.50)  The Lorentz invariance of the T can be inferred from the following expression for the W whose Lorentz invariance follows from equations (6.48) and (6.49). W  =  2 + 2p.q)/(2ir) T2MN(q 3  (6.51)  Equation (6.51) was found by making the following identification. 6(E The T will  turn out to  —  E 1 1 + Q°)  =  2 +2 6(q 11 2E p.q)  (6.52)  be simple combinations of the nucleon form factors and the  kinematical quantity 2 /q 4MN and lead to an illuminating definition of the matrix element squared in contraction with the various Lorentz invariant parts of the lel)ton tensor L. Defining the trinucleon tensors Te by, W we  =  4  f  dpp ( p 2 p)T6(E +  Eci  —  M3He  —  3 Q°)/(2)  (6.53)  have the link valid in PWIA, T  =  fT  (.  The relations between the Te and the T which follow from equation (6.54) are, ‘He  2He T 3He T  —  —  —  —  2 ( pI (1 N + 92 ‘N I I  j  )lVl  2N T -—’n’---’p 1 \ (MN 3  TN  LJ  / 2  ‘He  (  IVIN  1 ‘Nj  \ J—Jp Jin 5 m  ‘He  5 m  (MN  ‘N  “  j  ‘—‘n  J \  M i --p 1 —‘n  (6.54)  Chapter 6. Muon Induced Break-up: the Deuteron Channel  111  where the following identifications have been made. [dj3p° J 4rM 3 j3pYp,/  E P  —  MNM3He  —  + [l )P 72 M MHe  +  —  ,.  J  4ir M  9 3M  —  The T are found by evaluating the following sum of Dirac T =  () N  (6 56)  spinor contractions.  (6.57) SpSn  In equation (6.57), /3  =  gv  =  gv7a  +  91 gMiu  5 + +g  +  —  which follows from equations (6.50) and (H.6). The  gA77  —  gp 7 2 5  (6.58)  5 2 P  (6.59)  spinor u(p’, Si))  describes the struck  proton and a priori is not known. Two choices will be made for this spinor as described in the next two sections. The resulting theory will be referred to as on-shell and off-shell impulse approximation.  6.6.2  On-shell Impulse Approximation  The usual piactice is to set the energy of the struck nucleon by its three momentum according 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 standard trace techniques [128, p123]. These structure functions will be referred to as “on-shell”  Chapter 6. I4uon Induced Break-up: the Den teron Channel  112  since the four-momentum of the struck nucleon is constrained to lie on a three dimensional surface defined by equation (6.60) commonly referred to as the energy shell. It was found that, ml(ON)  2(ON)  TN  q  2 =  —  Aff2  ‘IIVI N  2  =  2 \2  [gv+gM) +gA  2 q  2  gv +gA 2 — g M  2  2  ±IV1N  3(ON) TN  =  2 g A(gv + gM)  4(ON) TN  =  —[g + g 2 vgM + g 2 Agp H-  m5(ON) N 1  6.6.3  2  —  —  q2 2 4 1 v1N  2  (g + gp  2  )j  m2(ON) ‘N  Off-shell Impulse Approximation  The struck proton is iot on its energy shell. It is a bound particle and thus does not have a single well defined value of energy because of its interaction with the other particles in the system. Further, the presence of the binding potential implies that the total weak current will consist not only of one body operators but also two body and in general many body operators. The vector part of the one body current is not conserved which is  in violation of the conserved vector current hypothesis (CVC). This is because the CVC applies 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 current are both symptoms of the same phenomenon, namely the fact that the proton which captures the orbiting muon is under the influence of a strong capture potential  )  (  with respect to the  binding potential.  In this section, an estimate is made of the correction due to the off-shell nature of the struck proton.  The term impulse approximation will be used to mean that only  the one body part of the current is included with no specification of how the current is  Chapter 6. Muon Induced Break-up: the Deuteron Channel  113  constructed 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 interaction allows the following identification for the energy of the struck proton. = M3He  (6.63)  Ed.  —  For small values of d i.e. d < md the kinetic energy of the struck proton is, E  —  1 m  —EB( H 3 e) + EB(d)  so that there is “missing kinetic energy” even at d  —  /d 2 2md  (6.64)  0 of 5.5 MeV. A useful dimensionless  parameter which measures the amount by which the struck proton is off-shell is  defined  by, = 1  —  14//MN  where,  W  (E  —  d2).  (6.65)  The effective mass W has been introduced and is a function of the three momentum of the struck proton 7  —d. The concept of effective mass is familiar from the theory of  electron motion in solids. In that case the effect of the strong binding Coulomb potential from the lattice on the kinematical properties of the electrons is taken into account by  using an effective mass found from the dispersion relation. Energy conservation has been used to give an effective dispersion relation for a nucleon inside a H 3 e nucleus, (6.63)  )  which in turn defines the effective mass. The parameter  ( equation  varies from 0.6% to  12% 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 the binding potential. In this work, the off-shell current is not calculated but rather estimated using a reasonable prescription as follows.  The paper of Naus et al.[104] identifies four choices that must be made in any pre scription for estimating off-shell curreiits. They are,  Chapter 6. Muon Induced Break-up: the Deuteron Ohannel  114  Figure 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 struck proton. If the particle was free then this would be the spinor obeying the free Dirac equation. Without a relativistic nuclear wavefunction some arbritrary choice must be made. • Vertex Operator: For an off-shell particle, the vertex operator for the current must be generalized and this introduces extra free parameters which have to be fixed. • Kinematics: In the usual approach, the energy of the struck particle is set by its three momentum to the on-shell value. There is thus a problem of energy imbalance at the hadronic W± vertex which needs to be corrected. • Current Conservation: The full vector current is conserved.  The one-body  Chapter 6. Muon Induced Break-up: the Deuteron hanne1  part is not.  115  Current conservation is usually put in by hand in calculations of  electron scattering cross sections and this corresponds to adding an approximate contribution from meson exchange currents. The four choices that are made in this calculation are given below. Wavefun ct ion The idea of using a quasi-free spinor is introduced. After finding the energy of the struck  by energy conservation,  proton  (  assuming no final state interaction  ),  and the  corresponding effective mass W() the spinor is taken to obey the following equation. 6 U p(p) =  Wu(p)  (6.66)  This is similar to the free Dirac equation but the mass MN has been replaced by the effective mass W. Multiplying equation (6.66) on the left by E  yields,  14/2  _2  (6.67)  which shows that the replacement of MN by W is consistent. The equation (6.66) cor responds to that of a nucleon moving in a scalar potential  and the time component  of a vector potential Vo where Vs and Vo are local in momentum  representation  and are  given by, Vo((7)  Vs((7)  =  =  (2  + W(q)2)  W()  —  MN  =  -  (2  +  MN2)  -6(MN  —6MN  + 2MN (6.68)  Vertex Operator The current is half off-shell. The term half off-shell means that only one fermion at the vertex is off-shell rather than two. In general, the half off shell vertex operator is given by [104], OFF  =  (g7a +gZ 2 +g ) A  Chapter 6. Muon Induced Break-up: the Deuteron Channel  (g7a  (g7a (g7a  where,  A =  +  116  gEa+  +g  q + gEa)7sA+ 2MN  2 +g  W+ pl 2W  +gZa)7 A 5 _  Za  and  =  2MN  (6.69)  (6.70)  The form factors g are functions of three scalars which are the four-momenta squared of each of the particles at the vertex. When both the fermions are on shell their fourmomenta squared equals their masses squared which are constants. Hence, the depen dence on three scalars simplifies to the familiar q 2 dependence. In the half off-shell case there is dependence on two variables which are q 2 and  1472.  The operators A± project out positive and negative energy solutions of the quasi-free Dirac 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 with coefficients gs and g’ being second class. In the on-shell case the vector second class current can be set to zero by current conservation. 0 where,  =  üqaI’iu F  =  =  gs  q 2 üu 2MN  gv7a + gMza + gs  (6.72) (6.73)  The conservation of the on-shell current follows from the gauge invariance of the interaction 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  117  With the quasi-free spinor this implies that, =  ,W 2 g(q )  2MN  6[l ) ,W 2 g(q ].  (6.75)  It is noted that the same condition which implies the absence of second class currents in the 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 approx imation 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 gv or g whose size is about one. Ignoring the W 2 dependence of gv and taking its q 2 dependence from appendix I, the size of the gs contributions are estimated to be, 6 1 Mx7’v  6  2MN  <3%.  (6.77)  A similar argument will apply to axial second class current using an extension of the PCAC hypothesis. The choice g  =  0 was made.  Finally, the W 2 dependence of the form factors is ignored i.e. the following approxi mation is macIc, , 2 g(q  ) 2 W  ,2 2 g(q MN ) .  (6.78)  Naus [129] has calculated the W 2 dependence of the electromagnetic form factors gv and g for the proton. Using the results therein [129, Figures 2.4,2.5 pp 46,47] the following estimate of the deviation from the on-shell values of gi and g was made. =  ) 2 ,W 2 g(q gv  —  (q,  , M) 2 g(q N)  <0.1%,  A(gM)  A(gv)  (6.79)  Chapter 6.  Muon Induced Break-up: the Deuteron Channel  118  Kinematics  By using the value of E implied by energy conservation the usual problem of energy imbalance at the hacironic vertex is circumvented. Current Conservation  to making an  Forcing current conservation is equivalent  currents and  only applies  to the vector  estimate of meson exchange  of the current. In this work such an estimate  part  will not be made and the results can said to be in “strict impulse approximation”. The largest meson exchange currents contribute to the axial part of the current and a simple estimate cannot be made since the axial current is not conserved.  Now that the four choices have been made, the off-shell structure functions can be calculated. A modified Gordon identity and modified spin summation are needed. 13  )[(l 11 ü(n,s  =  u(p  -  ,.Sp)U(p  ,s) =  sp  =  [T°  =  —gA  6J2)7  7 % +W 2Ep  Retaining only terms of first order in 6, the T  —  =  +  (6.80)  +MN(1— 6 1 ) 2Ep  structure functions were  + 6Tj  2  13  —  where,  (6.8fl found to be, (6.82)  2 q ff g 2 M(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  6.7  119  Matrix Element Squared  The quantity that will be referred to as the matrix element squared is defined by equation (H.20) and is given by, =  lvi  T  maP  = 2Te + Te  —  Te(1  where the dimensionless kinematical variables ‘a = 2 /q m,? in  The  =  mJM  it  —  it)  and  —  + Te + TheW  in  (6.84)  are,  0.88  ‘a  1  = 0.11.  (6.85) (6.86)  are given by equations (6.55),(6.82),(6.83) and (6.62). The values of the nucleon  form factors gv, gM, gA and gp which were taken are given in appendix I. In short, gv, g and g are measured in the spacelike q 2 region and gp is inferred from g using the PCAC hypothesis. Their values in the timelike q 2 region are found by analytic continuation which 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 6.8  for both the on-shell and off-shell impulse approximation. The off-shell contributions are seen to hold the value of M 12 roughly constant and enhance its value over the on-shell case by a factor of up to 3.8 at the high energy end of the deuteron spectrum. 6.8  Results  The deuteron spectrum obtained is shown as a function of deuteron momentum in figure 6.9 along with data from experiment E569 [121]. The off-shell impulse approximation IAOFF  (  IAON  —  —  solid line  dashed line  ) )  gives a larger value than the on-shell impulse approximation for all values of d and agrees reasonably with the data although  Chapter 6. Muon Induced Break-up: the Deuteron Channe1  43  -....  off—shell on—shell  -  2-  -  Deuteron Mo me nt u m MeV  100  120  I  I  100  200  -  -  300  400  Figure 6.8: The matrix element squared. there is room for further enhancement at the high d points. The higher result for IAOFF is clue solely to the larger matrix element for IAOFF as shown in figure 6.8. The difference between the off-shell and on-shell result is greatest at large deuteron momentum and this is because the amount that the struck proton is off-shell increases with d,  (  see figure  6.7). The total integrated rate was found to be 1627 s 1 for IAOFF and 1583 s 1 for IAON. The modal cleuteron momentum is 67 MeV. The total is roughly a factor four higher than the total i-ate found by Phillips [47j who included final state interactions employing the Amado model. How might the results for the deuteron spectrum change when the effects of final state interaction (FSI) and meson exchange currents (MEC) are included ? The following qualitative statements can he made. The effects of FSI will be largest at small d. This follows from the fact that FSI are  Chapter 6. Muon Induced Break-up: the Deuteron Channel  121  16  12 U)  >  0  8 -o  0 0  100  200  300  Deuteron momentum d [MeV]  10_i U)  >  0  -o  102  -o  Deuteron momentum d [MeVI Figure 6.9: Deuteron spectra for various ranges of d. The solid line is the off-shell impulse approximation. The dashed line is the on-shell impulse approximation. The data are taken from Cummings et al [121].  