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Radiative muon capture by 3He Ho, Ernest Chun Yue 2001

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R A D I A T I V E M U O N C A P T U R E B Y 3 H e by Ernest Chun Yue Ho B . Sc., The University of Br i t i sh Columbia, 1998. A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S (Department of Physics and Astronomy) We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A A p r i l 27, 2001 © Ernest Chun Yue Ho, 2001 In presenting this thesis in partial fulfilment of the requirements for an ad-vanced degree at the University of Br i t i sh 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. Department of Physics and Astronomy The University of Br i t i sh Columbia' Vancouver, Canada Abstract ii Abstract The rate of the nuclear reaction 3 H e + yT —>• 3 H + f M + 7 has been calculated by two different methods: the elementary particle model ( E P M ) approach and the impulse approximation (IA) approach. The exclusive statistical ra-diative muon capture ( R M C ) rate for photon energy greater than 60 M e V , T j J ^ , using the elementary particle model approach is found to be 0.2113 s _ 1 and the ordinary muon capture ( O M C ) rate is 1503 s _ 1 . Several trinucleon wavefunctions from different types of realistic nucleon potentials are used for the impulse approximation calculation and it is found that the capture rates calculated v ia the IA exhibit slight model dependences, possibly arising from differences in binding energy predictions, nature of potentials used or partial wave properties. The impulse approximation version of ranges from 0.1313 to 0.1387 s _ 1 and the corresponding O M C rate ranges from 1260 to 1360 s _ 1 . The difference in reaction rates between IA and E P M is larger in R M C due to some of the extra Adler and Dothan terms. Therefore, a meson exchange current ( M E C ) calculation of R M C seems necessary to account for this discrepancy. Contents i i i Contents Abstract i i List of Tables v List of Figures vi Acknowledgements viii 1. Introduction 1 1.1 What and why of radiative muon capture by 3 He? 1 1.1.1 What? 1 1.1.2 Why? 2 1.2 From ordinary muon capture to radiative muon capture . . . . 3 1.2.1 Ordinary Muon Capture 3 1.2.2 Radiative Muon Capture 6 1.3 Summary 17 2. The elementary particle model 18 2.1 Introduction 18 2.2 Kinematics and nuclear form factors 18 2.2.1 Nuclear form factors 19 2.3 Results 22 2.3.1 Ordinary Muon Capture 24 2.3.2 Discussion 27 2.4 Summary 28 3. The impulse approximation 29 3.1 Introduction 29 3.2 "Internal" and "external" degrees of freedom 30 3.3 Wavefunction 31 Contents iv 3.3.1 Antisymmetr izat ion and normalization of wavefunctions 32 3.4 Hamil tonian 33 3.5 Nucleon form factors 33 .3.6 LA o E P M translation . . 34 3.6.1 Operators 38 3.7 Results . 42 3.7.1 Ordinary M u o n Capture 42 3.7.2 Discussion 46 3.8 Summary and conclusion 50 Appendices A. Handling of radial wavefunction integrations in the IA . . . 53 A . l Introduction 53 A . 2 Integration of type: C(p, r) = ftp(p, q)ji(qr)q2+adq 53 A . 3 Integration of type: f((p,r)C(p,r)ji(sr)p2dpr2dr 54 A . 4 "Proofs" of the accuracy of spline interpolation 54 A . 5 Integration cutoffs 55 B. Second order terms for IA E P M translation 56 C. Wavefunction characteristics 60 C . l Channel specifications 60 C.2 Mode l dependent quantities of wavefunctions 60 D. Notations and conventions 62 Bibliography 64 List of Tables v List of Tables 2.1 A set of ( r 2 ) 2 values. The ( r 2 ) notation indicates that the above quantities are root mean square values of nuclear radii . Experimental data from Ottermann et al . [14] ( 3 He) and Beck et al . [13] ( 3 H) 20 3.1 Various O M C and R M C statistical rates when gP or FP is at the P C A C value. The numbers in the second column are the values obtained by using the " | " prescription up to O(-jjfr-) terms. The numbers i n the th i rd column are obtained v ia the correct approach up to 0(-^-) terms. The fourth column has values of the O M C rates using " | " prescription up to 0((^-)2) terms. Numbers in the fifth column are the values obtained by the correct approach up to 0((-^)2) terms; the sixth and rightmost columns contain r ^ ( « > 5 M e V ) and r » S t (K > 6 0 M e V ) respectively 46 C l Specifications for 22-channel wavefunctions 60 C.2 Some important quantities of trinucleon wavefunctions. B i n d -ing energy Eb in M e V . P{S) denotes the probability of S-wave and so on 61 List of Figures vi List of Figures 1.1 The Feynman diagram for ordinary muon capture 3 1.2 The Par t ia l Conservation of A x i a l Current Hypothesis 5 1.3 The vir tual pion exchange diagram, a consequence of the P C A C 7 1.4 The external radiating diagrams 8 1.5 Covariant derivative of the external weak hadronic currents illustrated. The Aa part, in one photon l imit , only affects the non-radiating weak hadronic current (i.e. the muon radiating diagram) 9 2.1 The E P M description R M C photon spectra using the full Adler Dothan amplitude for various values of Fp. The "Kl ieb and Rood" values are taken from the relativistic calculation of ref. [5] which are not shown in ref. [4] 25 2.2 Sensitivity of T ^ ( K > 60 M e V ) using E P M wi th respect to FP. 26 2.3 Photon polarization for various values of FP 27 3.1 Impulse approximation for O M C by 3 H e illustrated. The blobs represent the strong force that binds the nucleons. Please note that one has to sum all the contributions of the above process from every nucleon in order to get the complete rate 30 3.2 R M C photon spectra from two E P M calculations (one wi th the full Adler and Dothan amplitude and the other wi th only gauge invariant terms) and from I A calculations using various model potentials. A l l wavefunctions used have 22 Faddeev components and the permutation is projected on the same set of states. The infrared divergent part is not shown 43 3.3 Photon spectra from I A calculations for various values of gP. Wavefunction derived from Bonn-A potential is used for cal-culation 44 List of Figures vii 3.4 Same as figure (2.2) but for I A calculation. Using Bonn-A potential, r ™ ? ( « > 60 M e V ) = 0.1387 s" 1 45 3.5 Plot of [1]° vs. s for different nuclear potentials 51 3.6 P lo t of [a] 0 ' 1 vs. s 52 3.7 P lo t of [a] 2 ' 1 vs. 5 52 Acknowledgements viii Acknowledgements I would like to thank my supervisor Dr . Harold Fearing for his encouragement, guidance and support during my two years at T R I U M F . M y appreciation also goes to the kindness of Dr . David Measday of U B C / T R I U M F and Dr . Helmy Sherif