Chapter 6. Muon Induced Break-up: the Deuteron Channel  122  Figure 6.10: The neutron/deuteron momentum in the nd CM as a function of the LAB deuteron momentum. 400  300  200  I:  100  0  largest when the neutron and deuteron have small relative momentum in their CM and hence small relative speed in any other frame. Their relative momentum in the nd CM d 1 k  is given by, (6.87)  j.  d 1 k  By using the momentum conservation condition, equation (6.5), d 11 can be written k as a function of d and x  =  d.Q thus, 4/322  11 x) k (d,  =  [d2  +  9(1  —  2 dx/)  —  4/ k 3 x 1 3(1 dx/)j  1 2  (6.88)  —  The value of (d,x) 11 is shown in figure 6.10. The effects of FSI will be greatest when k 11 k  =  0 which occurs when d  =  66 MeV  (  and x  =  +1). At this kinematical point, the  neutrino travels at r radians to both neutron and deuteron which are likely to recombine and form a tritonc thus robbing the deuteron channel of flux and feeding the quasi-elastic H reaction does not violate energy momentum conservation since the 3 Note that the in-flight nd—* low k neutron deuteron state is a virtual state in this case. d 1 C  Chapter 6. Muon Induced Break-up: the Deuteron Channel  123  channel. 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 H 3 e where the cross section is reduced by a factor two when final state interactions are included. A counter example is provided by the proton-proton pion production reaction pp  In this case  attraction between the final state proton and neutron causes peaked enhancement in the cross section as shown by Dubach et al.[131]. In summary, final state interactions are expected to affect the differential rate considerably and especially at low deuteron momenta. An overall reduction in the rate is expected although there may also be a peaked enhancement near d  =  66 MeV.  In order to assess the probable MEC contributions to the differential rate, the results of Goulard, Lorazo and Primakoff [132] and Dautry, Rho and Riska [133] are noted. The former authors found that the matrix element squared for muon capture by deuterium received an extra 200% from MEC contributions at the zero neutrino energy kinematical point. The latter authors found that the total deuterium muon capture rate was enhanced by only 5%. The total rate is dominated by the large neutrino energy values since the phase space goes like the neutrino energy squared and so we may assert that the effects of MEC are large at small neutrino energy  Q  and small at large  Q.  This assertion is verified  by a calculation by Doi et al.[134, see figure 3] which showed that the MEC contributions to the differential muon capture rate by deuterium steadily increase with the neutron energy in the nn CM. The kinematics of the i 1 H 3 e of the td  —*  —  v,dn process are similar to those  vnn in that high deuteron/neutron momenta correspond to low neutrino  energy. Figure 6.11 shows the neutrino energy as a function of d. In summary, MEC are expected to increase the differential rate, especially at high deuteron momenta, by factors of up to three.  Chapter 6.  Muon  Induced Break-up: the Deuteron G’hannel  124  100 > 0)  LJ  75  E 50  0 100 200 300 Deuteron Momentum d [MeV] Figure  6.11: The  neutrino energy  Q  as a function of  LAB  deuteron momentum.  Chapter 7  Muon Depolarization and Hyperfine Populations  7.1  Overview  In chapter 3, the spin observables A, A and A were defined for the muon capture process  He +3  ,‘  (iF d(cos 0)  v  =  H. The rate F was written as, +3 /(F 0 2)(1 + APP (cosO) + 2 1 APtP ( cosO) + AP)  where F 0 is the unpolarized rate. The analyzing powers the nuclear Prol)erties of the 3 He  —*  (7.1)  and A depend only on  H transition. However, in any experiment intended 3  to measure the analyzing powers, the actual 1)lysical quantity observed will be a product of an analyzing power and a population parameter rate averaged over all angles  =  or P. For example, the total  is,  f  which depends on the product  d(cos 0)  dcos  =  F ( 0 1 + AP)  (7.2)  In order to use the measuremeilt of such an  observable to find the analyzing powers it is necessary to know the population densities N(f,f) and hence Ps., Pt and P. This chapter describes a calculation of the N(f,f). 7.2  Introduction  The life history of muons which eventually form muoiic atoms has been comprehensively treated by Rose and Mann (for the case of  12(  )  in  1961 [135]. The life history for the  He is similar and may thus be divided into five parts. case of 3 125  Chapter 7. Muon Depolarization and Hyperfine Populations  126  1. The muon life begins when it is created in the decay of a charged pion. A negative muon produced thus is 100% polarized along its axis of motion in the pion rest frame. This is because the accompanying anti-neutrino is right handed which by  the conservation of angular momentum forces the muon to have right helicity.  (  Any  orbital angular momentum has zero z-component because the momenta are back to back  ).  Polarized muon beams retain this polarization by selecting ‘backward  muons’, i.e. those which are produced in the CM at an angle r with respect to the lab 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 and small-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 it reaches the 15 state. The processes of atomic capture depolarization of the  muon.  (  part 3  )  and cascade  (  p’ 5  )  lead to substantial  The depolarization factor should be about 1/6 according to  the following simple argument. Assume that the muon  is  initially spin up. Upon capture, the muon spin g couples  to the orbital angular momentum 1 to form the total angular momentum points at an angle & to the z-axis and 1 > s. 1 and The time averaged vector a weak magnetic held  ),  will point along  j  .  precess around  and so its z-component,  j  .  and  (  Suppose 1  ..3  cosO.  as measured by  will be proportional to cos 2 0. Averaging aver all 0 shows that  Chapter 7. Muon Depolarization and Hyperfine Populations  127  the muon polarization has dropped by a factor three. Averaging over all 0 corresponds to taking the direction of the muon momentum completely randomized at capture. Immediately after capture, the populations of the two fine structure levels are given by 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  fact, the atoms in the j=l+ state cascade exclusively via Aj atoms in the j=l— state have just one Aj  =  =  =  0, —1 are allowed. In  —1 transitions and the  0 transition, all the rest being Aj  =  —1.  Considering these two types of transition: one has,  T+ J-f j In the first case, the change in second case we need  .  = ‘  +  j is i  (T-f)+’ (7.3) =  (l—1)+’  Aj=0  accommodated by the change in 1. However, in the  in order to preserve the magnitude of  j.  For an initial  spin up rnuoi it is possible to conserve the component of spin by ending up with a spin up muon and a spin sideways photon or a  Spin  down muon and a spin up photon. Both  of 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 causes the muon polarization to drop by a factor two. Combining the effects of atomic capture and cascade one arrives at a total depolarization factor of 1/6 as claimed above. He has nuclear spin Since 3  there is a hyperfine interaction between the nuclear spin  and the atomic total angular momentum  j.  To include the effect of this hyperfine  interaction, the atomic cascade is further subdivided into two distinct parts although in reality this distinction is not so clear cut. • The first stage is defined by setting the hyperfine interaction between the nuclear spin,  i and the total angular momentum of the muon atomic orbital, jto zero. Y  Chapter 7.  Muon Depolarization and Hyperfine Populations  128  Figure 7.1: Term diagram.  5  0  1  3  2  • In the second stage i and j are coupled to form mentum and the atomic states have good  .1  and  4....  f, f  the grand total angular mo .  This is the strong hyperfine  coupling limit and it is assumed that transitions to states of different  f  could be  distinguished by the energy of the photon emitted i.e. the width of the state is much less than the hyperfine splitting. Cascade Calculation  7.3 7.3.1  In the Absence of Hyperfine Coupling  In 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  129  Rose and Mann [135] have calculated the angular correlation (kE.kE ) of a decelerating 0 muon in carbon. They find that it drops with energy loss like 0 (E/E / 1 ) ’ with, 2m M (M +m, 2 ) 1  (7.4)  M is the mass of the scattering atom and E/E 0 is the muon energy divided by its initial value. For carbon, the correlation has dropped to less than 0.4% when the muon still has 90% of its initial energy left. hmerting the H 3 e mass into equation (7.4), it was found that the muon angular correlation has dropped to less than 1% by the time its energy is 72% of the original value. According to this calculation, the assumption of zero angular correlation is a good one since E/E 0 at capture is approximately 1% for H 3 e. With regard to the second assumption it is noted that higher order electromagnetic transitions 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. This diagram shows atomic states positioned according to energy, with a non-linear scale too coarse for hyperfine or even fine structure to show. Each line represents a state of fixed n, 1. If the scale were expanded so that the fine structure could be seen, two states with  j  =  1+  -  would emerge from each single line except for 1  By restricting transitions to “right to left” only Al any value  <  =  =  0.  —1 is allowed but An can take  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 the Al  =  +1 selection rule. Suppose we start at 1: the minimum number of transitions  that will take us to the ground state is then 1. If there were to he just one left to right transition  en route  then there would also have to be one right to left transition to get  back to the original 1, plus all the others which gives a total of 1+2. Every spontaneous transition causes depolarization in accordance with the second law of thermodynamics.  Chapter 7. Muon Depolarization and Hyperfine Populations  130  Hence, the presence of left to right transitions implies extra depolarization. As an example of the quality of this approximation, consider the fate of an atom finding itself initially with n=lo. This value is chosen for the sake of argument but it is suggested by the results of Haff and Tomnbrello [136] who have calculated Auger rates for 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, dubbed ‘circular’ if 1  n  =  —  (  an orbit is  1). Bohr [137] has suggested that the high 1 states would be  preferred 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 to a circular a =14, 1  =  =  15 and 1  =  14. Its oniy option is to decay  13 state and subsequently via circular orbits only, all the way  down to the iS state. Thus, every transition is right to left and the total left to right transition probability is 0%. This statement is also true for the a since its first transition must be to either an a  =  14, 1  =  =  15, 1  12 state or an a  =  =  13 state 13, 1=12  state. The latter state is circular and an atom finding itself in such a state can decay via circular orbits only. The former state is equivalent to the a  =  15, 1  =  13 state and may  only induce other right to left transitions. Consider now the a  =  15, 1  =  12 state. There is one possible left to right transition  and there are three possible right to left transitions. The transition probabilities Pg/are given by the simple dipole formula [116]. Da’l  — —  j-aw 3’ 1 L  2  -  T m m’  rn’  = where,  5 m 4 1/6Z i-ecia 11 f = =  (  l+ 1+1 21 + I  —  ) 1 f 3 1 (Rf) 2 for 1—> 1+1 forll—1  (7.6)  Chapter 7. Muon Depolarization and  131  Hyperfine Populations  Table 7.1: Transition Probabilities for the 1 He- Atom. 3 i Transition ( n,l) —+ (n,l ‘) (15,12) — (14,1:3) (15,12) —* (14,11) (15,12) —* (13,11) (15,12) —* (12,11) (15,11) —* (14,12) (15,11) —* (13,12) (15,11) — (14,10) (15,11) —p (13,10) (15,11) (12,10) (15,11) — (11,10) (5,2) —* (4,3) (4,1) (5,2) (5,2) —* (3,1) (5,2) — (2,1)  Rate s  3.43x 3.62x 2.15x 7.35x i.06x i.53x 3.13x 2.49x 1.52x 5.62x 1.61x 4.74x 1.08x :3.01 x  i0 i0 io 106 i0 iO 7 i0  io iO 106 108 iO i0° 10’°  and, (—1 )_1 4(21 — 1)!  n’i’ 71 R i  Fl(_nr 2 {  (4nn+i (n  (n + i)!(n’ + l — 1)! (n — 1 — 1)! (n’ — 1)!  —nr’;  21;  ()2)  —  (ii  (:  with, nT  The quantity  ni)  —  1)fl+n’_2l_2 71  + n’’’  (—nr 1 F 2  =  n—i—i  =  n  —  x 2, —nr’; 21;  (7.7)  (i;2  )}  —  is the reduced mass of the muonic atom, w is the energy of the emitted  11 ‘ recl  1 (a, 3; y; x) is the hypergeornetric function. The transition probabilities F photon and 2 for the  n  =  15, l  state and the For the  n  n  =  12 state are given in table 7.1 as well as those for the n  5, l 15, 1  =  =  while it is 0.2% for the  =  15, i  =  11  2 state.  12 term, the probability of a left to right transition is only 0.05% n  =  15,1  =  11 term. The preference for the (15, 12)  —*  (14, 11)  Chapter 7. Muon Depolarization and Hyperfine Populations  over the (15, 12)  —  132  (14, 13) transition is due solely to a larger dipole matrix element  between the states. As the atom decays to lower n states, the transition to the state of lowest energy becomes favoured as can be seen from the final series of entries into table 7.1. At low n, the enhancement of the /n  =  —1 dipole matrix element is the  same as it was at large n but here the (energy) 3 factor takes over. For the 5d state, left to right transitions represent 0.35% of the total transition probability. One can summarize the physical l)icture thus. The initial population is mostly at high  ii  and high I and it decays right to left on the term diagram with more and more  atoms falling into circular orbits as n decreases. In calculating the cascade depolarization the statistical tensor formalism, as has been expounded with refereice to this problem by Nagamine and Yamazaki [138] as well as Kunioa [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 the  populations themselves  However, one could choose any linearly independent  combination and a statistical tensor is one such combination with the peculiar property of transforming simply under rotations,  (  see [34, 1)109]  ).  The statistical tensor Bk(n,l,j)  is defined by,  Bk(n,l,j)  =  (_1)3’T , j,(mi)(k0 hiP, jmj; 2j + 1 1  j  —mi)  (7.8)  ‘n.j  which can be inverted to give the populations as a linear combination of the statistical tensors. ( ) 1 _J+nhi =  /2j + 1  Bk(n.,I,i)(kO jm;  k  aThe author is grateful for helpful discussions with Y.Kuno  j  —rn)  (7.9)  Chapter 7. Muon Depolarization and Hyperfine Populations  133  With this choice of normalization, Bo(n,l,j) equals the total population of the states  In,l,j,mj ). Also, 3 i(i + 1)  B ( 1 J) =  3.  2  mP, , 1 i,(m) Bo(J) x Polarization  (7.10)  where the I)olaIizatiofl is defined as  Polarization  =  mj  (7.11)  j  Thus, the first rank statistical tensor is proportional to the vector polarization of the states. In calculating the muon depolarization, it is necessary to consider first the atomic capture process.  According to the first assumption, a given 1 orbital must be filled  without prejudice towards any sub-state, m . This is accomplished by combining the 1 muon 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 probability of finding the state n, 1, j, rn.  )  is given by,  1 ( 5 P ) 3 ) P,j(m rn 1 ,7Ti fl  I  (jrnj  ; sin) 1 1m  2.  (7.12)  By inserting this expression in equation (7.8), using the inversion formula (7.9) to replace P(m), ) 1 , 7 P ,(m and performing some Racah algebra, a general equation for combining two statistical tensors was found. It is,  Bk(n,l,j)  [(2,sIl 1)]  Bk ( 1 s) 2 Bk ( l) x 1’  Chapter 7. Muon Depolarization and Hyperfine Populations  1 + 1)(2k [(2k 2 + 1)] (kO I k 0; k 1 0) 2  134  S  ij  $  1  1 k  j  (7.13)  2 k k  This equation does not agree with equation (5) of [139]. That equation is inconsistent with the normalization conventions used in the paper and is missing the Clebsch-Cordan  coefficient. From the Clebsch-Gordon coefficient in equation (7.13) and the fact that k 2 is re stricted to zero it is clear that k call the result for the levels n, 1, j non-zero irrespective of the  are zero. Since 2 k O ,  take on the values zero or one. One then has  that only the population and vector polarization are  value of  equation  Bk(n,l,j)  ),  only  j  (7.13)  i.e. all tensors Bk(n,l,j) of rank greater than one, simplifies to,  (-l) J 1 W(jj; ki) Bk(s)Bo(l)  =  (7.14)  Next, the evolution of Bk(n,l,j) during the cascade will be traced. Consider a transi tion from the state  ii,  i, j, rnj  )  to the state n’, 1’, j’, rn  ).  Using the second assumption  given at the start of this section, the transition operator can be written as a first rank tensor operator in orbital angular momentum space T . The relative transition rate 1 for the various rn. , rns is calculated taking the same reduced matrix element for all the 1 transitions. If  is the rate for a transition from the state n, l,j, 7n to the state  n’, 1’, j’, rn then,  (l’,j’, rn T 1 1, j, rn) (j’rn l’77i; rn) (jrnj irni; m)  8 nil ,rn’nr  ,  br;  ) 1 un  [(2j + 1)(2j’ + 1)j W(jj’ll’; 1)(1m -rn j’m;  j  -mi) (7.15)  Chapter 7. Muon Depolarization and Hyperfine Populations  135  The new population 1 iP ,i(m) is given by.  P , 1 (in)  ,P 1 (nr) x  =  Normalization constant)  (7.16)  1,J ,?nj  By inserting this in an expression for Bk(n’, j’, 1’), replacing the populations using the inversion formula (7.9), performing some Racah algebra and finally normalizing, one finds that the statistical tensor Bk(n,l,j) suffers the following change. Bk(n’,j’, 1’)  W(jj’ll’; 1 )2 Uk(jlj) Bk(n,l,j)  (21 + 1)(2j’ + 1)  (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 “right to left” transitions emanating from a particular term are equivalent with respect to depolarization. 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. This  route is illustrated in figure 7.2. A question exists as to the dependence of the residual muon polarization on the initial  1 distribution. To answer this, the depolarization of a 100% polarized ensemble of muons was calculated for distributions supporting a single 1 value and the results are shown in table 7.2, (the route for 1  =  0 is (n,l)  —*  (2, 1)  —*  (1,0)). The following salient features  are noted. • For large 1. the value of p argument, (1/6  =  16.7%  ).  residual  is near 1/6 as expected from the simple classical  The depolarization follows the pattern of 1/3 at capture  and 1/2 clue to the cascade.  Chap tei 7. Muon Depolarization and  Hyperfin e  Populations  136  Figure 7.2: Effective cascade route.  1  >  Table 7.2: Muon Depolarization with 1  25 23 21 19 17 15 13 11 9 7 6 5 4 3 2 I 0  presidual%  p after caPture%  16.9 17.0 17.1 17.3 17.5 17.8 18.1 18.5 19.2 20.2 20.9 21.9 23.4 25.7 30.1 40.7 40.7  33.4 33.4 33.4 33.4 33.4 33.4 33.4 :33.5 33.5 33.5 :33.7 33.9 34.2 34.7 36.0 40.7 100.0  =  1 Populated Only.  p residual/ p after capture ><  50.6 50.9 51.2 51.8 52.4 53.3 54.2 55.2 57.3 60.1 62.0 64.6 68.4 74.1 83.6 100.0 40.7  100%  Chapter 7.  Depolarization and Hyperfine Populations  Muon  137  Table 7.3: Muon Polarization During the Cascade. 1 9 8 7 6 5 4 3 2 1 0  P% 33.5 33.1 32.6 32.0 31.2 30.0 28.2 25.1 19.2 19.2  • This picture is reasonable down to low values of 1, • The 1 • The 1  =  1 0  1  —  —  I  The values of p  =  (  1  5, 6 say  ).  0 transition causes no depolarization. I transition causes extensive depolarization,  residual  for low 1 are calculated in the  real population starting entirely with 1  =  2 would have  An  (  2/5  ).  =max. approximation. A  PILresidu  less than 30.1% because  of the presence of transitions to n S with n >1 which then give rise to extra left to right transitions. However, the purpose of table 7.2 is to indicate the depolarization due to specific paths rather than to make any definite statements about real muon depolariza tion. 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 stages of the particular cascade starting at 1  =  1  =  9. This general feature is found for all values  of 1. The Rose and Mann 1  distribution is  shown in figure 7.3. It is clear from table 7.2 that  any distribution of this basic shape will give  . 6 presicluall/  In other words, the residual  Chapter 7. Muon Depolarization and Hyperfine Populations  138  Figure 7.3: Rose and Mann 1 distribution.  2500  I  I  I  I  2000-  -  -  1500-  U  -  .  C  c1000  -  500-  -  0I • 1• • 0 2 4 6 8 10 12 orbital angular momentum quantum number I  muon polarization is rather insensitive to the initial distribution of the muonic atoms among atomic states. Any reasonable initial distribution will lead to  Experimental results for I  =  6 / 1 presidua1  0 nuclei are listed in Table 7.4 [140]. The anomalous  value for H 4 e 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 Souder  was at 7 and 14 atmospheres. It should be noted that H 4 e was the only gaseous target used in the experiments reported in [140]. Since H 3 e and H 4 e have the same atomic number one would expect similar physics in the absence of the hyperfine interaction, least at the atomic level  ), and  (  at  the above mentioned anomalous result must be taken into  account when making predictions for the N(f,f) in 3 He. 7.3.2  With Hyperfine Coupling  Generally speaking, the effect of coupling between the spin of polarized nuclei and the atomic total angular momentum is expected to increase the value of the residual muon  Chapter 7. Muon Depolarization and Hyperfine Populations  139  Table 7.4: Residual Muon Polarization P, for Spin Zero Nuclei. Element 4 H e i2C 16Q  24 Mg 5i 28 S Zi 64 Cd Pb  P % 6+1 14+ 4 15+ 4 19+ 5 16+ 4 15+3 19+ 5 19+ 5 19+6  polarization. This will be true when the nuclear spm  is pointing in  the direction of the  incident muon’s spin. Of course, when these spins point against each other there is the possibility of a cancelling out effect. This effect can be realized by the following simple argument which uses the vector model for the total  spiis. spin  .1.  Suppose the nuclear spin i and the muon  For a l)olaIize(l nucleus, i will tend to point  us call this direction up unpolarized  spin  ).  The resultant  f  are coupled, forming  spin ill  a certain direction  will also tend to point  is added. The coupling interaction causes i and  and when we then project because it is following  1  up in  .  (  let  the case that  an  to precess around  f  onto the ‘up’ axis it will be found more often up than down  which itself is more often up than clown.  The hyperfine splitting of levels goes like the expectation value of 1/r 3 multiplied by 1(1+ 1)/j(j ± 1). l3ethe and Salpeter [116, page 17] give a result for hydrogen-like atomic wavefunctions, 3 Z  \/ where a!. is the muonic Bohr radius  anl(l + 1)(l + =  132.8  fm.  (7.19)  )  In general, the hyperfine splitting gets  smaller as n grows. Thus it is expected that at some value of n,  (u1) say  ),  the hyperfine  Chapter 7.  Muon Depolarization and Hyperfine Populations  140  splitting is so small as to be smeared by the natural width of a level. Although there may  be a  region  where the hyperfine splitting is comparable to the natural width, the physical  situation will be modeled by assuming zero hyperfine coupling for ri Iiyperfine coupling for ri  j  to form a state of good  ni. This is done by combining the populations of the nuclear levels,  Bk ( 1 i)  ),  and strong  The theoretical description consists of coupling the  < fl )• 1  nuclear spin i to the atomic total angular momentum =  > 71 hyp  with the populations of the atomic levels,  (  (  at  described by  described by Bk , l,j) 11 2 (n  the following equation which is equivalent to equation (7.13).  f,f, ),  using  The statistical tensor  Bk(ri,l,j,f) will be non-zero for k=0,l.2 since both the atoms and the nuclei have vector  polarization, (k ,k 1 2  =  1).  Bk(n,l,j,f)  [2i +1)(2+ 1)]  =  1 (i) 2 B Bk ( n,l,j) x 1 ,k k 2  1 + i)(2k [(2k 2 + l)] (kO k 0; k 1 0) 2  i  if  j  i f  1 k  2 k  (7.20)  k  The evolution of Bk(n,l,j,f) down to the iS state is then traced assuming El traii sitions as before. The following equation describes the change in Bk(n,l,j,f) due to the  transition  1, j,  f)  —  1’, j’,  Bk(n,l,j,.f)  =  f’). [(21 + I)(2j + 1)(2j’ + 1)(2f’ + l)]uk(f if’) x W(j, j’, 1,1’; 1  W(f, 1’ j, j’; 1 ) Bk(n,l,j,f) 2  (7.21)  This is derived in the same way as equation (7.17). It is helpful to consider the case where  illlyp  1,  ‘He  =1 and P  nuclei are always spin up other half half spin down  T  =  0 to understand the basic idea. In this case, the  but the muons are found half of the time spin up  ..  Half the time one finds the configuration  f=1,f=1. The other half, one finds the configuration  fl,  fl  ‘  and the  which is all  which is half f=1,f=0 and  Chapter 7. Muon Depolarization and Hyperfine Populations  half  f=0,j=0.  141  The latter two states contribute nothing to the polarization of the f=1  state which takes the value one half. In this sirnj)ie case it equals the muon polarization after coupling so we started with a totally unpolarized ensemble of muons and now they have polarization  .  The 15 hyperfine statistical tensors B(f) will be parameterized l) tile four constants A,B,C and D which are defined by tile following equations. B(0)  = 025— AP P3He/4 11  B(1)  = 0.75+AP,tP3He/4  B(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 is of tile  SfllC  rank k.  • Tile Clebsh-Gorclon coefficient in equation (7.20) limits the values k 1 and k 2 can take on for a given value of k. Take B(l) for example. Only tile form these ) 2 , 1 (k k  can  tensors  Bk(n,l,j,f)  with  k  1 contribute and to  take on the values (0,1) or (1,0). Thus, B(i) has two terms: one  proportional to PR aildI tile other prOI)ortional to PHe. Tile equations (7.22) imply the following identifications for  1v  BPR+CP3Ne  Pt  DPRP3He  P  =  AP j 1 P3  and P.  (7.23)  Chapter 7. Muon Depolarization and Hyperfine Populations  142  Table 7.5: Residual Muon Polarization per Unit Target Polarization for Various Values of  presiclual%  The value of  p residual  p Table 7.5 lists at  residual  p residual  1 59  2 28  3 2:3  4 21  5 20  61 19]  is given by,  N(1, 1)  —  N(i,  in the case where P,  1)  P = B P + C PHe•  1 and  PHe  (7.24)  = 1. The population densities  are taken from the case 1 = 13. This choice is motivated by the results of Haff  and 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 as expected, an effect which decreases as n 1 increases.  7.4  Attempt to Include Effects of External Collisions  As mentioned earlier, the anomalous residual muon polarization for H 4 e indicates that a simple cascade theory is inadequate. The extra depolarization is presumably due to collisions between the muonic ion and the surrounding H 3 e atoms which induce Stark and external Auger transitions. The effects of external collisions will be included by making 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 a depolarization factor Ddepol. This depolarization factor is the same for both j=l+ and jzil—.  