of the University of Alber ta for being my second readers and giving me lots of constructive comments. This work would not have been possible without the tri-nucleon wavefunctions and instructions on how to use them given by Dr . Wolfgang Schadow. He, and Dr . Tae Sun Park, also provide me a lot of expert advice on setting up my own home L i n u x computer on which some of the present work is performed. Finally, I wish to thank Dr . Javed Iqbal for his introduction to me my present supervisor and both Michael Forbes and Alexander Busch for wri t ing two very nice WF£fi.2e thesis templates. I have chosen to use the one writ ten by the former because I received it a few days earlier than the latter one. To study in spring is treason; And summer is sleep's best reason; If winter hurries the fall, Then stop till next spring season. A Chinese poem for bibliophobes. Poet unknown. English translation by Yutang Lin [1]. ix 1. Introduction 1 1 Introduction 1.1 What and why of radiative muon capture by 3 He? 1,1.1 What? Radiative muon capture ( R M C ) by 3 H e is the process 1 and the corresponding non-radiative process is called the ordinary muon capture ( O M C ) by 3 H e . Here / i ~ stands for the muon that is captured; and 7 are respectively the muon neutrino and photon emitted. The capture takes place from a muonic atomic orbital , similar to an electronic state except that it is about 200 times smaller. Whi le it is in this orbit the muon can decay in the normal way with rate of 0.46 x 10 6 s _ 1 . Thus, as one wi l l see later, that muon capture is a fairly rare occurrence. Both capture processes (1.1) and (1.2) can be studied in at least two levels: the level of the nucleus and the level of the nucleons2. A t the nucleus level, one takes 3 H e and 3 H as whole entities and calculates the capture rate based on phenomenological nuclear form factors. This method is called the elementary particle model ( E P M ) and wi l l be discussed in chapter 2. One way to visualize processes (1.1) and (1.2) at the nucleon level is to regard the capture taking place on the constituent nucleons. B y recognizing 1 The "breakup" reactions: 3 H e + ^ ~ -» d + n + v^ + 7 and 3 H e + /i~ -> p+n + n + 1^  + 7 are not considered here. 2 not to mention the level of quarks! 3 H e + pT -> 3 H + Vf, + 7 (1.1) 3 H e + p~ - r > 3 H + i/p (1.2) 1. Introduction 2 that 3 H e ( 3 H) is some antisymmetric combination of | ppn) (| pnn)) and its permutations, one sums the following reaction that happens inside the nucleus to obtain a first order approximation to the capture rate: P + H~ ->• n + + 7 (1.3) for R M C and P + H~ -> n + Vp (1.4) for O M C This method is called the impulse approximation (IA) and w i l l be dis-cussed in chapter 3. Notice that since one is summing the contributions of the rate from constituent nucleons, the interactions between the nucleons are ignored by this method 3 . 1.1.2 Why? There are several motivations to calculate R M C (and O M C ) rates. Below are some major ones: 1. Experiment E592 at T R I U M F [2, 3] is an experiment to measure the photon spectrum of the radiative muon capture by 3 H e . A n accurate theoretical prediction seems necessary to interpret the experimental results. 2. W i t h the advent of computing hardware and software and methods of trinucleon wavefunction calculation, a more precise impulse approxi-mation seems overdue. Indeed, the most recent (and the only other) impulse approximation calculation was done about twenty years ago by K l i eb and Rood [4, 5] yet the Hamil tonian they were using d id not seem to completely satisfy some low energy theorems and they also handled some proton momentum terms in a crude way. The present impulse approximation calculation w i l l hopefully straighten out some of the errors and improve the approximation they made. 3 Please see section 3.1 for justification on ignoring the interactions between the nucle-ons. 1. Introduction 3 1.2 From ordinary muon capture to radiative muon capture Since there are many similarities between radiative muon capture and ordi-nary muon capture 4 , it is worthwhile to take a step back and have a glimpse at the simpler problem of ordinary muon capture. The relationship between different form factors of O M C derived from various hypotheses wi l l eventually be applied to the problem of R M C wi th some modifications. 1.2.1 O rd ina ry M u o n Capture In ordinary muon capture , the leading order Feynman diagram is shown in figure (1.1) F i g . 1.1: The Feynman diagram for ordinary muon capture. In figure (1.1), Pi stands for the four momentum of in i t ia l nucleus or nucleon (i.e. 3 H e for the E P M or proton for the IA) and Pf is the same quantity for the final nucleus or nucleon (i.e. 3 H for the E P M or neutron for the IA) ; \i is the four momentum of the muon and v is that of the neutrino produced. The above diagram follows from the Fermi's original current-current coupling of the weak interaction. M O m C = ^ ^ ^ ^ - ^ d r o o i c (1-5) 4 See Mukhopadhyay [6] and Measday [7] for reviews on the topic of ordinary muon capture 1. Introduction 4 where the leptonic current J k p t o m c [ s jleptonic = u{u){la{l - ^)}U^) (1.6) and the hadronic current ^hadronic * s hadronic = *(Pf)Wu{PJ ( 1 7 ) The Wa is the weak hadronic vertex, which can be parameterized by four form factors. Wa = G ^ + G M ^ ^ + G ^ V + G P T 5 — (1.8) 2,Mn m where 75 = ^ V i V (1.9) A l l the GiS (i = V, M, A, P) are functions of Q2, the square of the momentum transfer at the hadronic vertex. 5 Note that Mn denotes the mass of the nucleus in E P M but the mass of nucleon in I A 6 . Relationship between various form factors The Isotriplet Vector Current Hypothesis ( I V C ) 7 stipulates that Gy(Q2) should be of the form, GV{Q2) = efGfv(Q2) - e^Q2) (1.10) and GM(Q2) GM(Q2) = HfGfM(Q2) - HiG^iQ2) (1.11) 5 Indeed, the form factors also carry the Pf (Pj) dependence if the initial (final) hadron is off-shell, but this dependence wil l , as usual, be neglected. 6 Please see appendix D for definition of m and for all other expressions that are not denned. 7 I V C is a statement saying that the vector current Jia = 4'W^'0'(Q)vector7"**, * = +,0, - , $ = (| 3 H e or p), | 3 H or n))T, forms an isovector. Since r ' already transforms like a vector during isospin rotation, all that is required is W , Q(Q)Vector forms an isoscalar. The magnitude of it can be found for the i = 0 case. 1. Introduction 5 where Gy (Gy) is the electric form factor of the in i t ia l (final) particle and G'M (GM) is the magnetic form factor of the in i t ia l (final) particle. Notations of ei etc stand for charge and anomalous magnetic moment respectively. These form factors are supposed to be found experimentally from electron scattering experiments on the respective particles. The Conserved Vector Current Hypothesis ( C V C ) states that the vector current is conserved. This statement can be writ ten in equation: Q • ^hadronic = 0 (1.12) where Jhadronk ^S the vector part of the hadronic current. Notice that equation (1.12) is automatically satisfied (up to a small isospin breaking) given the form of Wa in equation (1.8). The Partial Conservation of Axial Current Hypothesis ( P C A C ) relates the divergence of the axial vector current to the pion field. A pictorial rep-resentation is shown in figure (1.2). Fig. 1.2: The Par t ia l Conservation of A x i a l Current Hypothesis Algebraically, it i s 8 : Q • Jhadronic = «ir"4</ I <KQ) I •> = a » m » T O 2 l n 2 < / l * r ( Q ) l » > ( L 1 3 ) mTt W 8 It is a slight abuse of notation not to differentiate between Ja (a current operator) and ( / | Ja | i) (a current amplitude) but the context in which the above notations are used will usually clear the confusion. 1. Introduction 6 where 4>{Q) is the pion field which operates between the in i t ia l and final states and an is the pion decay constant (av = 0.9436). In the E P M , i is 3 H e and / is 3 H while i n the I A , i is the proton and / is the neutron. Note that (f\JAQ)\i) = -—Gw{Qr')u{P,)>j*u{Pi) (1.14) vn „• where Gn(Q2) is the pion-i-f coupling constant. Since, Q • ^hadronic = u(Pf){2Mn GA + GP^}^u{Pi) (1-15) equating (1.13) and (1.15) using (1.14) one wi l l obtain a relation between GA and Gp, which is: Gp = J ^ O * { l + e(Q 2 )} (1-16) Note that a„ disappears by virtue of the relation GA(0) = - o » G w ( 0 ) (1.18) This is just the P C A C at Q = 0. The e part is small and is approximately a constant over the range of Q2 concerned [4, 5]. Therefore the dominant part of Gp looks as if it were caused by a vir tual pion exchange (Figure (1.3)). Please do not confuse e wi th e, the latter wi l l be defined as the photon polarization four vector later in the chapter. 1.2.2 Radiative Muon Capture Let us shift the focus to radiative muon capture. There are several differences between O M C and R M C . These include: 1. Unlike the case of O M C where Q2 is a constant, Q2 for R M C varies approximately between — ra2 and m2. Moreover, there are two kinds of momentum transfer at the hadronic vertex: QL, signifying momentum transfer when the muon is radiating, and QH, the momentum transfer 1 . Introduction 7 Fig. 1.3: The virtual pion exchange diagram, a consequence of the P C A C when one of the hadrons is radiating. Specifically, QL = Pf-Pi = fl — K — V QH = (p} + n)-Pi . = p - v (1.19) where n, u and K are the four momenta of the muon, neutrino and photon respectively. 2. Gauge invariance (GI) . Let M = e • M be the R M C amplitude (e is the polarization vector of the emitted photon), gauge invariance means that K • M = 0 where « is the momentum of the photon. Gauge invariance essentially says that the longitudinal photon polarization is unphysical and would not affect physical observables. O f course, there is no such thing as gauge invariance in the case of O M C since the photon is absent i n the first place. 3. Using minimal substitution Qa -> QQ — eAa, C V C and P C A C now read (compare equations (1.12) and (1.13)): (Q - eA) • J h v a d r o n i c , R M C = 0 (1.20) (Q — eA) • ^ h a d r o n i c . R M C = (1-21) Note that AaJa l i terally means "sticking a photon on the vertex of current J Q " . 1. Introduction 8 Our aim is to construct an R M C amplitude that is gauge invariant and .satisfies both the C V C and P C A C . This method of constructing the R M C amplitude, to be shown below, is due to Adler and Dothan [8]. The method essentially checks the P C A C , C V C and G I of the currents wi th a photon attached to either the muon, proton ( 3 He), neutron ( 3 H) or the pion and adds in counterterms so that the sum of al l the currents (plus counterterms) satisfies G I , C V C and P C A C . Firs t consider the four external radiating diagrams in figure (1.4) and let the sum of their hadronic currents be ^hadronic.RMC-F i g . 1.4: The external radiating diagrams Let Mi be the value of the diagram on the upper left hand side of the figure. Enumerating the diagrams clockwise, one has, after suppressing the constant ^ § V u d for convenience: M x = « H 7 Q ( l - 7 5 ) « ( ^ ( ^ / ) ^ Q ( Q H ) 5 F ( / P i -M 2 = u(u)la(l-15)u(fi)u(Pf)QfSF(/Pf+ fi)Wa(QHXPi) : ^ h a ( l - 7 > ( ^ ( P / ) { m , _ ( Q V _ K ) 2 1. Introduction 9 {-ie){2<^ - *) • eGft^uiPi) M4 = u(v)7a(l-75)SF(li- jk)(-ie 4)u{n) u{Pf)Wa{QL)u{P^ (1.22) where SF is the Feynman propagator for spin | particles and Qi(j) = ie-i(f) ~p-aXpKpe\, Hiy) being the anomalous magnetic moment of the in i t ia l (final) hadron. U p to one photon emission, (QQ — eAa)^hadronic.RMC c a n be calculated schematically as shown in figure (1.5). F i g . 1.5: Covariant derivative of the external weak hadronic currents i l -lustrated. The Aa part, in one photon l imit , only affects the non-radiating weak hadronic current (i.e. the muon radiating d i -agram). 1. Introduction 10 The result of the calculation 9 is: (Q — eA) • </hadronic,RMC = fi(P,){G? JQH + G" JQ^5 + G ^ L F S F ( ^ ~ A ) Q i u ( P 0 + *{P/)QtSF{ff+ £){G» Q" + G" JQHJ5 + G » ^ V M P 0 -m 2 , - (C} w - K ) 2 ml-Qjj m u(Pf){Gv jQL + 2MnGLA75 + G ^ 7 5 M P ) -G ^ 5 i l ! 7 R } « ( P 0 + fi(P/){G^^^i^K/,eQ}u(PO (1.23) where e has been defined in equation (1.17). Equat ion (1.23) can be reduced to the following fo rm 1 0 : (Q — eA) • ^ hadronic.RMC = s (P/K (G? - Gv) > + (G? - G f r / 7 5 } « ( P i ) -u(Pf){GLMia^^ea + G L P ^ ^ M P i ) + u{Pf){2MnGHA+GHP^-}^SF{Pi- £)Qiu(Pi) + u(Pf)QfSF(/Pf+ / ; ) { 2 M n G 2 + G | ^ } 7 5 u ( P , ) -m m 2 - (Q« - K ) 2 m \ - Q \ m u(pf){2MnGLA + G ^ y y p , - ) + m fi(i'/){G?f^«^/e/,ett}tt(Pi) (1.24) 9 O n e calculates (Q - eA) • J h a d r o n i c . R M C i n o r d e r t o find t h e counterterms needed to satisfy equations (1.20) and (1.21). 