Chapter  7.  Muon Depolarization  and Hyperfine Populations  143  If it is true that the extra depolarizatiou is due to Stark transitions the first asslimp tion should be valid. Placci [143] has measured muonic x-rays in 3 He at 7 atmospheres and found a fit to the intensities using a rate Vv’ 1 for complete Stark mixing at principal  quantum number  71  given by, =  1010  with 1 W 3  w 1 3  5 ) 3 (f  s. This rate drops rapidly with n. and at n  (7.25) =  5, W  8 x iO s which  should 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 W . The Stark transition 5  probability 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 crude approximation rather than the feature of some particular model.  The calculation begins at  n = ni  at which point we need to decide what val  ues of 1 Bk(n1 , l,j) to use. The 1 Bk(n1 , l,j) should reproduce the experimental value of  p  residual  =  4.4±0.3 % if they were simply cascaded down to the iS level with no hyper  fine interaction. There are four quantities of interest, the two populations and two vector polarizations of the j=l+ states at n  = fl ). 11  Remember that the higher rank tensors  are 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)  Using the second assumption, tile value of the vector polarizations are set by  (7.26) Ddepal  which equals tile experimental residual muon polarization divided by the value that tile vector polarizations would have given in the absence of the hyperfine interaction. Having set  tile  Bk(n1 , 1 l,j) tile muon population is then cascacledi down to the iS state  using equation (7.21). The values of A,B,C and D thus found depend on tile value of  ni  but also on the initial 1 distribution. However, it is expected that the latter dependence  Cli ap ter 7. Mu on Depolarization and Hyperfin e Populations  144  Table 7.6: Values of the A,B,C,D Parameters with External Depolarization. 1 n  1 2 2 3 3 4 4 5 5 6 6 will be small since for 1  p,sidua1  1 14,20 14 20 14 20 14 20 14 20 14 20  A .0440 .0440 .0440 .0440 .0440 .0440 .0440 .0440 .0440 .0440 .0440  B .0220 .0173 .0169 .0170 .0165 .0170 .0165 .0170 .0165 .0171 .0166  =  D .0440 .0283 .0285 .0201 .0201 .0168 .0169 .0150 .0151 .0139 .0140  itself is rather insensitive to the initial 1 distribution. Results  14, 20 are given in table 7.6 for values of  0.246 for 1  C .5000 .2083 .208:3 .1609 .1609 .1396 .1396 .127:3 .1273 .119:3 .119:3  14 and 0.255 for 1  =  up to 6. The values of Dol are  20.  The results of this method can be tested by comparing the residual muon polariza tion with a recent result of Newbury et al.[144j. This experiment measured the residual polarization of an originally unpolarized muon captured by a 1)olarizecl 3 He target at 8 atmospheres. It was found to be 7.2 + 0.8% per unit target polarization. In this experi mental situation. p  residual  is given by the CP3He so that the residual muon polarization  per uiit target polarization is C. Looking at table 7.6 it is clear that there is some depo larization due to collisions after the hyperfine interaction becomes important and so the clear distinction made in the first assumption above is not valid at this pressure.  7.5  Summary and Conclusions  The intra-atomic processes prior to nuclear muon capture have been understood and it is clear that the anomalous residual muon polarization found in H 4 e is an inter-atomic  ( or  Cli apt er 7. Mu on  rather ionic-atomic  Depolarization and Hyperfine Pop ulat ions  )  effect.  145  Any reasonable initial distribution amongst atomic states  will 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 external depolarizmg effects occur before a point where the hyperfine interaction becomes im portant. By  the predictions for p  comparing  residual  with a recent measurement using  a polarized H 3 e target and an unpolarized muon beam it is concluded that this simple approach is umeliable. In order to further understand the hyperfine population densities, the procedure used by Landua and Klempt [145,146] could be applied. These authors calculated the effect of atomic collisions on x-ray  intensities for muonic  and pionic 4 He and were able to reproduce  the density dependent anomalous pionic x-ray intensities with few free parameters. 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Lett. 63 (1989) 1025.  153  Appendix A  Natural Units  In nuclear and particle physics it is convenient to adopt the natural system of units. In this system all quantities have dimensions of energy  (  or alternatively length  )  raised to  some power and cumbersome scale factors which often occur are eliminated by setting them to one. The two scales used to define the natural system are the speed of light in vacuo c and the reduced Planck constant h. The speed of light is an upper limit set by special relativity and sets the scale for speeds. The Planck constant indicates the size of quantum mechanical effects as in, for example, the uncertainty relation for energy and time observables, /Et > h/2.  (A.1)  The mechanics of the natural system can be understood by considering the following two examples of conversion from natural units to Système Internationale (SI) units. Suppose we are given a momentum in MeV. The  (  non-natural  )  dimensions of mo  .T [M.L ] . It is therefore necessary to mentum are [M.L.T’] and those of energy are 2 insert a factor of the speed of light in order to convert from natural to SI. 1 “MeV”  =  e x 106 “Joules”  =  e X10 6  1 Kg.m.s  (A.2) (A.3)  In the above equations e is the magnitude of the electronic charge and the units in quotation marks are natural units. The second example concerns the equivalence of energy and inverse length in the 154  Appendix A. Natural Units  natural  system.  155  What is the conversion factor which takes us from inverse length to  energy ? The dimensions of c and h are needed to perform the following manipulation. They are [L.T’] and j 1 . 2 [M.L T respectively. 1 “fn ” 1  =  1015  =  ic “nit” 15 l0  =  hc Joules 5 10’  =  1 o’ h C MeV  “m”  106 e  dimensions  =  [L’l  dimensions ] .T 2 [M.L  (A.4) (A.5) (A.6)  197.327 MeV  (A.7)  Appendix B  Model Calculation of  This appendix details the model calculation of  VAd(q’, q)  VAd(q’, q).  A simple form of the deuteron  wavefunction is used which generates the general behaviour of VAd(q’, q) and explains why 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. =  2 —(q/d)  (B.1)  00  2 N  =  j  2 H/(qH dqq = 2  F(3/2) (d2/2)  3/2  (B.2)  The only free parameter in this function is d, the size of the deuteron in momentum representation. This parameter is fixed by demanding that the model calculation of gives the same result as the full calculation which used a realistic deuteron wavefunction with both s-waves and d-waves. The wavefunction given by equation (B.1) has the right shape for the s-wave part of the deuteron i.e. it is a monotonically decreasing function of 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) the following expression was found for the s-wave Ad potential. The range of the AN potential AAN  is written simply as A for the sake of notational clarity.  where,  VAd(q’,q)  =  —2J(q’,q)  J(q’, q)  =  q q _ / 2 A ’  and J( q)  2 _2sx/A  156  f  (B.3) 22 s 2 dss2e_2 K(s,  d([3  +  q2  + sqx])  q) K(s, q’)  (B.4) (B.5)  Appendix B. Model Calculation of  157  VAd(q’, q)  Performing the integral over x yields the sinhc function which is the hyperbolic equivalent of the sinc function: sinhc(x) (q’, q)  =  q / q / 2 a ’  =  sinh(x)/x. One then finds, sinhc(sq/g2) / 2 ths2e_28  2 N  1  W 1-.  .L— 2 cr1  ‘  —  ) 2 sinhc(sq’/g  2  .1_2 A  j__  4d 2  (B.7)  + 2 g  —  (B.6)  2 A  2 d  The value of .1(0, 0) can be found by noting that Iim_o sinhc(x)  =  1 and performing  the resulting gaussian integral which yields a gamma function cancelling the gamma function from the normalization coiistant. J(0, 0) Equating J(0,0) with 0.847 fm , h 1  =  0.615  1 fIn  =  f  2 N  /hi = 2 ths2e_23  /2A yields d and g  =  =  3 (h/d)  (B.8)  0.633fm’ and this in turn gives o  =  0.561 fm.  Now, VAd(O,q)/VAd(O,O) can be found by evaluating J(0,q)/J(0,0). If the model is to generate a reasonable approximation to VAd(q’, q) then the range of this ratio should be close to 0.96fm’. Using equation (13.6) it was found that, J(0, q) J(0,0)  —  /1 2 fdsse_2s  2 (g e q  —  )  ) 2 sinh(sq/g  Ih 2 f°°dss2e_23  (B 9) (B.10)  1  where,  2  T 7 c  —  1 2 cTi  h 2 8 g  .  )  Thus, with the simple form of the deuteron wavefunction, the Ad potential is exactly a gaussian with range  T  =  1.04 fm . Since this result is in close quantitative agreement 1  with the full result one has confidence that the qualitative features of VAd(q’, q) will be  158  Appendix B. Model Calculation of VAd(q’, q)  3  C0  x  Figure B.1: The sinhc function. well represeiited by the model calculation. Now let us calculate a “slice” of the potential at fixed q’ in order to see what happens to the range of the Ad potential. J(q’, q)  —  —  J(q / ,O)  // 2 2 Jdse_23 sillh($q/g sinh(sq’/g ) ) 2 /a (g 2 e -q 00 1 1 2 1h —2s 2 q j J dsse / sinh(sq F /g)2  (B  12 )  To evaluate the integral in the numerator of this expression the following identity was used. sinh(A) sinh(B) With,  =  1 [cosh(A-bB) 2  —  [00  Jo  2 cosh(2ax) e  =  —  cosh(A—B)]  (B.13)  2 F() ea  (B.14)  one finds,  J(q’, q)  J(q’, 0)  ) 2 214 sinhc(2qq’/a e  a  2 22g =  1.56 fm 1  (B.15)  The function sinhc(x) is a monotonically increasing function as x increases from zero to infinity, see figure BA. Thus, when the gaussian with range function  one  is multiplied by the sinhc  expects the result to have a range somewhat larger than  is exactly what is found in the full calculation.  uT.  This feature  Appendix B. Model Calculation of  159  VAd(q’, q)  The sinhc function spoils the gaussian shape of the potential at non-zero q’ in the model calculation. However, in the full calculation, the shape of the potential was best represented by a gaussian at q’ Since .J(q’, q)  =  J(q, q’),  (  l.Ofrn’,  (  see  figure 1.2  ).  which follows from inspection of equation (B.6)  )  one can  immediately write down the full expression for the model VAd(q’, q). —  VAd(q’, q)  The  non-separable part  =  e r e’ 2 4 / 2 sinhc(2qq’/a )  (B.16)  of this is the sinhc function which is a function of qq’ and cannot  be written as a function of q multiplied by a function of q’.  Appendix C  Derivation of Two Body H Pionic Decay Amplitude  In this appendix, two details of the calculation of the two body pionic decay amplitude for H are given. First, the effect of the Pauli principle is made clear and second an identity for the reduced spin matrix element is proved. We wish to calculate the matrix element  (‘3He  Heff  ’H) 1  where 1 ’He and  are  fully antisymmetrized wavefunctions and Heff is given by equation (2.14). The effective Hamiltonian is the sum of three terms and can be written, Heff  11(1)  H  =  P H ’P + P H’P 3 +2  (C.l)  i=1,2,3  The form of the last two terms follows from requiring that (1 H l 1) 1  =  (2111(2)12).  The fully antisymmetrized hypertriton wavefunction is given by equation(1.6). Let us write  ‘I’3He)  = NAAI ‘/‘-e )for the Helium wavefunction where NA is a normalization  constant and A (3He  =  1 +P 2 + P . Thus, 3  HeffI  3H)  =  (  =  /3(  A He  [H(1)  H(1)P] A 3 H’P + P 2 + P  )  W’)Al /)  where the latter equality follows from P A 2  =  A 3 P  (C.2) (C.3)  =  A and P  =  , P 3 P  =  . The 2 P  creation and annihilation operators associated with H’ are a1)ta and so only that part of  J “H )  j which the lambda particle is labelled (1) will contribute to the matrix  element. Therefore,  ( ‘He “effj  )=  \/3( 160  M’fP’l j,(1) )  (C.4)  Appendix C. Derivation of Two Body H Pionic Decay Amplitude  161  which is equation (2.21).  The operator M’ has both an s-wave and a p-wave part, the latter of which contains the nucleon spin operator  (1)•  Due to the particular form used for the hypertriton  wavefunction, the matrix element of the p-wave part is simply related to matrix element of 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 matrix  elements is, (aJMj T bJ’M) = 1  (JMjI J’M; KQ)(aJTKQUbJ’)  where TKQ is any tensor operator of rank K and projection  (C.5)  Q and a, b represent the  structure of the states with total angular momentum J, J’ and projection M, M. The definition of the spatial tensor operators RKQ(Iclk ) is, 2 RJQ(A k ) 2  =  (4)Y(, ).  The wavefunctions are expanded in channels =  (y-) 6 x-)  [Y  (C.6)  where the channels are defined by, )®x(1)](1).  For the hypertriton there are only two channels which have support in  (C.7) They  correspond to channels 2 and 6 of the 8-channel wavefunctions of Kamemeya et al. and have the properties shown in table C. I. The matrix element of  11(1)  (3He; nij Heff(q)I H;  is found by partial wave expanding the plane wave.  )= m; kmk)( 3HeRk(0k)jk(2qy/3))  (C.8)  162  Appendix C. Derivation of Two Body H Pionic Decay Amplitude  Table C.i: Hypertriton Channel Specifications. 10, 0 2  C  2 6  ).  0, L  0 0  0 2  s 1 1  The Clebsch-Coiclan coefficient immediately  ,$,1/2 3/2  0i, 0 0  constrains  k to be 0 or 1 and the reduced  matrix element further restricts k to be even only because of the even parity of the states with the result that oniy the zeroth order multipole contributes.  