1 0 T h e photon normal izat ion factor ^ k V w i U he dropped for convenience as it wi l l always appear in the phase space factor when one calculates the capture rate. 1. Introduction 11 The two tricks involved in the above calculation are the observations that {ft- fi)SF{P>- A) = (A- A)-B , M ft- fi - Mn = i + MnSF{Pi- fi) SF{Pf+ fi)(Pf+ fi) = i + MnSF{Pf+ fi) (1.25) and the Dirac equation for the in i t ia l and final hadrons. Notice that the second last term of equation (1.24) has already been worked out for the case of ordinary muon capture. In other words, equations (1.20) and (1.21) imply that, u(P,){(G£ -G$)t + {GHA - GLA) tf}u{Pi) - u{Pf){GLMia^-^-eQ +GpQ—i^}u{Pi) + u{Pf){2MNG% + GP'C^h5SF{Pi- fi)Qiu{Pi) + m m s(P,)Q,SF(f/+ mM„G«A + G g ^ h 5 « ( P , ) - ~ / l K e ) 2 (Q - eA) • A J h a d r o n i c , R M c = o 7 r m 3 T ( /7 | <f>{Q) \ i) (1.26) provided the relationship between GA and G p is enforced by equation (1.16). In the above equation A J h a d r o n i c R M C is the extra piece of R M C hadronic current whose terms are to be derived from the Adler and Dothan procedure. {fl I 4>{Q) I i) is related to the pion photoproduction amplitude. The Born terms of {fy | <f>{Q) | i) are given by: </7 I .1 «•> = u ( P / ) { - ^ G f 7 5 } ^ ( ^ - fi)Qiu{Pi) + u{Pf)QfSF{Pf+ / , ) { - ^ L G f 7 5 H P ) + rn v ™ n „ L -i {-ie{2QL - «) ) • e 7 5 } ^ ( P i ) -u{Pf){-2-^G>nL{2Q» • e ) 7 5 } " ( ^ ) (1-27) 1. Introduction 1 2 In the above equation, G'£ = dG^qT^\Q^=Q2L- The first three terms correspond to the three external radiation diagrams and the last term is the gauge term to ensure the amplitude is gauge invariant when the pion is on shell. A look at equation (1.26) will reveal that the third, fourth and fifth terms almost fit the first, second and third term of the above expression respec-tively by virtue of equation (1.16) yet there are still two problematic spots: missing a -u(Ps)^{2Q" .£)7»«(P<) ~ - u { P I ) ^ ^ ^ ^ { 2 Q « • O r y * ) = - u ( P / ) ^ ^ P ^ ( 2 Q f f • O T M P O = + 2MnG'iS"'K)){2QH • e)75u(Pi) + 0(e2) term in the fifth term of equation (1.26) and the mismatching of momentum transfer between G„ and Gp there. The first problem can be solved by some rearrangement of terms in equation (1.26) and for the second one, one can expand G% as G% = G% + G'?{Q\ - Q2H) + 0(Q4). After all the work, equation (1.26) can be written as11, u(Pf){(G$ - Gy) 4 + (GHA - GLA) muiPt) - u(Pf){GLMio"e^eQ _ G £ ^ L £ 7 5 W p . ) + u{Pf){2MnGHA + G^WSrift- A)QMP) + m m ml - (QH - K)2 ml-Q2H rn u(Pf)2MnGt{2%'K\2Q» • e)^u(Pt) - u(Pf){£-^(2Q" • e)75}u(P) + « ( P / ) { G ^ ^ ^ « a ^ ^ e a } « ( P 0 + {Q- eA) • AJhadT0IiiCtRMc = a„ml(f1\4>{Q)\i) (1.28) and (/7 | jn(Q) | i) as </7 I J»(Q) I 0 = fi(P/){-—G^Sr'Pi- £)QAPi) + 771 TT u(Pf)QfSF(/Pf+ m ~ G ^ } u { P i ) . + TTl-jr 1 1 Notice the gauge condition of the photon is chosen to be such that K-e = 0; therefore, Q L c = QH • e. 1. Introduction 13 ( - i e ( 2 Q L - K ) ) - e 7 5 } u ( P l ) -u{P{){-2-MllG'^2QH • 6) 7 5}tt(P.-) + 7 8Mfi) (1.29) Assuming the linearity of Gn(Q2) over the range of Q2 concerned, the last two terms of equation (1.29) can be combined to form That is to say the C V C and P C A C conditions for R M C are: u(P,){(G"v-0Lv) it + (G«-GLA) ^ W P J - i l P j K G i i ^ l - e , m mv - LJL u(Pf){£-^t(2QH • 6 ) 7 5 } « ( P i ) + ^ Z ) ^ ! ^ ^ ' 4 ^ } ^ ) + (Q — e^4) • A Jhadronic.RMC provided equation (1.16) holds true. Now turn to gauge invariance, let Mi = e • Mj and if one then calculates K • M{ the results of calculation would be: K • Mi = e i u i ^ i l - ^ u ^ u i P ^ W ^ H P i ) K-M2 = -efu(u)7a(l-75)u(^u(Pf)Wa(QH)u(Pi) K-M3 = u(v)ya(l - y5)u(n)u(Pf){-—% l m l - ( Q H - K y (-ie)(2QH • K)GHP}^u{Pi) m K-M4 = -u{v)la(l - 7 5 ) ^ ) u ( P / ) W Q ( < 2 I > ( P i ) (1.31) The tricks used in the above calculation involve the following: 1. Introduction 14 1. Using the antisymmetry of al3x wi th respect to switching indices to conclude that a®XKpK\ = 0. 2. Using equations like Pi fi + Mn fi MP) -2Pi • K 2Pi • K- fi Pt + Mn fi u{Pi) -2Pi • K = u(Pi) (1.32) to simplify expressions. Since ei — ej = e, i t is easy to show that, "t^Mi = u(u)ja(l - y5)u(n)u(Pf){(Gy - Gy)ya + i= l (GZQg - GLMQ%) 2mMnG'A(l + s) 1 2Mn a + - ° A h 7 + ml-Ql 2 9 ^ Q H ^ + ^Vw*) a-33) To satisfy gauge invariance up to O ( K 0 ) , one needs A M such that 4 K-iJ^Mi + AM) = K • M i = l = 0 + O(K2) (1.34) Postulate12 A4 Q adronic,RMC = -2u(Pf){G'vl» + G\1rf}{g»aQH-e + Kae>>-K»e«}u(Pi) - u(Pf){GLMiaa^ + G LXj5}u(Pt) 2,Mn m 1 2 K l i e b and Rood did not have - u ( P f ) { 2 m " f ^ } + e ) 2 ^ e Q a H i 5 + G'v(uf -Hi)^-i<J0iiKcefi}u(Pi) in their R M C calculation. 1. Introduction 15 u 2mMnG'A(l + e) 2QH • e 5 M P f ) ml-Ql ™ ~ Q h 1 U [ P l ) ~ G'v(»f ~ / x i ) | ^ - t a ^ K C € / J } t i ( P j ) (1.35) will satisfy both equations (1.30) and (1.34) up to, but excluding, terms of P r o o f Let A M = J l e P t o n i c • A J h a d r o n i C i R M C . The fulfilment of equation (1.34) of A M can immediately be read off once one expands the form factors of equation (1.33) to first derivatives. Note that all the pole terms are treated exactly. To demonstrate that the A J h a d r o n i C ) R M C in equation (1.35) satisfies equa-tion (1.30), one can calculate Q • AJhadronic,RMC- The — eAa part does not enter since it is higher order than required. The result of the calculation is: Q • A J h a d r o n i c , R M C = u(Pf){G'v 4 + G'A tfHQl - QnMPi) -u(Pf){2MnG'A(2QH • e) 7 5}«(Pi) -u(Pf){GLMia^^-efi + G £ ^ 7 5 M P ) -1QL • K «(Pf){G'v(l*f ~ »i)-j^i^^}u(P>) (1-36) Putting it back into the left hand side of equation (1.30), one would get for the left hand side: -u(P,){2MnG'A(2QH • < r ) 7 5 } « « ) + u(P,){GLv^^-i^< ^^(P,) + 2 ^ ( 2 Q " . . - ( 1 + £ ) Q ^ ) . _ " rnl-Ql £-^(2QH • e ) 7 5 W ) (1.37) m _ 1 3 Equation (1.30) determines terms up to 0(Q°KN) and equation (1.34) determines terms up to O(K0Qm). Here K° means "photon four momentum up to order zero" and n, m are arbitrary positive integers. 1. Introduction 16 which equals --^Ln(Pf)2MnG'A(2Q» • €) 7 5u(P t) + ^PfHGy^^ ^ K ^ M P O - u(PM£2™Mn^QH'eQW + • eh5}u(Pi) m~ — Q L m m (1.38) Using to eliminate G'A in favour of G'w and e in expression (1.38) one would end up having, ^ W } « ( P i ) + C(c: 2) (1.40) The first term is just the right hand side of equation (1.30). The magnitude of the second term can be approximated as follows: ^ < ° > < ^ ' + <&fc> ~ ^ ( 0 ) ( ^ 5 ) 2 + ^ ) ) < 3 2 ~ 2KG"Q4 (1.41) keeping i n mind that a* ~ 1 (0.9436, to be exact); in the above equation, G", \G"\ « 1 is the small second derivative of either GA or G„ and K is some constant with magnitude \K\ ~ 1. The third term is annoying but the Adler and Dothan procedure is not sufficient to determine uniquely counterterms to get r id of it. To see this point, notice that a term in A Jh adronic, R M C W ^ the format -u(P^GLvYa^=^za^KCe,}u(Pi) (1.42) 1. Introduction 17 with YA = -^-^ or (and probably among many others) can both do the trick but both have some undesirable pole behaviours. Notice that the latter is formally a term of O(KQ) but it is exactly these terms that the Adler and Dothan procedure cannot determine uniquely. A more detailed study of these structure terms should probably involve a model that incorporates the exchange currents of mesons. Please see section 3.7.2 for possible implications of these undeterminable terms on the resulting capture rates. O(KQ) terms aside, the Adler and Dothan procedure does indeed generate terms of A J h a d r o n i c as in equation (1.35) that satisfy the gauge invariant requirement up to O{K0) and P C A C up to terms of 0{eG") and O{Q0). 1.3 Summary After a brief introduction of the process of ordinary muon capture by 3 H e and its various hypotheses involving the O M C hadronic current, the correspond-ing radiative process is compared against the ordinary one. For the R M C amplitude to be gauge invariant and its hadronic currents to satisfy the C V C and the P C A C , an extra current A J £ a d r o n i c R M C must be added in addition to a naive insertion of photons to external particles. The Adler and Dothan procedure is employed to generate those extra terms as far as possible. This procedure guarantees the uniqueness of those extra terms up to O(K0) by gauge invariant requirement and O(Q0) by the C V C and the P C A C except the pion pole terms which are treated exactly. Terms of O(KQ), however, are not determined uniquely by the procedure and these terms can only be revealed by a detailed examination of the structure of the weak hadronic vertex. 2. The elementary par t ic le model 18 2. The elementary particle model 2.1 Introduction The elementary particle model is probably the simplest method which can be used i n the calculation of the R M C rate. It was first used by K i m and Primakoff [9, 10] in calculating the beta decay of complex nuclei. A s its name suggests, the central idea of the method is to completely ignore the internal structure of both the 3 H e and 3 H and regard them as "elementary particles". Since both 3 H e and 3 H have spin | , and isospin \ up to a small isospin breaking, the calculation is relatively simple compared wi th other methods.. The drawback is one sti l l has to rely on experiments done on these nuclei for the phenomenological form factors parameterizing the reaction. For example, one can perform electron scattering experiments for determining the vector current form factor and beta decay experiments for the axial current form factor. Previous E P M calculations of R M C by 3 H e include Fearing [11], K l i e b and R o o d [4, 5] and Hwang and Primakoff [12] but only the calculation of Kl ieb and R o o d included some Adler and Dothan terms. Moreover, some physical quantities were obtained by Kl ieb and Rood v ia a non-relativistically reduced amplitude. The present calculation differs from those previous ones in that the full Adler and Dothan terms are included and the resultant amplitude is not non-relativistically approximated in the hope of getting a more accurate reaction rate. 2.2 Kinematics and nuclear form factors For the elementary particle model description of R M C by 3 H e , the degrees of freedom are the four momenta of 3 H e ( P » H E ) , 3 H (P>H), photon (K), neu-trino (v) and muon (/J,), together wi th their respective spinors. The four-2. The elementary particle model 19 momentum conservation is written as: PSH + V + K = P3He + P (2.1) The quantity that is of interest is the differential capture rate, which is given b y 1 dT = E y d ( P W 5 p a c e ) T r ( p M t M ) ( 2 2 ) photon polarization for R M C , and the total capture rate Tomc = j d(phase space)^ Tr ( p i k T ^ M 0 ™ ) (2.3) for O M C . Here M (Momc) is the R M C ( O M C ) amplitude that was discussed i n chapter 1. One just needs to recognize the following name changes u(Pi) -> u(Psne) u{Pf) -> u{Pm) Mn —> Mt on the hadronic vertex d -»• Fu i = V, M, A, P (2.4) for M (denote it as M(Psiil P ^ j e , Mt)). The name change from G —> F is merely a change in accordance with the usual convention. p is the density matrix which describes the ini t ia l spin configurations of the muonic atom. The detailed implementation of p w i l l be discussed in the next section. d(phase space) (d(phase space) o m c ) is the differential phase space factor of the emitted particles for R M C ( O M C ) . The detailed calculation of these wi l l appear in the next section. Also note the integral sign of equation (2.2) does not apply to K since what one needs is the differential rate 2 . 2.2.1 Nuclear form factors One can conveniently parameterize the nuclear electromagnetic form factors linearly in terms of Q2 when Q2 is small, as in the case of R M C by 3 H e . FyHe(Q2) = 1 + ^r 2)3He,electricC} 2 + 0 ( Q 4 ) (2.5) 0 P M H 6 ( Q 2 ) = 1 + ^ 2 ) 3 H e , m a g n e t i c Q 2 + 0{Q4) (2.6) 1 Despite its notation, d { P h a s ^ P a c e ) j s actually a differential of angular variables. 2 One integrates K in equation (2.2) to get the total capture rate of R M C . 2. T h e elementary par t ic le mode l 20 and analogous equations also hold true for the case of 3 H . The above equa-tions relate the electromagnetic form factors to (r 2 ) where ( r 2 )^ is some form of root mean square charge or magnetic radius of the nucleus which are sup-posed to be found from experiments (see, for example, refs. [13, 14, 15, 16] for related experiments). {r2)h ( f m ) He, electric 3 H e , magnetic 3 H , electric 3 H , magnetic 1.976 ± 0 . 0 1 5 1.99 ± 0 . 0 6 1.68 ± 0 . 0 3 1.72 ± 0 . 0 6 Tab. 2.1: A set of ( r 2 ) 2 values. The ( r 2 ) notation indicates that the above quantities are root mean square values of nuclear radii . Exper-imental data from Ottermann et al . [14] ( 3 He) and Beck et al [13] ( 3 H ) . B y the I V C discussed in chapter 1, one has Fv = 2FvHe-Fvn FM — fi3neF^e - / ^ H - F J 1 (2.7) where / U 3 H e is the anomalous magnetic moment of 3 He wi th a similar definition for 3 H . Their values are A^He — —8.3689 n.m / i 3 H = 7.9173 n.m (2.8) Equations (2.7) imply that Fv(0) = 1 FM(0) = / i 3 H e - / / 3 H (2.9) The functional form of Fv and FM are chosen to be Fy = l + UyQ2 0 FM = 0u3He - M» H )(1 + \RMQ2) (210) 0 2. T h e e lementary par t ic le m o d e l 21 with Ry = 1.94 fm and RM = 1-72 fm. These values are taken from a grand average of various experiments and (with RA) they produce "good" numerical values for O M C . The function FA(Q2) is not so clear cut since there does not seem to have any direct experimental determination of that form factor. One has to resort to the impulse approximation