f  dx dy x jo(2qy/3)( y 2  He  Y)(Y  (1))  (C.9) 1)fl(1)  To calculate the matrix element  it is necessary to combine the spin operator  with spatial tensor operators and to that end the following definitions are made. 5o  u,  ) 2 1 ’ TKQ(kl k  +1k/2(u + iu) =  ml ,?fl2  The product  (KQ I 1 rn 2 k ; k 17711 ) m Rk (ll)Sk 2772  (C.10)  )H(’) yields operators with well defined tensor properties thus, krnk; 1m)°TJQ(k1)  Rk,lk(0k)Sl?fl =  (C.11)  KQ  The value of A’ is constrained to be 0 or 1 siice the states have total angular momentum which implies that the value of k may be 0,1,2. The even parity of the states restricts k to be even anci so in this case both the zeroth and second order multipoles contribute. The zeroth order multipole is, ) 1 (3He°°TO(01)7  (3He  ) ( ‘U’) ( ‘I k) 0,  0,  (C.12)  Appendix C. Derivation of Two Body H Pionic Decay Amplitude  163  where the spatial integrals and Bessel function have been suppressed. This form should be compared to, (3HeH°°TO(OO) b(’)  = (P3He  I) (  1 ‘)  a  where (awl a’)  (C.13)  ( 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,  S(sa,Sa,Sa’,La)  where,  =  (i)L+sa+S  12  I the matrix element of momentum is If La’  =  (1)•  However, if La’  =  2, so that  a 5  J I  1 2  Sa Sa’ a  (C.15)  J  Sa’ which was not allowed in the case of =  this implies that in turn that  2 then La  Sa’ La  (C.14)  Sa’, La)  + 1)(2Sa’ + 1)]  11112  2  Sa  and so there exists the possibility that Sa  [( 2 Si $ a  5’a  0 then La  S’  =  S(l,,2)  =  0 and since the total angular  and 5 a’  a 5  ,  =  and hence =  and so for the specific case where j) has no L  a 5  (_) =  =  and hence Sa a’ 5  =  Sa’.  Further, (C.16)  1 component and only s  =  1  components, =  (-)6(a, &)  (C.17)  The second order multipole is likely to be much smaller than the zeroth order multi(2qy/3). From the fact that F(q) is oniy reduced by 30% 2 pole because of the factor j  from the value of F (0) it is deduced that the values of y for which the nuclear density 1 is appreciable are such that qy small for x cantly,  (  1. The second order spherical Bessel function j (x) is 2  1 and so will reduce the major contributions to the matrix element signifi  in the case of the 3 He  —*  H isovector transition the second order multipole was 3  Appendix C. Derivation of Two Body H Pionic Decay Amplitude  164  found to be 0.2% of the zeroth order multipole). Thus, if the contribution of the second order multipole is igiored, (/3( which is equation (2.23).  3He  11)  k)  (C.18)  Appendix D  Derivation of Inclusive H Pionic Decay Amplitude  In 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 exchange integrals are found. Using closure x, 11 X, rn  ) ( X, rnf  1 the sum of matrix elements between the  hypertriton and general three nucleon states becomes an expectation value of the operator Heff( ) tHff( ) for the hypertriton. (D.1)  = 2 (X;rnjHeg(H;rnj) rn  ?n,mj,X  The expectation value can be simplified by using the properties of the permutation op erators.  (  HHeff ‘I)  =  (b t H’ H ’ /‘) +  The first term is easily evaluated using pertriton state which implies that  ( j)  (i) (i)t 11  (‘)  )  =  (/ =  (P + F t H ) 3  rn  —  )(s 0 p(’)/q  —  (D.2)  1 and the normalization of the hy  1. (D.3)  =  1  )  p(1)/q) (1);  m)  =  2 +2 s (/qo) p  The second term of equation (D.2) can be further simplified to twice the real part of H’ by writing the matrix element involving P 2 3 as the the expectation value of H(1 P  165  Appendix D. Derivation of Inclusive H Pionic Decay Amplitude  complex conjugate of the adjoint operator and using P H /‘ 3 P t IH’  =  166  . 3 P )*  H(1))tI /‘ 3 (H(1)fP  ) =  (D.4)  >*  H(1) 2 P H ’ (k t  (D.5)  2 fl(1)fl(2)l(L)(2)p  2 and Assuming that oniy the zeroth order multipoles of H  &(‘)  (i)t 11  )  (D.6)  are non-negligible,  this permuted matrix element reduces to two terms: one invo1viig the s-wave A decay strength s and the other involving the p-wave A decay strength. The integration over d eliminates the cross terms.  J  (  i(1);  ‘i  )  2(1)()  +  (D.7) 1)()  where,  =  (‘jo(x/2)jo(y)  ()  2P’)  (D.8)  The spin and isospin parts of these reduced matrix elements are given below and the spatial parts are expressed in terms of two exchange integrals  i()  and  77d(q).  The  subscript indicates which hypertriton chaimel contributes according to the value of L. 2 are, The matrix elements of the permutation operator P (xy P 2 ‘x’y’)  =  6(S,  (s, s, Sa)112(ia, 1 12 Li)S i !:,:,X(x, y, x’, y’) ) (D.9)  where,  12 S) S (s,s,  =  (_l)’[(2c, + 1)(2cr’ +  I (a,a’) 2 1i y, x’, y’)  =  =  )ii( 1 (_ j 2  + 1)(2i +  1)]  {  2  2  -  Sc  ZcrI  P X’y’(l\c’)LcxML) (xy (lccx)LcxML 2  ‘  }  (D.1O)  Appendix D. Derivation of Inclusive H Pionic Decay Amplitude  Using (1, 12 1, S  )  —1/2  12 1, S 112(0,0) and (1,  )  =  167  1 the reduced matrix element  can be written,  ()  =  too  where,  I  (q) 3 i —  —  )  (D.11) (0)  dxdydx I dy / x 2 y 2 x 2 y 2  i Jo  (0) (X)’/d  (x  ,  )‘/‘A(y)bA(y)  xjo(x/2)jo(y) X (x, y, x’, y’) 2  f and,  (x, y, x’, y’) 2 X  dx dy dx’ dy’  To evaluate  (2)()  x2y2x/2y/22)(x)2)(xI)A(y)A(y/)  xjo(x/2)jo(y) X (x, y, x’, y’) 2 a’) dt 2 (t) ) 1 P  (D.13) (D.14)  —  =  x’ + 2/3xx’t +2  =  a’  (D.12)  =  x2  /4 + xx’f 2 + x’  the matrix elements of  (D.15)  12/3  [2/32  —  1] are needed. S 12 is  the total spin operator for particles labelled 1 and 2. Recoupling tile states to have good total spin of particles 1 and 2, a 9 (i)( S  1.2/3  ‘5  )  (s, s, Sa)S 3 S (s’, s’, S) 32 ss,  )(8S[2/3S 3 X( 2  6(Sa, Sn’) where tile identity (s, 12 ‘, 5) S and tile reality of = 1 and S,  =  =  —  (D.16)  1]’Sa’)(3)  12 s, )S S (s, 01 s,S)[2/3s(s + 1) S , 01 (s 12  —  1] (D.17)  13 s, S) has been used which is proved using P S (s’,  =  3 P  For tile hypertriton there are only two cases to consider since s or  .  and including 2112(0,0)  Using (1,0, 12 1/2) S =  tile factor (—)[3/4 + 1/12] i.  =  (0, 1,1/2) 12 —S  =  \//2, (1, 12 1,3/2) S  =  =  1  —1 from the permutation of the isospin vector one arrives at —5/6 for tile s-wave part and —1/3 for the d-wave part. e.,  i(2  =  5() —  r/d(q)  (D.18)  Appendix D. Derivation of Inclusive H Pionic Decay Amplitude  168  Gathering all the results together, f(X;mfHeff(H;mjH2  -  ,K  =  s2[1 +  —  d()]  (/qo) +p { 2 1  —  —  (D.19)  which is equation 2.42 and holds when only the zeroth order multipoles in the expansion of H(2)H(1)t contribute. The integrals ?1s() and  ij()  were evaluated using Gauss-Legendre numerical inte  gration. The deuteron wavefunction in configuration representation is given in Machleidt [26] and the lambda part of the wavefunction was found from the Fourier transform of its momentum space representation thus, 2°°  A(y)  () G)  = =  j  2N(Q)i  N(QA)  =  jo(qy)A(q) 2 dqq  ()  (D.20)  qsin(qy)  j  /Q] 2 exp[-q  exp[_Qy2/4] 2  x— {cerfe(—  — —  cerfe(- +  )}  (D.21)  The last result is taken from Oberhettinger [44]. For values of y > 10.0 fin the following asymptotic form was used,  A(y)  =  N(QA)  ()  exp[  /Q] 2 exp[a  (D.22)  Appendix E  Three Body Basis and the Permutation Operators  E.1  Three Body Basis  In a non-relativistic theory, three particles each with  spin  -  and isospin  can be repre  sented by normalized states, (E.1)  iii s sis 23 ) 3 i 2  where k denotes the momentum of the particle labelled i and  are the spin and isospin  components. The centre of mass motion can be split off by making the following transformation, K  —  1 k + 2 3 k  -  (E.2)  =  where (a/3’y) is a cyclic permutation of (123). The above transformation is pertinent when all three particles have the same mass, aii approximation that will be made for j5*. the trinucleon systems. The pair coordinate is the momentum of particle in the CM. 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 K.R +  JLX+  which are reciuired to satisfy, 1 +2 k.r .r + 3 k .r k  169  (E.4)  Appendix E. Three Body Basis and the Permutation Operators  170  so that the plane wave states of three nucleons are invariant under the transformation which splits off the CM motion. By writing the equations (E.2) in matrix form and taking the inverse of the transpose of the transformation matrix, it was found that the required configuration space transformation is, 1  -*  R  =  =  r—r3  =  +  —  ).  (E.5)  The Pauli principle requires the trinucleon states be antisymmetric under interchange of aiy two particle labels. States ‘) satisfying this criterion are formed by permuting partially symmetrized states ) 23 which satisfy, b P23H!’23)  =  (E.6)  I’23)  The operator P 23 interchanges particle labels 2 and 3. The state I II’) defined by, +3 2 )=(1+P 2 ) 3 P  (E.7)  is antisymmetric under interchange of any two particle labels. The operators P 2 and P 3 are permutation operators and are defined in section E2 where the total antisymmetry of  IIi)  is also proved. The states  &23)  are expressed as sums of chaiine1 states a which  in the LS coupling scheme are defined by, (xy23)  =  (x,  y)(  (E.8)  a) M  (:a)  where,  l  =  =  orbital ailgular momentum of pair in their CM  =  orbital angular momentum of spectator  (E.9)  Appendix E. Three Body Basis and the Permutation Operators  La  =  total orbital angular momentum  =  spin of pair  171  total spin  Sa =  total angular momentum  =  total angular momentum projection  =  isospin of pair  ‘a  =  total isospin  Mia  =  total isospin projection  MJQ  (E.1O)  The symbol ® indicates the coupling of La to Sc. to form total angular momentum , with projection 0 J  The function Y and the vectors X and  L() M y  (LMi lc.mi;  are defined below.  ()Y() 1  (E.11)  n m  xs:r5(1)  8 sm (ScyM ; 5  =  I 3 mi)(sc.m  ; m 2 rn ) 3  yrnlyrnsyrn3  (E.12)  7775  rn 2 m 3  i’v’  rni)(ic.mj rn 2 ; m 2 ) 3  771772173  (E.13)  rn 2 m 3  Note that for the spin and isospin vectors, particles 2 and 3 are first coupled to some I and then the third particle is coupled according to s 0 The requirement that  ‘23)  and not  0s  be antisymmetric under interchange of particle labels 2  and 3 is satisfied by including only channels for which (_1)15  =  —1. Further, the  requirement of positive parity for the 3 He and 3 H states is satisfied by demanding that a 1  +  ‘a  =  even.  Appendix E. Three Body Basis and the Permutation Operators  E.2  172  The Permutation Operators  The permutation operators P 2 and P 3 are defined by their action on three ordered objects thus, (.o*) 2 P  (o*.)  (.o*) 3 P  =  (*.o).  (E.14)  Along with the unit operator defined below these cyclic permutations form an abelian  group G according to the group product table E.2. 1(.o*)=(.o*)  (E.15)  The operators P 2 and P 3 can be written in terms of the non-cyclic permutation P 23 and P 12 thus, 3 P where  interchanges the objects in positions i and  that the states  matrix  )  elements  of  =  j.  13 2 P 2  (E.16)  This identification is used to prove  2 and P P 3 between the partially antisymmetrized channel  are equal as shown below. c’) 3 (cP  =  23 P 12 (cjP c’)  (E.17)  =  23 (cP 2 2 P cV) 3  (E.18)  =  12 P 23 (—1)(—1)(P ’)  (E.19)  =  c’) 2 (cP  (E.20)  1 3 2 P 1 1 2’3 3 2 P 1• 3 1 P 2  Table E.1: Group Product Table for the Group G.  Appendix E. Three Body Basis and the Permutation Operators  where the trivial property (P) 2  =  173  1 is used as well as the fact that a) is antisymmetric  under interchange of particle labels 2 and 3. The total antisyminetry of the vector ‘), which was introduced in equation (E.7), is proved as follows. There are three choices of particle labels which may be interchanged. They correspond to the actions of ,P 12 and . P 23 31 Consider first . P 23 P =  2323 P )  +3 2 (1--P 2 ) 3 P  23 P ’P)  =  +23)  (E.21)  =  23 + P (P 22 1 3 + 23 )R& 1 2 P ) 2 3  (E.22)  =  23 + 2 (P 13 P 2+  (E.23)  =  23 + 3 (—)(l+P ) 2 ) Ii P  (E.24)  —HP)  (E.25)  )I 1 2 P ) 23 2 3  The action of P 12 on ‘I’) is found similarly. To find the action of P 31 on ‘) it is useful to make the identification P 31  =  23 Using the group product table it is found . 3 P  that, ( P ) 3 + 2 ) =(1+P 1+P P  (E.26)  so we have, 31 P i) This concludes the proof that  =  23 P I’)  —‘I’)  =  (E.27)  Ii) is antisymmetric under interchange of any two  particle 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 to  the following transformation. 1 2 p  3 P  1 —  p t 2 p  (E.28)  Appendix E. Three Body Basis and the Permutation Operators  174  The above identification follows trivially after finding the irreducible representation of C. Since C is abelian, Schur’s first lemma implies that the irreducible representations are one dimensional. Defining a basis by,  (. o *)  (o *.)  1  24 e  (* • o)  (E.29)  =  (E.30)  the irreducible representations are,  1  =  2 P  1  =  e23  3 P  The tilde indicates that the quantity is a representation and not an operator.  The  irreducible representation of P is given by, =  and  the  (t  correspondence given  (e2/3)* = e_i23 =  _  in  equation  =  3 P  (E.31)  (E.28) follows from the equality of the irre  ducible representations of P and P 3 etc. 3 can be verified using the definition of the hermitian adjoint and P  The equality P  calculating a matrix element between two three-particle states which are simple tensor products of three orthonormal single particle states 0,1 and 2. The definition of the herniitian adjoint is, (BPA) Consider the state I and particle  *  )  =  B)* 2 (AP  V A), B).  (E.32)  with particle labelled • in the state 0, particle o in the state  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 ket indicates the state.  Appendix E. Three Body Basis and the Permutation Operators  ( 012012 )  Using the normalization of the single particle states  175  1.  Taking the  permutation operators to act on the particle labels, we have 1  012) 2 (201P (012P201)  t20l) 2 P  =  =  1*  =  012)  . 