to estimate the dependence of FA on Q2 over the range of Q2 concerned. Delorme [17], using impulse approximation, expressed the nuclear form factors in terms of nucleon form factors. This method is also employed by Kl i eb and Rood [4, 5] and Congleton and Fearing [18, 19] in their E P M calculations. Fy(Q2) = gv(Q2)[l}° (2.11) FM(Q2) = 3{9v(Q2) + 9M(Q2)M- - gv(Q2)[i}° -^ ( Q 2 ) ^ ] 1 ' 1 (2.12) FA(Q2) = gA(Q2)[a]~ (2.13) M2 FP(Q2) = VgP(Q2) + §V2^gA(Q2)[o]2l (2.14) where [ l]° , [a]+, [a]~, [ff]2'1 and [ i P ] 1 , 1 are the impulse approximation reduced matr ix elements resulting from integration of internal momenta. They wi l l be defined in section 3.6.1. The Q2 dependence of FA is given by FA(Q2) = 9A(Q2M-(Q2) , . FA(0) 9A(0M-(0) { • } Notice that Fy(Q2) + FM(Q2) -> 3{gv(Q2) + gM(Q2)}[o}- (2.16) when [zP] 1 ' 1 —>• 0. Since [ i P ] 1 ' 1 is indeed small, one can use the above l imit ing equation as an approximation and put it back i n equation (2.15). The result is: FA{Q2) = 1-Q2/M2 2 FV(Q2) + FM(Q2)} ( 2 U ) FA(0) 1 - Q2/M\ Fv(0) + FM(0) * { - ' where MV = 0.710 G e V 2 and M\ = 1 . 0 8 ± 0 . 0 4 G e V 2 [18]. Th i s follows from the expansion g{ = ^ ( 0 ) / ( l - Q2/M2)2 i = V, A. Upon expansion to 0(Q2), 2. T h e elementary par t ic le model 22 FA can be written as FA = F A (0 ) (1 +jU*Q2) (2.18) FA(0) = 1.212 ± 0 . 0 0 4 [18] (2.19) w h e r e R\ =  Fv%ttZ{(o)Rlf + KBI - -k) w h i c h m a k e s ^ = i - 7 0 3 f m -The P C A C hypothesis discussed in chapter 1 relates FP to FA- The relationship is shown as equation (1.16) with GA replaced by FA and Gp replaced by Fp. e 2.3 Results When al l the relevant Feynman diagrams are evaluated and the gauge free-dom is exploited so that the photon is made transverse (i.e. K • e — 0 and 0 = 0), the relativistic amplitude M (M = £ 4 = i + J l e p t o n i c - A J h a d r o n i c , R M C in equation (1.35)) can be written in the following non-relativistic like form, which operates on the product space (see ref. [20], chapters 3 and 4 for in-formation on direct products of matrices) of the hadronic spin (via a) and leptonic spin (via di): M(P^P^Mt) = W^Vud(l-c?L-0){f1oL-ex + f2a-ex + ifz^x x &i • ° + -exa- s + -^-v • ex + 2m 2m . fl -. - _ , /8 _ /9 -. , i—s x eA • cr + —OL • so • eA + — aL • au • ex + 2m 2m 2m /io -. — _ . . / n - s - - . -o • sv • ex + i^z^s x v • ooL • ex + 4 m 2 4 m 2 ' /12 _ — _ , / l 3 _ -a • vv • ex + —d • VoL • ex + Am2" " A ' 2m ' i-p^sx ex-oaL • s + ip^sx v • ov-ex} (2.20) 4mz 8 m J where S = U + K and N ' is the normalization factor for the particle spinors, 1 N ' = P^ + Mt 2\ 2Pb (2.21) 2. T h e elementary par t ic le mode l 23 The coefficient of | in N' comes from the normalizations of the photon and the neutrino. The photon polarization unit vector A = 1,-1 is defined as, k x e A = -iXnex ( 2 . 2 2 ) That means the Sphoton polarization m equation ( 2 . 2 ) should be replaced by S A = - I , + I - /t) z = 1 — 15 are some very complicated scalar functions independent of a, CL and t\ and will not be denoted. , Define d(phase space) to be the differential phase space factor which has the form „ , \ i / /r\\ 1 9 d3p3H d3u 1 d3k d(phase space) = C \ 4>^{0) (2TT ) 3 (2TT) 3 « ( 2 ? r ) 3 (2TT)4<S(4) (p + P3He - Pm - K - v) ( 2 . 2 3 ) where < (^0) is the muon wavefunction at the origin and C (close to 1, chosen to be 0.9788 [18]) is the correction factor which accounts for the non-pointlike nature of the nucleus (an averaging of the muon wavefunction near the origin over the small but finite size of the nucleus). Some standard manipulations of equation ( 2 . 2 3 ) lead to: 1 2 P ° d(phase space) = C | < (^0) | 2 — — | 2K(1 - cos(0)) - 2(m + M t ) | _ 1 u2(K,9)di>K2dKdk ( 2 . 2 4 ) where cos(f?) = 0 • k and - Mm + Mt) - m2 - 2mMt [ K ' °> " 2«(1 - cos(0)) - 2(m + M t ) l ^ 5 j expressing the four-momentum conservation of the 5 function. Define p^ - as the elements of the density matrix which takes care of the initial spin of the muonic atom. Since p is a density matrix, it has the property Tr(p) = 1. p is chosen to be diagonal in the | / , fz) (coupled lepton (s),) spin and nuclear (sn u ci e a r) spin state; f = s^ + s n u ci e a r) basis rather than the | s ,^ 5 n u c iea r ; 5^, s„ u c l e a r) because of the hyperfine splitting of the muon in the muonic IS orbit. In the special case when the initial spin mixtures 2. T h e elementary par t ic le mode l 24 are equal, p{j = ±6^ and is basis independent. In that case, equation (2.2) reduces to dTstat _ s-^ f fd(phase space) dK ~ x=k+J dK iTr(M(PsH, P^ e, M^M(Pm, Pane, Mt))} (2.26) The quantity d I j ^ g t is commonly called the differential statistical rate and the corresponding photon spectrum is plotted in figure (2.1). The total R M C rate is obtained by / ^dn. The E P M description of R M C statistical rate for photon momentum higher than 5 MeV 3 is T^(K > 5 MeV) = 0.8189 s _ 1 (2.27) <rmc stat when Fp is set at the P C A C value. Figure (2.2) shows the variation of T with respect to changes in FP. Perhaps a more important quantity is the R M C statistical rate for photon momentum higher than 60 MeV because experiments cannot (yet) measure photons resulting from R M C with momentum lower than 53 MeV because of bremsstrahlung4. Its " P C A C " value is: FSZ(K > 60 MeV) = 0.2113 s _ 1 (2.28) Figure (2.3) shows different photon polarization 5 P 7(K) predictions vs. FP 2.3.1 Ord ina ry M u o n Captu re The calculation of ordinary muon capture (OMC) is also done for comparison. Since R M C is the major topic of this work, the presentation below will be brief. Let M0"10 be the amplitude for the ordinary muon capture as shown in section 1.2.1. The momentum transfer squared at the hadronic vertex 3 The K integration cannot go to zero because of infrared divergence. 4 The two step process fj,~ —> e~ +ue+vli fo l lowedbye~+ 3 He —> e - + 3 H e + 7 produces photons up to 53 M e V . 5 To obtain £P 7 («), simply replace Ephoton polarization i n equation (2.2) by EA=- 1 I + 1(-I) ( A + 1 ) / 2-2. T h e elementary par t ic le mode l 25 40 60 photon momentum [MeV] Fig. 2.1: The E P M description RMC photon spectra using the full Adler Dothan amplitude for various values of FP. The "Klieb and Rood" values are taken from the relativistic calculation of ref. [5] which are not shown in ref. [4] Q2 = -(!