3 i4=p  1  (E.34)  Appendix F  Expansion of a Function in the Gaussian Basis  We wish to express a smooth function f(x), fast as i/x  ),  (  which tends to zero as x  —*  oc at least as  as a sum of Gaussian functions of range x. where the x, can be chosen to  be any values thus, flmax  f(x)  exp[—(x/x 1 Aflx J 2 ) 7 .  =  (F.1)  fl= 1  The 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/x ] and integrating one has, 2 j 7 nia  j  dx f(x) exp[—(x/x 1 2 ) 7  7 A  = n= 1  1  where,  dx x 12 n) 7 exp[—(x/x ]  (F.2)  1 1 —i-- + —  =  ——  j  X  (F.3)  X  This can be written as a matrix equation thus, F  XA  where,  (- ),  =  =  (X)F  )fl/rn  =  1 1 f(l r)(xm i ) 1+1 + 7  , 7 (A)  =  , 7 A  ,  =  A  (f  (F) =  j  (F.4)  dx f(x) exp[-(x/x ] 2 ) 77  (F.5)  and the problem of finding the expansion coefficients A has been reduced to tile mversion of a real symmetric matrix of dimension  umax  and also tile evaluation of  umax  integrals  involving the function f(x). Tile IMSL library routine DLASF was used for matrix inversiOn ill tile application of  the above method to the hypertriton wavefunction. 176  Appendix G  Matrix Elements  To calculate the rate for muon capture by 3 He to the 3 H final state, three reduced matrix elements need to be evaluated. They are, [11°  =  H 3 (  =  HW 3 (  =  HU 3 (  [jO1 []21  He) 3 Ijo(vr)  (0.1) (v 2 Ij  ®]1  He)/v.  The reduced matrix elements are reduced in the spin-angular co-ordinates but not in  the  isospin  space because the wavefunctions which were used were not exact isospin  partners. The  are the spatial co-ordinates of nucleon  thus functions of the Jacobi co-ordinates £ and B, iE and  j  j  in the three body CM and are  and not R. Equation (B.5) defines  by, D  1  1  1  0  1  —l  1 r  (0.2)  .  By matrix inversion we have, l0  1  r2  =  =  expressions for  —ii, (2, Y2)  y  and  (, y)  in  1  1  3 x  =  (G.3)  —  1  73  2 and since x  1  I  =  terms of 177  -*  hi  , 2 —r  (, )  =  y3  = —1+2),  (, )  the following  are easily derived.  178  Appendix G. Matrix Elements  =  X—Y 2 — 3—  Y2  =  3 X  =  1—  __  The operator in the coupled to the  []21  (04)  reduced matrix element has the spherical harmonic Y () 2  spin operator 3 j  to give total angular momentum 1. The precise definition  is,  (i) 0 2 {Y  (1M 2 )S m 2 rni; 1mg) Y  =  (0.5)  fl2q7fl  where,  S =  +(a+iu).  (0.6)  Before deriving the form of the matrix elements, an expression for the norm of the wavefunctions will be expounded.  The matrix elements are then obtained by minor  modifications of the expressions for the norm.  G.1  The Norm  The full  wavefunction  ‘4’  )  is  the sum of a Faddeev component ‘/) and its permutations. (G.7)  The norm is, (0.8)  Appendix G. Matrix Elements  179  , P 3 It is possible to simplify this expression using P = P  2 and (1 + P P 2+P 2 = ) 3  :3(1 + P 2+P ) but this will not be done in preparation for the matrix element calculations. 3 In that case an operator sits between the permutations and does not commute with ,P 2 P . The expression (G.8) is partially simplified however by using the properties of 3 the permutation operators. = (b1 +4P 3 +2PP 3 +2PP Rb) 3  (G.9)  This simplification will not quite be possible in the case of the matrix elements but the expression (G.9) is general enough to explain all the techniques needed. There are three  parts;  DIRECT, PERMUTED and DOUBLY-PERMUTED. DIRECT = PERMUTED  (G.1O)  = 4(iP Ib) 3  (G.1l)  DOUBLY-PERMUTED = 3 + 2IPP I /’) PP  (G.12)  A useful check on the calculation of the doubly-permuted parts was to verify numer i cally the following identities. ) 3 KPP  =  ()  (G.13)  ) 3 (PP  =  ) 3 (P  (G.14)  DIRECT The Faddeev component is expanded into channels as described in chapter 4, equations (4.4-4.8). M  ()  =  x,y) [Y()®x (i)] 1  >  Cy)(  (i  a)La,sS;  1  I(1)  ) jc)  (G.15) (G.16)  Appendix C. Matrix Elements  The  180  (l\)La, sSa; M) will be a useful shorthand for the spin-angular  notation  part 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)  =  (G.17)  AN(x)(y) nN  q(x)  1 NiaX  [()  1+3/2  2 F(l + 3/2)]  2  =  exp(—x / 2 x)  qj(y)  =  2  NN =  ( (la)L, sS;  =  ca’  [() 2  Expanding both bra and Ret,  ( I)  /y) 2 NNy exp(—y 2 f( + 3/2)]  )L sa’S’; A 1 (l ,  (G.18) 2  I  n]Vn’]V’  f  dx dy x 2y (x), (x)(y),(y) 2  (0.20)  The spin-angular and isospin part of this expression yield a factor 6(, ‘) so the direct part of the norm only receives contributions when the channel in the bra matches the chaimel in the Ret.  (/ /‘)  A ANf nNn’N’  (  2xxflt X  +  la+  1 7 X  2ynyn’  y+  (0.21)  The total number of contributions is 405,000 for the 8-channel wavefunction and 545,000 for the 22-channel wavefunction. PERMUTED The permuted part is the overlap of the Faddeev component /‘) with its permutation  3 ‘/‘). The spatial P  ) 3 (IP  representation  =  where the property P  of the  (X2, y2)  =  permutation is,  2 [@  y2)  (3)] 1 0X  ‘(3)  (0.22)  2 has been used. The spin and isospill vectors are symmetrized P  with respect to interchange of particle labels 1 and 2 as indicated by the label (3).  Appendix C. Matrix Elements  181  The spin vector X’(3) is expanded in terms of the basis X(1) as shown below. =  S(s,  ,  S)  XMs(l)  (0.23)  Sr S  =  Si ( 3 s, s,S)  where,  XMt (1 )X(3)  (0.24) 11  =  (—1)[(2s + 1)(2s + 1)}  2  j  1 2  Sc  So  J  (G.25)  The isospin vector is expanded similarly and having the same SU(2) algebra the result for  ‘13  follows immediately.  13 ‘(1) I ‘i’(3) = (i,i)  (G.26)  {  13 i) = t 1 (i, (—fl { (2i + 1)(2i + 1)]  where,  1  1  1  (0.27)  11.1 2  )  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  /xh11yl22  Yj(2,  i/2)  R(l i A 1 )  =  11  where,  11 1) R(l  +  12  +  ‘2  =  1J 2 X  N  —  ‘\2.  LM  (0.28)  Y (,l;)  ,‘  =  (—1  )Ai  I h1+2()’\1(i0  (2)  (\  is set to  (21)! (2A)! (2l + 1 )(2 + 1) (2l )!(212)! (2 )!(22)!]  [  110;  o)(o  120; 20)  {  lI  12  ly  1 A  2 A  A L  J  I 2  (0.29)  Appen clix C. Matrix Elements  182  This expansion is found using some standard recoupling theory  (  see [34]  )  and also the  following two identities. 4 (21 + 1)’ 2 [(2l + i)!(21  2 1 l1b  Yi( b) -  =  2)  y@)  [(2li+i2i+ 1)]  —*  (G.30)  2  (laO I hO; iO)  Identity 1) follows from partial wave expanding x  2  =  (G.31)  eieib and regarding the  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 x , 2  Y2  has a simple  form and this feature exemplifies the beauty of the Kamimura wavefunctions. =  (G.32)  )(y AN(x ) 2 nN  a  12 2 2 NX1 exp[—(x 2 )/x j ] + y + x.y 7 —i--  (Y2)  =  Nniy 2 exp[—(x + y 2  )/y)]  —  (G.33)  Notice that tile unpleasant xy dependence cancels with the xmQya dependence of  , 2 yM(  Y2)  as shown in equation (G.28). This is a consequence of the basis functions  q’4(x) having sensible x  —*  0 behaviour. There is an angular dependence on  and  coming from 2 )q(y which is partial wave expanded as shown below. (x ) exp(2c)  where,  =  ik(2cxy)(_1)k[2k+ 1]4y()  ik(Z) =  jk(iz)/(i)k  2c  =  —  x  (0.34) (G.35)  + YN  Expressing tile permuted Faddeev component as a sum over the dummy index a’ rather than a, tile PERMUTED part of the norm is, 4( k  Ii) 3 IP  =  4  ANAjNIN?1Nflh1iNNANNIA, aa’ nIVn’IV’  X  Appendix 0. Matrix Elements  I  + +11 2 x  183  12+\2  2 1y  k  [x (_i_ +  3 1  Jdxdy  I  k  exp  I  L  2(i+i+9)2(i+i+ \x YN  N1(X  jj  J  l + l = lal +A = Ra’l 21  1 ‘)\al)S13(5, 1Y  so’, Sa’)I (i, ia’) 13  (_1)k[2k + 1]( (l\a)La, saSa;  4irY@,  X  )( i  (la’i)La’, sSa’;  i). (G.36)  The spin-angular matrix element which needs to be calculated is thus, (_l)k[2k + 1]( (1  a)La,saSa;  I(kllaa’La) where,  I(klalaALa) =  (la’’)Lai,sSa’;  =  (G.37)  6(s, Sa)(Sa, Sa’)6(La, La’)  X  k 2 (_l)k+L(  (G.38)  + 1)[(21a + 1)(2a + 1)]  (laO kO; iO)(AO kO; O) The isospin part yields a simple  )  {  la  l  X  ,.(G.39)  L  J  i). The final task is to evaluate a double integral  H with the following form which follows from equation (0.36) after a scale change of variables. (k, m, ri; c) 1 H  =  f  dx dy  e x2+2rny2+2fl+ke_s i 2 k(2cxy) _Y  m, n  = 0, 1,2... (G.40)  The integral is performed by expanding the bessel function [147, —  ik(2cxy) —  In the above (k +  ,  /7r  (cxy,  2 F(k +  )  1 (k +  1’ 2 (cxy)  p348]. (0.41)  is the Pochammer symbol defined by, F(a +p) F(a)  (0.42)  Appendix U. Matrix Elements  184  Performing the gaussian integrals yields F functions and so, F(m+k+)F(n+k+) Fi(m+ k + ,n+ k+ ;k+ ;c 2 Ck ) 2 8 F(k+) ck ir(2rn + 2k + 1)!!(2n + 2k + 1)!! F(—m, —n; k + ; c) (043) 2 4 2 +,n+n+k (2k + 1)!! (1 2)rn+m+k+  (k, rn, n; c) 1 11  =  —  The last equality follows by using F(n +  )  /ir(2n + 1)!!/2’ and also a linear  =  transformation for the hypergeometric function [147, p143] which yields a finite (Jacobi) polynomial. The final expression for the permuted part is, H/)) 3 4(/P  =  4 (La,Lai)6(Sa,Sa’)  X  t a  x  N’  G’ (kl ) )H 1 (k, n, n;  cp)  ,n+k+3 3 2m+k+3,  (G44)  k l +l  =  la’  A + ,\  where,  ,‘a a’  (klA)  I(klaAala’Aa’L a)lRa’1lIA/l 21 J ) x  = lot ‘‘a  313(Sa, ba’,  2m  =  la+l+Aik  2n  =  ‘a  + 12 +  1  1 =  1  Cp Cp  k  —  2y  (cp<1) (0.46)  1  9  x2+4x2+16y2  1  1  =  YN  x,  =  1 3 ----+—x, YN’ 4  1  (0.45)  (ia, ia’) 13 Sa’)1  1 4y, (0.47)  Appendix G. Matrix Elements  The coefficients G’, S 13 and  185  ‘13  were calculated separately using routines for Clebsch  Gordan, 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 and 29,985,000 for the 22-channel wavefunction. DOUBLY-PERMUTED In principle, the calculation of the doubly-permuted parts is no more complicated than the calculation of the singly permuted parts. This is because of tile gaussian nature of tile basis functions (x) and qSj(y). The unpleasaiit  =1  —  and  —  =  Y2  dependence arising from  in both bra and ket is combined simply and partial  —  wave expanded as in equation (G.41). ) exp(2c 1 exp(2c x.y) 2 Tile filIal result for  tile  1 +c exp(2(c )) 2  exp(2c)  (G.48)  doubly-permuted parts of the norm can be written down  directly. 3 + PP 2(l/PP ) 3  2(La,Lci)6(Sa,Sci)  =  X  ANANlN?1aNflh1iNNNNlXi X nArn’ f\J’ 33 C  dDp)  2m+k+3 2n+k+3 anp IDP  (k,rn,n; , 2 23 dDp) c + G)(kllilA)H 11 + 12 = 1  +  2  1! 11  +  2  =  ) + )  =  where,  1!  = 1  G  I(ki 10 _)0 1 cy -‘‘  i 11 R(l  )R’(l  ) 1 1Y  x  (.)  Appendix C. Matrix Elements  186  x S  ) 1 Ii3(, ia)Ji (, i 3 23 Li  =  >  -  (0.50)  L) R(l A 1 I(kii l A 1  a tl \  )R(l  ) x 1 iJ  C1  X S  12 i)Ii 1 (i, (i, i 3 ) 1  (0.51)  2m = l ++l+)—k 1 2n  33 c  +,\ l + + — 2 ) k l  1  1  2 4x 1  1  = 1  = 72  1 2  4x,  9  +  1  +  2  YN 6 ‘  1  273 DP / DF 3  273  (0.52)  9 16y,  1  224242 xnl  1  1  +  23 C =  DP 3 DP/  2 7 X  1  1  3  3  1  3  3  —---+-—-+—— 1 x 4yr’ YN 4 (0.53)  =  The cIefiiition and value of (s, 12 s, S) are, S 12 aj S) S (s,  xSMt(l) XSOM( ) 2 Sar /  ( — 1)$ 13S,Sa,Sa). The symbol R(l i) arises from the expansion of A 1 (—1)  1+12  ) , 3 yM(  (0.54) (0.55) and is equal to  1 i ). R (11 A  The total number of contributions is 35,336,250 for the 8-channel wavefunction and 190,931,250 for the 22-channel wavefunction. The evaluation of the norm for the 22channel wavefunction required 20 hours of CPU time on a VAXstation 3100-M76/SPX.  Appendix C. Matrix Elements  G.2  187  The Non Spin-Flip Matrix Element [lj°  There are two extra features in the matrix element [i]° as compared to the norm. They are 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 the sum of  three operators is proved. The sum of three operators 0  0 + 02 + 03  can  be  written,  O  1 O  P+ P 1 O OP 3 +2  where the identification of 02 = P 0P has been made to ensure that (1 2  (G.56)  IOiI1)  =  2 I02 2). By noting that, (l+P ) + ) ) ) 3 ( 1 + 2 ) =3(l+P (l+P 1+P (O O O P  (0.57)  we see that the matrix element of 01 + 02 + 02 between symmetrized states is three times the matrix element of 0 between symmetrized states. Thus, [110 = 3 H3Ijo(iiri) ( H 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 of the norni. The bessel function  jo(vr)  = jo(2iy/3) modifies the double integrals over x and y as  shown below. The direct part integral is straightforward, the permuted part integral is not. a)  Direct Part.  In this  case,  the integral  f dy y2+2e  /YNl 2  becomes  /vNljO(2I,y/3) 2 f dy y2+2e_Y  Appendix G. Matrix Elements  where  YNN’  188  is defined by, 2  YNN’  =  (0.60)  + YN  i/N’  The latter iitegral is performed by expanding the Bessel function using equation (0.41) and using Kummer’s formula [147, p 98j to reduce the confluent hypergeometric 2 series F 1 to a  finite  polynomial.  N1jo(2vy/3) 2 J dy y2+2e_Y > e’ + 2 f° dy y  —  exp  ) 2 (—p  NN’  where,  p  (—A; ; F 1  (0 61)  ) 2 p  (G.62)  = 1JYNNI/: 3  a) Permuted Part. In this case the integral which needs to be performed is H . 2 (k, n, 2 H  n;  c, d) =  dx dy  j  0 e x2+2flky2+2ke_X i 2 ( (2cxy)j 2dy) _Y  (0.63)  where d and c are real and the magnitude of c is less than one. The fact that c C  Cp, , 33 c  33 c  can be shown by expanding  4_2_2 —  <  1 for  y and noting that only positive  terms remain. Expanding both  k  and  0 j  using equation (G.34) yields a form for H 2 containing a  generalized hypergeometric fullction of two variables (k,m,n;c,d) 2 H =  r(2m + 2k + 1)!!(2n + 2k + 1)!! 4+m+n+k (2k+1)!! 2 (n + k + ; m + k + ; k + 1 c  where,  ,  1 Ji  (0.64)  ()pq()pxPy  (0.65)  ;x,y) 2 i(a;;7i, p,q=O ()p(72)q  The fuiction  c2 , —d ) ;2 p. q.  is the confluent form of the Appell series F 2 [1481,[147, p192] and  has been discussed briefly by Humbert [149]. The complicating aspect of  ‘IIi  is the mixed  Pochammer 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 in  the field of generalized hypergeometric series.  Appendix C. Matrix Elements  189  Theorem m  + m; i, 72; , x, y) = (1  ()  x  —  (y 1  p=o  XI  —  iF@ + 7);  i  —  (0.66) where,  Ix<  rn=0,1,2... and,  1, Re( ) >0, 71  (x) F 1 ; 7 a;  mC  = q=O (7)q  p!(m  —  p)!  (0.67) (0.68)  ¶7!  Proof Need, i)  (—rn)  =  ii)  1 (1_x)a  =  ( +p + q)  iii)  iv)  —  (—1)  rn!  (rn—p) x< 1  (c)p(c+p+r)g  7’  (0.69)  (c)p(c7+p)q  —  Fj(a,;7; 1) 2  Re(7  — —  —  v)  ()p(  =  + p)q  )  >  0  (a)pq  The identities i) ii) iii) and v) are easily proved using the definition of the Pochammer symbol, equation (0.42). The identity iv) is proved in Wang and Guo [147, pl 1. Let 56 us work on the right hand side (R.H.S.) of equation (0.66) by expanding the confluent hypergeometric function and expressing 7 C in terms of a Pochammer symbol. /  00  R.H.S.  (7)  p=Oq=O  Now expand l/(1  (—x)°  =  —  x)+P+  (—1  = pO q,rO  Now use iii) to express  (+p+ 00  R.H.S.  (y  1  (0.70)  q! (1  using ii).  m R.H.S.  p!  ( + p)q yq  = p=op,r=o  ((—m) p+r p!r! (71)p  +p + q) r  + p)q  y  (0.71)  q).  ()p+r( + p + (71)p(72)q  r)q  y p!r!q!  (0.72)  Appendix G. Matrix Elements  190  Consider the term with q fixed i.e. write,  q=O  m  then,  (—m)P@)P+r@  Tq =  (71)p  p=Or=O  yq  1  R.H.S. = Tq  (0.73)  (72)qq!  + p + r)q p+r p!r!  3 (a) A ( 3 s=O  (0.74) )P(Th1)P 1 (_  3 A  where,  s!  = 2 (—m, —s; 7i; 1). 1 F  (‘Yi)p p!(s —p).  (‘ii)  0  (0.75) Applying iv) and checking that Re( 71 + m + s) > 0 we have, (—rn, —s;7; 1) 1 F 2  and so replacing the dummy index  s  +  +  3 Tn)  7n) x  (0.76)  q 1  (G.77)  p! q!  (71)p(72)q  p,q=O  =  (71  by p and using v), (a)p+q(71  R.H.S. =  =  1 2 ‘I1i(a;7 , 1 +m;7 ; x,y). -y  QED  (0.78)  The integral H 2 can therefore be reduced to a finite polynomial. ir(2m + 2k + 1)!!(2n + 2k + 1)!! x (2k + 1)!! 24+rn+n+k  (k,rn,n;c,d) 2 H  (G.79)  3’  (1  _c2)7  mC 2  (m+ I/ (k+)  iFi(n+k++p;;  _C2)  lr(27n + 2k + 1)!!(2n + 2k + 1)!! 4+m+n+k (2k + 1)!! 2 ck  /(l 2 exp[—d  (1  —  1  —  c2)7  )j 2 c  m  3’  mCp 0  + k + r)  (k +  /  (\1  c  1  _2)  (0.80)  x  p  2  —  2 —d  C2)  (—n F 1  —  k  2 d —  1  —  2)  where Kummer’s formula has again been used. The final expressions for [110 are, [1}°  =  2 + 1 )O 3 P ( 1 +F 2 + P3)3He) (‘3HW(1 + P  (0.81)  Appen clix C. Matrix Elements  (3HWO  =  191  3He)  [K  +2  W3He) + 3 OP  HeO 3 ( H)] 33  +2( 3HW10l13 + 3 I/) 14OlP H e)  (G.82)  DIRECT(l) + PERMUTED(l) + DOUBLY-PERMUTED( ) 1  DIRECT(l)  =  Oi  (3HHOl3He)  (3—4i)x  =  H)A,N,( AN( H 3 e)  (  =  3Ijjo(iiri)  ’ x 7 2x  +  nI’/n’IV’  )1a+  ,  2 )Fi(—; 2 exp(—p , p) .  PERMUTED(l)  2 =  [(‘3H  2  3He) + 3 OlP  (G.83)  (  (G.84)  2  +  2 yn,!  (G.85)  (G.86)  (3HeOP3W’3H)]  6(Lc, La’)(Sc, Scx’)Si (c, a’,Sc)I 3 (ic, i’)(3 3  —  4ic)  X  ca’  [A (H)AN, ( He) + AN(He)AN, 711 3 H)]N , 3 ( 7111 NNNN1, X N nNnl Al’  G(kl)H ( 2 k,7n,n;c,dp)cp2m+k+3  2n+k+3  (G.87)  k 11  +  )4  +  7/  12 =  = Ac,’  DOUBLYPERMUTED(l)  = =  nNn’N’  2  [(  U/’3He) + 1 0 3 P  28(Lc,,Lc,’)6(Sc,,Sa’) X c,c , 1 Nc, NNIA,  (3H  X  iRb3He)] 3 POlP  Appendix G. Matrix Elements  F  192  1  k,rn,n;c )(kllllil)H 1 ( 33 ( 2 ,dDP)  2m+k+3 2n+k+ DP DP 3 /  (0.88)  k,rn,n;c 23 ( 2 [+G)(kllll)H dDp)j ‘1  + 12  —  2 + .A  1  =  ‘  I  + 12  = lcxl  I  + )  = ‘“cr’  d  where,  j) G(kl A 1  =  > a  = 3 p/ v/  I(klcycjaiArLa) R(l  lcy)’) x  t a  (i, ii)(3 13 •513(s, s, S)I ) l A 1 (kl 33  (0.89)  dDp = v/ Dp/3 3  —  4i)  (0.90)  lci)t.cyi) I(klc\cy1crJc’Lcr) R’2(li\ilcx\ )“P’l’ 2 1 )‘1  =  X  Ta a Tat  3 s, S)S (s, 1 S (s, 13  x  S  13 i)I 1 (i, (i, i 13 )(3 t l) 11 G’’ 23(i) (kl  —  (0.91)  4i)  \ C) R(l i \ J(klct\cslcti,\aiLa) 1  21  1  X  0 T, i  s,s,S)S 13 ( 2 Si (s,si,S) S  i, i i)Ii ( i, 3 I ( 2 )(3 t  —  4i)  (0.92)  There are exactly tile same number of contributions to each part of [110 as there are to  tile  G.3  corresponding part of tile norm.  The Spin-Flip Matrix Element  The presence of the spin changes the value  operator  []0I  in the matrix element [6]0t1 has two effects. Firstly, it  of each channel-channel contribution. Secondly, it increases the number  193  Appendix C. Matrix Elements  of channel-channel combinations which can contribute since the S, = S’ selection rule no  longer holds. The fundamental result that is needed is the reduced matrix element of  i  between the LS coupled channel states. ((la, )La, sS;  i  (lc,  )  c’Sa’;  )  = (la, la’)6(c, a’)6(L, L’)(c,  S(8aLaSSa’) =  where,  (  {( l)L+So+S_SI $ 2  + 1)(2Sa’ + 1)] X  1  I  8a  1  J L  S’  (0.93)  S Sa’  L  (G.94)  J  The convention that has been used for reduced matrix elements is that of Brink and Satchler [34] i.e. JM) = (J’M’ JM; KQ)(J’WTjJ) 1 (J’M’T where T 1 matrix  is  a tensor operator of rank K and projection  Q.  (j-) is the reduced  element of the rank one spin operator between two spin  The expressions for the various parts of []Oi  [5]01  (G.95)  states and equals \/.  are given below.  DIRECT’)’ + PERMUTED + DOUBLY-PERMUTED  DIRECT’  (/)3HO1  Oi  IhHe)  (0.96)  = Ijo(iii )i 1  (G.97) X  cyc’  (3 — 4ia)S(ScL,Sc,S,’) AN(H)A:N,(He) ,iNn’N’  X  (2x x 7 i!) ••+-  2 exp(—p 2 )iFi(—;r;p)  (  2YYi) —I— y,  2  x  (0.98)  Appendix G. Matrix Elements  (0,1)  PERMUTED()  —  —  2  194  [(  (G.99)  0lP3Wb3He) + (1i3HeOP33H)]  x  =  3 FA He) + 3 I nN H)A:N, (  71 NN NN’, N ,  X  nNn’N’ 2rn+k+3  (k,m,ri;c,dp) 2 G’(klçA’jH  2n+k+3 / p 3  (0.100)  k 1  + 12 = lf  A +A =  1 DOUBLY-PERMUTED°  2 [(?/)3HIIPOlP3Wl/’3He) + (h/’3HII1 0P3W1/’3He)] 4 =  ) x 1 26(L,L,  AN(H)A:N,(He)N?ll  NNNN’,  f\J’ N’  G’ (k,7n,n;c 2 AllAc)H ,dDp) 33(e) (ki 1 33 lA)H A 1 (kl ( k, rn, n; G’ 23(a) 2  k 11  l’l  dDp)  j  27n+k+3  DP  2n+k+3 IDP  (G.101)  la  + 12 2 +A  C23,  1  _  + l2 = lc’  A +A  =  where,  G’ (klcAc) ()  I(klaAala’A’Lcx) R’(lcAcic’Acr’)  =  X  cr a 1 1  a’ Sc,r)S(aLaS,ySoi) x 5  1 I ) 1 (i,i (3 3 C, 33 ()(k1IAI1IAI)  = a 1  —  4i)  I(kii \L) R(liAii 1 0 Ctl  , 0 :5c  (0.102) )R’(lcAciaii)  X  Appendix 0. Matrix Elements  195  Si3(S,a,Sa)S13(S,5a’,Sa’)S(SLaSaSa’) 13 a)Ii 1 (’i, (, a’)(3 3 , G(kl l 1 ,\) \ =  I(kl  —  X  4i)  (G.103)  la’Ac,’Lc,)  X  la)kalai\ai  12 s, Sy)S S (s, (5, so’, Sa’)S(LaSaSa’) 3  2 G.4  12(Z  (z, zi)(3 3 z,)I  —  X  4z)  (G.104)  []21  The Spin-Flip Matrix Element  For this matrix element we need the reduced matrix element, (y) 0 2 ((laa)La;saSa; [Y =  (la’&)La’;sa’Sa’;  6(la, la’)’(Sa, 2 (SaLaLa’SaSa’)RY Sa’)S ( laz\a)iLaL&)  (0.105)  where,  (i)2[3(2La + 1)(2La’ + 1)(2Sa + 1)(2Scx’ + 1)]  (aLaLa’Sc$ar) 2 S  { (laaa’LaLa’) 2 RY  =  (  }  Sa Sa’ 2 l)1a+LQ[(  La  La’  2  S  Sa’  1 1  + 1)(2Aa’ + 1)]  (20 )a0; )‘a’O)  X  I  ,\  X  2  1  (0.107)  .  I  La’  La  la  J  The spatial integral differs due the second order spherical Bessel functions which replaces the zeroth order spherical Bessel function appearing in  [1)0  and  [8]01.  For the DIRECT part, the y part of the integral is,  j  /y,)j2(2vy/3) 2 dy y ’ exp(—y 2 /ir(2\  y++Aal  =  +3)”  (0.108) e p ) 2 iF(—Aaai xp(—p + 1; ;p ) 2  Appendix G. Matrix Elements  196  where,  Ai =  a  + 2  (G.109)  The spatial integral for the permuted parts is H 3 with rn = 0, 1,2... and n ‘00  (k, 3 H  rn, n; c,  J  d)  dx dy  i(2cxy)j e x2+2ky2+2ke_X ( 2 2dy)  1,2,3... (0.110)  0  —  —  2 ckd  /( 1 2 exp[—d (1  —  —  c2)fl+k+  ir(2rn + 2k + 1)!!(2n + 2k + 3)!! x 3+L+n+k 15(2k + 1)!! 2 c2)P c 21 m (n+ k+ (k +  (0.111)  (  —  k—p+ 1;;  1  _2)  The final expressions for [121 are, []21  DIRECT’ + PERMUTED + DOUBLY-PERMUTED  1 ’ 2 DIRECT (77)  (v 2 j  =  (3HWO1U7!3He)  — —  , )(3 7 6(j 771 i  6(lc,  —  () 0 2 i)v[Y  ) 77 4i  (0.112)  (0.113)  X  (SaLc,La’Sc,S •$ ( 2 )RY 771 Lc,l) 77 1a\c,Acx’L X  2x,x,’ ANA:N,NNNN/,  2  nNn’ N’  + 3)!!  YNN’  [K /)3H  2  21 PERMUTED =  30  2  2  )  x  (0.114)  77 ) 2 exp(—p ( 1 —) iF + 1; ; p p2 i ) 2  (0.115)  OP3U’3He) + K3HeO3W3H)]  [AN(H)AN,(He) + AN(He)AN,(H)] x 7777’  X  (klA)H ( 3 k,m,n;c,dp)cp2’,n+k+3 G’ 3(,.21 k 11 + +  177’  2  =  ‘‘‘  (0.116)  197  Appendix C. Matrix Elements  1 ’ 2 DOUBLYPERMUTED (u)  [(  — —  2  =  2  b3H  !hHe) + 3 POIP  b3He)] 3 3HPOlP  AN(H)A:N,(He) x  cc’ nI’Jn’I\J’  (0.117)  NfllQN?’1,NNNN’, x  [+  k  l’l +  1 ,v  +  33 C  dDp)  1  I  2m+k+3 2n+k+3 aff, DP 3 /  1 l 11 (kl 2321 ( 3 )H k, 1 rn, n; , 23 dDP)] c  lc  11 + 12  +  l)(kllllAi)H 2 ( 33 ( 3 k m,n;  -2  l  =  lyl  =  where, l)(Lli\i) 2 ( 3 0  lc\L) R’(llc’)RY (1a’) cx’LcLc’) 2  I(klc  =  X  IQI  13(Sa, 5  ‘,  2(SaLLiSaSal) 9 Sa1). —  rac’ )l) 1 3321 (kl Li  (G.118)  4i)  I(klalcLc) R(liilc  =  X  )R(lAccy’)  X  TQ )\(1Ql  (l’\c’LaL’) x 2 RY (s, s, S)S 1 S (s, so’, 3  X  S  Ii(, a)Ii(i, )(3 1t c ,‘  \il)) 1 (1cl  =  I(1la  Tc  —  4i)  la#\La) R(liilc a)R’(l  (0.119) ii)  X  ,\  (lc’AAa’LaL’) 2 RY 12 9 . (S,  cy, 5  (5, 13 Sa)S  X  Sc’,  (sLoLcr’SoSa’) 2 Sa’)S  )<  S  I(i, i)Ii (i, ii)(3 3  —  4i)  (0.120)  198  Appendix G. Matrix Elements  G.5  The Form of (p, q)  In the calculation of the two body break-up momentum distribution p (d) the following 2 channel decomposition of the full wavefuiiction in momentum space representation is required. M  (pqIW3He;MJ)  =  (1)  (p,q) [Yh()ØX (l)] 1  (G.121)  In this section the momentum space representation of the Faddeev component is found and then an explicit form for I1a(p, q) in terms of the Faddeev components is presented. G.5.1  Momentum Space Representation  Given the relationship of the basis kets,  (Iz7)  =  (0.122)  exp(—i)exp(—i  and the expansion of the Faddeev component  ‘/‘  in channel states, Mj  =  (1) Mj  (7)  ii  (x,y) [Y@)®x!(1)]  =  (p,q)  (i)] 1 [Y()øx  (0.123).  ii  (0.124)  (1)  we have the fo1lowiig transformation formula for the channel wavefunction i(p,q)  (_1)1  ()  jdxdyx2y2ji(px)j(qy)i(x,y)  With the sum of gaussians form for  y)  (0.125)  the momentum space representation  b(p, q) has a very similar form to 7/’(x, y). q)  =  (—1)  /p) exp(—q 2 /q) 2 q exp(—p 1 p  (0.126)  nN  where, (2 =  )  3/2  , 7 p  2 F(l + 3/2)  =  2/x  qN  =  and,  /YN 2  2  (2 =  (0.127)  +3/2  2  2  f( + 3/2)  (G.128)  Appendix G. Matrix Elements  199  An Expression for (p, q)  G.5.2  The function Jia(p, q) has a direct and a permuted part. =  (pq(lak)L;saS; (1 + P 2 +P ),b) 3  (G.129)  =  L’g(p,q)+(p,q)  (G.130)  The function /‘(p,q) equals the function (p,q) given in equation (G.126). The permuted part will eventually be taken in an inner product with a wavefunction antisymmetric under interchange of particle labels 2 and 3 and so the identity “P 2 weakly equal to F ” can be made. The term “weakly equal to” is applicable when two operators 3 are equal only in a restricted sub-space, in this case the restriction being to the sub-space spanned by vectors antisymmetric under iiterchange of particles 2 and 3. 2(—l)  (p,q)  1 cr ()21  I  x  2  n]V  G’(kl) ph1+1ql2+2 exp(—p /N) 2  /)ik(pq/7) 2 exp(—y  (G.131)  1 + -i--  (G.132)  k 1!  )I’  + +  II  I —  ‘2  where,  1 —j--  nN 32  1 =  —  P 4  +  1  1  --  -—  q  nN 7  62+2  Pn  N  =  3 —  —  Pn 4  N  (G.133)  Appendix H  Deuteron Spectrum in the PWIA  In this appendix, equation (6.17) is derived and an expression for  p(q)  is found. The  starting point is Fermi’s golden rule and the safe approximation that the weak interaction is pointlike at the values of q 2 encountered is made so that the matrix element can be written in a current-current form. Normalizing the states to one per unit volume, the differential rate is given by, =  dF  CF2  L 2 C ( 2) W’ 2 3 d 1 V  (H.1)  The factor 2 underneath CF comes from the definitioii of the weak coupling constant and the factor 47r underneath the atomic wavefunction evaluated at zero separation comes from the use of the s-wave Bohr orbital i.e. (O)  =  p (i= 0). La is the lepton  current-current tensor. L  (;s7(1  =  );s) 5 —7  si ,sv  [iv +  —  (.v)g + i }/(mv) v 6  (H.2)  W is the hadronic current-current tensor combined with some of the phase space factors. =  (i J t  The currents J and  f)  J  +d (2)3 (236  -  P  -  q)(  fJI)  (11.3)  are the sum of three single particle currents from each of the  He. nucleons in 3 =  I(j)j(j) 200  (H.4)  Appendix H. Deuteron Spectrum in the PWIA  201  The fact that a proton in the initial state must become a neutron in the final state is taken 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 W can be written in terms of an analogous nucleonic tensor W thus,  W =  ()  (H.5)  (II) 2 fd3W  where the nucleonic tensor is defined by, W  =  (p,spjfl,Sn) S  and  p2  Sp  f  (fl 6 3 ) 2 (  — q)(n,S jp,sp). 1  —p  (11.6)  is the two body break-up momentum distribution.  The spin projections of the struck proton and ejected neutron have been written as  and s, respectively. Equation (H.5) has an appealing form. It states that the total nuclear tensor is that of a single proton averaged over the range of proton momenta found in the nucleus weighted by the probability of finding a proton-deuteron with the relevant relative momentum. The proof of this equation follows. The plane wave form for the final state is,  j  where,  )  +P 2 (l+P ) 3  ( n; s,  =  (  dJd  —  (263(*  —q  (H.8)  (p)[(j3) ØX j. 1  (J7d;Jd)(l)  (11.9)  2  1=0,2  The permutation operators which affect antisymmetrization of the final state are written 2 and P P 3 and  is the neutron momentum in the neutron-deuteron CM,  =  (— -d).  The correct normalization of the final state, i.e.  ( .1’ f)  (i* 6 3 (27r)  —  *)6(si s j6(J, 1  Jd)  (H.10)  in the PWIA  Appendix H. Deuteron Spectrum  202  follows upon the assumption of incoherence. This assumption makes the approximation that 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 body current in the three body Hilbert space is needed. To this end the decoupled spin/isospin basis  {,  y} is introduced. The set of discrete quantum numbers  €  describes the spectator  particle and the set -y describes the pair. ,i 1 {s } 1  —  (H.11)  ,, 3 in 23 rn} i  =  where .s and i are the spin and isospin projections of particle 1,  and i 23 are the  total spin and isospin of particles 2 and 3 and rn 3 and rn are the total spin and isospin projections of particles 2 and 3. The one-body current has the following representation in the the three body Hilbert space.  (p, q,  •a  €,  -I  p ,q -  I  I  , € ,  )  c  -I  =  The current imparts three momentum  ,  q  Q  )3 (p  -  -*1 p )63 (q  -.1  —  —  —  q  —  / 2/3Q)(7,7)  (H.12)  to nucleon 1 and does not change the momenta  of the other two nucleons. The spin and isospin of nucleons 2 and 3 are not changed which explains the  ‘,“)  factor. The factor  of a nucleon with momentum with momentum  7’  =  +  Q,  j,(7, q)  is the current for the transition  spin projection .s, and isospin projection z, to a nucleon spin projection s and isospin projection i.  Using the antisymmetry of the initial state and the properties of the permutation operators it is found that,  (fJi) =  ; dJd J) + J(2) + J 1 (‘)( ns  3He)  (H.13)  The assumption of incoherence is applied again this time to the overlap of the neutron : H e wavefunction. i.e. with the 3  (j)(flhlJJ3He)  =  0. This corresponds to neglecting the  Appendix H. Deuteron Spectrum in the PWIA  203  direct cleuteron knock out process.  (f IJ i)  d; J Ijl)I  = (1)(  The current thus decouples into a proton  3He)  (H.14)  neutron one-body current and a deuteron  —f  helion overlap. +  jd3p(2:3(+  (fii)  )  ja(q)  ISp  -  (H.15)  where, (iJ 1m ; 1  =  I  ) 8 un  0,2 1 71l  =  {l,ms,0,0}W3He;Ji)  (H.16)  The hadronic tensor is now easily evaluated and found to be, W  fd3pP()*Q)  =  J Ji 811 Sp8  (p,q)6(EP + 88 (p,q)j 83 xj  =  -  dsP()Id3P()*  Performing the sum over J and Jd on zero unless  Q°  )/(2) 1 E 3  (H.17)  yields a quantity which is  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-body break-UI) momentum distribution  p2(p)  in the standard way, JiJis()  12  (11.18)  (p)W 2 fd3pp  (H.19)  P2(P) P ‘d 8  we have, W =  ()  Appendix H. Deuteron Spectrum in the PWIA  204  where the nucleonic current-current tensor is given by equation (H.6). Extracting the energy conserving delta function from W/ to define the matrix element squared by,  ()  2 ( JLW =M 2 ) 6(Efl 3  -  Q°)  the expression in equation (H.1) reads, 7 J p2dp(E  dF  +  Q°  —  )p(p) M 1 E (p,Q). 2  (H.2fl  This is ow written in terms of d, x and Q(d, x) using, ii  d i 3 i 47r  i1  J—=-]  Q  =  1 2  dx]  —1  dQ 2 Q  0  p=d E 11 E  The integral over  Q  is  =  M3Re_(d2+md2)  =  2 2+d (Q  =  rn,  —  —  2dQx + mi2)  Q.  (H.22)  eliminated using the following identity for the delta function,  6(E + Q°  -  Performing the trivial integration over =  7 ( 2 p d)d  E)  Q,  f,  =  8([Q()).  (H.23)  equation (6.17) is arrived at.  dx  (Q-dx)  M  2  (d, x)  (H.24)  Appendix I  The Nucleonic Weak Current and Nucleon Form Factors  1.1  General Introduction  The charge changing weak interaction current of quarks and leptons has a pure vector minus axial vector (V-A) form. For example, the muon to muon-neutrino current is, (iii,  IJwK)  ü(v)(1  =  —  (1.1)  )u() 5  where the coupling constant GF/\/2 has been omitted. The nucleonic current does not have the  form because nucleons are composite objects and thus their total  same simple  current is a matrix element of the sum over currents from the constituent particles. The nucleonic current can, however, be conveiiently pararneterized in terms of form factors by writing it in terms of independent quantities with known Lorentz properties, •a,i3  a (n J< p)  gv7 =  a  u(n)  a  q 1  LIVIN a  —  (gA  where,  gM  --  qa  pa  =  a  +  +  u(p).  (1.2)  2 gp  pa  =  4-rnomeiitum of neutron  =  4-momentum of proton  —  75  =  3 2 l 0 7 j  =  (1.3) 205  Appen clix I, The Nuc]eonic Weak Current and Nucleon Form Factors  gv  =  vector form factor  g  =  weak magnetic form factor  gs  =  induced scalar form factor  g  =  axial form factor  206  induced tensor form factor  g gp  =  pseudoscalar form factor  (1.4)  In equation (1.2) q is the 4-momentum transfer to the nucleon and the form factors are Lorentz scalars. In general, the th are functions of the scalars 2 2 but ,p and n q when the nucleons are on-shell  (  i.e. p 2  =  =  M  )  this simplifies to the familiar q 2  depence. The current splits into a vector and an axial vector part according to the properties under improper Lorentz transformations  (  i.e. those transformations containing an inver  sion of the space axes or “parity” transformation  ).  The vector part has parity +1 and  is given by, (‘  Jw,v)  =  (n) [gv7a + gMi 2  + gs 2 u(p).  (1.5)  The axial vector part has parity —1 and is given by,  (“AL  WIK,AI)  =  ü(n) [gA7a75 +  +  2 gTi  5]  u(p)  (1.6)  The fact that both vector and axial-vector currents contribute implies parity viola tion,a feature peculiar to the weak interaction. The current may be further categorized into first and second class parts using a clas  sification due to Weinberg [150]. The current is said to be first (second) class if under a G-parity x parity transformation it is even (odd). The G-parity transformation is a rotation of 1800 in weak isospin space combined with the charge conjugation operation  Appendix I. The Nucleonic Weak Current and Nucleon Form Factors  207  ). The second class currents correspond to the scalar and tensor form 2 C, G=Cexp(i7r1 factors and standard practice is to set these form factors to zero. At low q 2 their contri bution to the current is small and so this may be a reasonable approximation. Another approach is to hypothesize that the weak current is invariant under GP, where P is the parity transformation. In that case only one sign of GP can contribute to the current and first class currents are kept in favour of secoid class currents by the properties of neutron beta decay. To have complete knowledge of the current then, it suffices to know the values of the form factors gv,gM,gA and gp. 1.2  The Vector Form Factors  The isotriplet vector current hypothesis [151] provides a simple link from the vector part of the weak current to the electromagnetic current of the nucleons.  This hypothesis  states that the isovector part of the electromagnetic current and the vector parts of the charge raising (n  —*  p) and charge lowering (p  —*  n) weak current form an isotriplet of  currents corresponding to the components 13, 1+ and 1 respectively, where 1+ This hypothesis  (  as it was in 1958  )  I + ‘2.  follows from the standard model of electroweak  interactions given that the proton and neutron form a good weak isospin doublet. Writing the isovector part of the proton electromagnetic curreit as, ] u(p) 2 + Fiu  (1’JMp)  and the  (1.7)  neutron current as,  (JEM)  =  u(n’) [F 7 + Fiu 2  with Dirac and Pauli isovector form  factors  ]  u(n)  (1.8)  1 and F F , the following identification can 2  Appendix I. The Nucleonic Weak Current and Nucleon Form Factors  208  be made using the isotriplet vector current hypothesis and the Wigner-Eckart theorem.  —  g  n  p 1  —  =  1  (1.9)  —  The form factors F are normalized to one at q 2  =  0 and  1 c ’(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 q 2 from electron scattering. In the range of q 2 relevant to muon capture (—m <  2 q  <  m) gv and g are very close to  being linear in the variable q 2 and a convenient parameterization is in terms of radii and  M• 7  ) 2 gv(q  =  Equation (1.9) implies gv(O)  =  /6] 2 gv(0)[1 + rq gM(0)[1 + i’q /6] 2  ) 2 gM(q  1 and gM(0)  =  (1.11)  3.706. Höhler has made a thorough  analysis of nucleon form factors for space-like q 2 and provides the following values of  and  TV  TM,  TV  see [152, Table 5,fit 8.2]. =  0.576fm  =  0.771 fm 2  (1.12)  The linearity of the form factors in the region —m < q 2 < 0 was tested by using the dipole form for the Sach’s form factors GE and GM from Kirk et al [59] to find gi and g. The Sach’s form factors are related to the Dirac and Pauli from factors as shown  Appendix I. The Nucleonic Weak Current and Nucleon Form Factors  209  below. F  =  2 F  =  (142)(GE42GM) 1 \  —  A1I12 ‘±lVIJ  (GM  —  GE)  Fitting a straight line so that it passed through the  2 q  (1.13)  0 and q 2  =  —m points, the  maximum deviation from linearity was found to be 0.01% for gv and 0.02% for  g.  The  linear form is thus completely adequate. For the 3 He(,u,iij reaction H  2 q  is fixed at —0.954m  =  . Equation (1.11) 2 —0.273fm  gives the following values of gv and gj at this q . 2 gv(—0.954m)  =  0.9737  gM(—0.954mj  =  3.576  (1.14)  For the H 3 e(, i’d)n reaction, q 2 varies from —0.84m to n. It is impossible to measure the electromagnetic form factors of the nucleons for 0  <  2 < 4M using electron q  scattering or electron positron annihilation and so some educated guess must be made in this region.  The values of gv and g were found by analytic continuation of the  expressions in equation  (1.11) i.e. the straight line is simply extrapolated into the 0 <  2 < m region. q  It is not  clear  F for q 2 > 4ni  a priori that this procedure is valid due to the presence of a cut in the 7m.  The procedure was tested by comparing the values obtained to  those from a fit to both space-like and time-like form factors incorporating the correct analytic properties by Houston and Kennedy [153]. The fit used used poles from the p(T70) and p(l2SO) isovector mesons placed on the second Riemann sheet as well as a fourth order polynomial to simulate both more distant poles and also the crossed single nucleon exchange cut on the second sheet. The fit allows a realistic analytic continuation  Appendix I. The Nucleonic Weak Current and Nucleon Form Factors  210  1.04  1.02  C  •1.00  C’  0.98  0.96 —1.0  0.0 q / 2 m2  Figure 1.1: Comparison of realistic continuation with simple linear extrapolation of form factors.  Appendix I. The Nucleonic Weak Current and Nucleon Form Factors  211  into the 0 < q 2 < 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 the 0 <  1.3  2 q  <  m region is perfectly adequate the maximum deviation being 0.3%.  The Axial Form Factors  The particle data group [41] give the value of the axial form factor at zero q . 2 —1.261 + 0.004  g(O) =  (1.15)  This value is found from the correlation between the electron momentum and neutron spin measured in neutron beta decay. The q 2 depeudeice of by observing the ip  g  for spacelike q 2 is measured  nt reaction for various neutrino energies. The analysis of the  experiments assumes a dipole form for g and fits the parameter MA. gA(q  2  — )-  gA(0) /M)2 2 (1_q  (.116  Taking a weighted average of various results [154]-[162] the best value for M is given by, M  =  1.08 + 0.04 GeV . 2  (1.17)  The expression in equation (1.16) was used to analytically continue g into the time like q 2 region. The equivalent linear form for gA(q 2 found in muon capture ) valid in the range of q 2 is, ) 2 gA(q with r  =  gA(0)[1 + rq /6] 2  (1.18)  2 which can he compared to the radii of the vector form factors given 0.433 fm  in equations (1.12).  Appendix I. The Nucleonic Weak Current and Nucleon Form Factors  212  The value of the pseudoscalar form factor gp is not well known experimentally. In principle, its value can be found from the muon capture rate by hydrogen but the exper iment is difficult to perform, both final state particles being electrically neutral, and also there arising problems of interpretation due to p  —  —  p molecule formation.  As a starting point the value of gp will be taken from theoretical considerations. One of the aims in this research is to develop the theoretical understanding of muon capture by 3 He to such a stage that the only uncertainty  ii  the theory is the value of gp. In that  case experiment could set its value. The theoretical value of gp arises out of the partially conserved axial current hypoth esis (PCAC) as well as knowledge of  and  g  the pion-nucleon form factor. The  gNN,  application of PCAC in the following form is due to Primakoff [163] and the analogous derivation of F, the nuclear pseudoscalar form factor for the 3 He  —*  H transition, is 3  detailed in the latter half of section 3.3.1. 2 4M  ) 2 gp(q  where,  ) (2 2 gA(q  ) 2 (q =  The value of  €  () [  [1 + E(q )] 2  ) 2 q —  g(q )/gA(0)  —q  (1.19)  can be estimated using a monopole form for gN(q ) as suggested by 2  Thomas and Holinde [164]. =  A  ) 2 gNN(q  For small q , 2  €  800MeV  (1.20)  is given by, €  (  =  —0.006  (1.21)  —  The smallness of this value is due to the similar small q . 2  =  2 q  dependence of g and  NN  for  Appendix 1. The Nucleonic Weak Current and Nucleon Form Factors  213  A useful quantity to consider is gp which is the combination of kinematical factors which apl)ears with gp in the matrix element squared for the rp  ,‘  un reaction. Using  equation (1.19) we have, gp(q2) 2 =  For gA 3 l.  2 q  in the range —m <  2 q  ) 2 gA(q  /q 2 (i  -1)  <m the magnitude of gp is in the range  (1.22)  gA 4 °•  to  and so the following rule of thumb can be employed which will prove useful in the  expansion of the effective muon capture Hamiltonian. 2 q  2N 4M  gp  g  (1.23)  The 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  214  3.85 3.80 3.75 3.70 3.65 3.60 3.55 .0  I  I  —0.5  0.0  0.5  2  —q /m —100  2  )  I  —200 —300 —400 —500 —t29—1.0  —  —0.5  0.5  2  —q /m  2  1.0  —  .0  —0.5  Figure 1.2: The nucleon form factors.  0.0  0.5  / m,L 2 —q 2  1.0  

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