§S^)2 = -0-2763 fm" 2 . Upon expanding M^P^, Pme, Mt) and rewriting in a non-relativistic like format, one has: M ° - c ( P 3 H , P 3 H e , M t ) = K'^Vud(l-oL-v){gl+g2a-v + g3aL-o} where gx = Fv(l + v M t + pOj mv M 2 M t ( M t + P ° I ) 92 — (Fp — FA — Fy — FM) M + P4 #3 FA - {Fv + FM) v Mt + Pb (2 .29) (2 .30) (2 .31) (2 .32) 2. The elementary particle model 26 Fig. 2.2: Sensitivity of r™(K > 60 M e V ) using E P M wi th respect to FP N ' = 1 M t + \ 2 2 P & The differential phase space factor for O M C d(phase space)^ is: (2.33) d(phase space) c Ci^(°)|2f7r7 ^ ^ ^ ^ + /0 = C|^(0)| 1 P° V 2 (2TT) 2 m + M t where i / is evaluated at, due to the 5 function m 2 + 2 m M t v -(2.34) (2.35) 2(m + M t ) The value of P ^ for E P M description of O M C is therefore: P\ = s]M? + v2 (2.36) using the value of v from equation (2.35). The statistical rate for O M C is obtained via equation (2.3) with pij = \8ij. The result, wi th FP set at the P C A C value, is: TZt = 1503 s- 1 (2.37) 2. T h e e lementary par t ic le m o d e l 27 i * photon momentum [MeV] F i g . 2 .3 : Photon polarization for various values of Fp. with F y ( - 0 . 2 7 6 3 fm- 2 ) = 0.8267 F M ( - 0 . 2 7 6 3 f m - 2 ) = -14.067 F A ( - 0 . 2 7 6 3 fm~ 2) = 1.050 (2.38) which should be compared wi th the most recent calculation by Congleton and Fearing [19] who obtained a value of 1497 s _ 1 . The closeness of the above result to theirs is not so surprising given that the values of the formfactors (i.e. FV, FM and FA ) at the Q2 for O M C are approximately the same as the ones they were using (which also gives a reason why a linear parameterization of formfactors in section 2.2.1 is sufficient). Also interesting to mention is the most recent experimental result of statistical O M C rate of 1496 ± 4 s _ 1 [21]. 2.3.2 Discussion Below are some points to note wi th regards to the E P M description of R M C . Please refer to section 3.7.2 discussions on the comparison between E P M and I A . 2. The elementary par t ic le mode l 28 1. The photon spectrum (figure (2.1)) and total R M C rate agree wi th the results of K l i eb and Rood [4, 5] who obtained a total rate of 0.814 s _ 1 via a non-relativistically approximated amplitude (also see figure (2.2) for variation of R M C statistical rate vs. FP) , showing that the extra terms not included by Kl i eb and R o o d do not contribute significantly to the photon spectrum. 2. The photon polarization P 7 ( K ) obtained in this work does differ from that of Kl i eb and Rood quite significantly (see figure 2.3). The reason behind that is while a non-relativistically reduced amplitude used by Kl i eb and Rood produces the correct spectrum within a few percent, it cannot produce P 7 ( K ) accurately. Fearing [22] noted that the first order contribution of P 7 ( K ) actually comes from 0(-^) terms in the squared Hamil tonian but K l i eb and Rood compromised P 7 ( « ) ' s accuracy by truncating many 0(-^) terms when they squared the already non-relativistically reduced amplitude (see section 3.7.2). 3. The Adler and Dothan terms have a great influence on the resulting photon spectrum. They contribute about 10% of the spectrum at the high photon energy range (K > 60 M e V ) . The large size of G'v and G'A because of the large root mean square nuclear radii of 3 H e and 3 H compared to proton or neutron is pr imari ly responsible for the contri-bution. 2.4 Summary In the framework of the E P M description of R M C by 3 H e , the " P C A C " value of the exclusive statistical capture rate, TRS^(K > 5 M e V ) , v ia a fully Adler and Dothan amplitude, is 0.8189 s _ 1 and the O M C statistical rate is 1503 s _ 1 . Whi l e these results agree wi th K l i e b and R o o d [4, 5] and Congleton and Fearing [18, 19] respectively, K l i e b and Rood's photon polarization P 7 ( K ) disagrees wi th the result,of this work, possibly due to the fact that they threw away some terms in the squared amplitude which contribute significantly to P 7 ( « ) . 3. T h e impulse approx ima t ion 2 9 3. The impulse approximation 3.1 Introduction In this chapter, the method of the impulse approximation w i l l be used to investigate the R M C by 3 H e . Even though the elementary particle model is a convenient approach to the reaction, it does not give the detailed mecha-nism of "what is happening inside". Indeed, each of the 3 H e and 3 H nuclei is a three-nucleon bound state. Perhaps it would be a good idea to look at the nuclear reaction using a nucleon perspective. The idea is, using a t r in-ucleon wavefunction, to integrate out al l the degrees of freedom relevant to the internal coordinates of the trinucleon system and to extract quantities that are dependent on the E P M degrees of freedom. In other words, one wants an expression similar to M(P3ne, P S H , Mt) in the E P M but one with al l the information regarding the internal degrees of freedom hidden in some formfactors resulting from internal integration 1 . The first step is to regard the three bound state nucleons as free nucleons and the Hamil tonian of the nuclear reaction as the sum of the Hamil tonian of its nucleon counterparts. This procedure is called the impulse approxima-tion. Naively speaking, the approximation of bound state nucleons by free nucleons in this case is a good one since the binding energy (~ 8 M e V ) is about 0.3% of the mass of the nucleus. However, the impulse approxima-tion deliberately neglects the contribution to the R M C from meson exchange currents between the nucleons, which may be large. Figure (3.1) shows the impulse approximation for O M C by 3 H e . 1 The method of finding this expression will be discussed in section 3.6 3. T h e impulse app rox ima t ion 30 F i g . 3 .1 : Impulse approximation for O M C by 3 H e illustrated. The blobs represent the strong force that binds the nucleons. Please note that one has to sum al l the contributions of the above process from every nucleon in order to get the complete rate. 3.2 "Internal" and "external" degrees of freedom In describing any non relativistic three-particle system, there exists a coor-dinate transformation which separates the centre of mass coordinates and the internal coordinates of the three particles. This enables one to treat the three-body problem as an effective two-body problem. Let ki be the (three) momentum of the of the ith nucleon and define: P — ka + hp -t- fc7, Qa = — 3^/9 — 3^7, (3-1) Pa = \kp ~ where (a, B, 7) are any set of numbers obtained from the cyclic permutation of (1 ,2 ,3) . 2 2 From now on, when no confusion arises, g^and <£,will sometimes be denoted as q and 3. T h e impulse approx ima t ion 31 It is now obvious that P is the centre of mass momentum vector and qa is the momentum of particle a (spectator) with respect to the centre of mass momentum of the other two particles (subsystem), while pa is the momentum of particle /? wi th respect to particle 7 (or vice versa). Also note that the Jacobian of the transformation I d^'kp'kS1} I is 1. 3.3 Wavefunction The tri-nucleon momentum space wavefunction, provided by Schadow [23, 24, 25], is obtained by solving the Faddeev equation (see, for example, ref. [26]) with various model potentials. It can be written as a sum of channel wavefunctions where each channel represents a specific angular momentum and spin eigenstate. \^) = J2MP,<l)\ic)\P) (3.2) The I P) represents the centre of mass momentum of the tri-nucleon system and is independent of the internal interactions of the nucleons and so it can be safely omitted for convenience in later calculations. The coupling scheme 3 of the channels is: \ic) = \(((0La)(0la))Ca,(SasQ)Sa)J)\(IQia)l) = \ic(J))\ic(X)) (3.3) I ( ( ( 0 L Q ) ( 0 Z a ) ) £ Q , (Sasa)Sa)J) is the spin and angular momentum part: LQ (SQ) is the angular momentum (spin) of subsystem (^7) , la (sa) is the angular momentum (spin) of particle a (the spectator particle) which are coupled to form Ca (Sa) and then to J = \. The | (Iaia)X) is the isospin part: Ia, being the subsystem isospin which couples wi th ia, the spectator isospin, to form J = | . Note that the angular momentum part and the isospin part are not coupled together. The zeros in | ic) are "dud" couplings. They are merely used to facilitate computation. <f respectively. 3 Indeed, the LS coupling scheme that is used here is not suitable for calculating the wavefunction, the jj or the channel coupling scheme was used to solve the Faddeev equa-tion and the resulting channel wavefunctions were then transformed to LS coupling scheme. 3. T h e impulse approx imat ion 32 3.3.1 An t i s ymmet r i za t i on and normal izat ion of wavefunctions The full wavefunction | ip) is antisymmetrized wi th respect to the labelling of nucleons. That is, \1>) = | V'Fad) + P | ^Fad) (3 .4) where | ipvaA) is the "Faddeev component" of the wavefunction obtained by solving the wave equation wi th one particular choice of nucleon labelling scheme, and P is the necessary permutation operator to make the full wave-function antisymmetric. The normalization is defined as: (ib\tb) = 3 { # a d | rb) = 1 (3 .5 ) The fact that the wavefunction is antisymmetric with respect to nucleon labellings makes the second expression of the above equation legitimate. It is also numerically possible since it consists of a finite number of terms. The first expression is, however, numerically impossible in general since permuting the nucleon label ( P | ^Fad)) w i l l project the Faddeev component onto a set of eigenstates which is usually much larger, i f not infinitely larger, than the original (finite) set of eigenstates comprising the Faddeev components. It is therefore necessary in practice to restrict the set of projected states to one wi th a finite number of elements. Suppose one defines Pl^Fad) to be the permutation projected on a restricted set of states, then the nu-merical normalization of the wavefunction (V'lV')num = ( (V 'Fad l+^Fad lP^d^+PIV 'Fad) ) (3-6) wi l l not be 1 because of this restriction. For example, a 8-8 channel wave-function (i.e. | i/>Fad) has 8 channels and the permutation is projected on the same 8 channel) has a normalization of 0 .93 and a 2 2 - 2 2 channel wave-function typically has a numerical normalization around 0 . 99 (see appendix C ) . 3. T h e impulse app rox ima t ion 33 3.4 Hamiltonian The effective Hamil tonian is defined by: 3 (3.7) where Hi and T7 are the weak interaction Hamil tonian and the isospin low-ering operator acting on ith nucleon of the trinucleon system. The relevant Feynman diagrams are exact duplicates of the E P M ones except that the 3 H e is replaced by a proton and the 3 H is replaced by a neutron. Since the wavefunction is fully antisymmetric with respect to interchange of particle labels, one can write equation (3.7) as where j is 1 or 2 or 3, provided H is acting on the above wavefunction. The relation between different nucleon form factors is an exact duplicate of the nuclear ones. In other words, al l the C V C , P C A C , I V C hold for the nucleon case once one replaces 3 H e with a proton and 3 H with a neutron. One has, 3 ri = Y,TrHi = ZrjHj (3.8) i=i 3.5 Nucleon form factors 9i(Q2) = 9i(0)(l + lr2Q2) (3.9) where i = V,M,A and 9v{0) = 1 9M{0) = Hp ~ gA{0) = -1 .267 ± 0 . 0 0 3 5 [27] (3.10) with r2v = 0.576 fm 2 , r2M = 0.771 fm 2 and r\ = 0.433 f m 2 [18]. 3. The impulse approx ima t ion 34 3.6 IA E P M translation The a im of this section is to find the correspondence between the E P M amplitude and the I A amplitude. . Let Mia be the R M C transition amplitude obtained by the impulse ap-proximation. This means Mia is the I A equivalent of M(Pme, Pm, M t ) in the E P M . More explicitly, one would use the following formula in order to have the quantity ^ (compare wi th equation (2.2)): fK = t J ^ J ^ e space) ( 3 . n ) and = j d{phase 5pace) o m c Tr(pMr c t M^ c ) (3.12) for O M C . Define M.epm as 4: Mepm = (27T)45 (4)(P3H + (y + K - fj) - P3He)M(P3He, P^ M t ) (3.13) It then follows that the I A equivalent of M.epm, M.ia, would be: Mia = 3/(2ir)W>(k'a + {y + K - /i) - fcQ)(27r)4^(^ - ^)(2TT) 4^ 4>(A:; - fc7) ^tu{k'Q, le-, fc7)^3He(fcQ, kp, ky)u)(ka, kp, ky, fcjj, k°)M(k'a, ka, Mp) d4k' d4k-( 3 1 4 ) In words, the above equation merely says the capture takes place on (spec-tator, in the sense of equations (3.1)) nucleon a 5 and so one integrates out the degrees of freedom of the two subsystem nucleons. The 3 comes from the antisymmetrization of the wavefunction. u>(ka,k0,kj,kp,k°), loosely speaking, specifies the "dispersion relation" of the subsystem particles in the phase space. There is a degree of ar-bitrariness in choosing the functional form of ui since al l the nucleons are 4 The S function merely enforces the four momentum conservation of the reaction. 5 That is why M(k'a, ka,Mp) appears. 3. T h e impulse app rox ima t ion 35 off-shell and therefore the non-relativistic wavefunction cannot enforce the energy-momentum relation. Suppose one were to integrate 6 Mepm over the phase-space of and P^, that is, one does the following: / d4Pm d4Pme e P m ( 2 7 T ) 4 (27T)4 ( 3 - 1 5 ) The corresponding thing in I A is: (3.16) can be writ ten as: (3.16) = 3 J(2n)35{3){k'a- ka + v + it - fl)(2Tr)5(k'° - k°a + u + K - ^) (2ir)3*<s>$, - k0)(2Tr)6(k'° - *J)(2,r)3*<s>(*; - k,)(2v)5(k!° - k») ^n(k'a, k'p, k'JipiHeiha, h, ky)u(kay kp, ky, k°p, k°)M(k'a, ka, Mp) . 1 1 ^ (27T) 3 (27T) 3 ( 2 7 T ) ( 2 7 T ) ' Using the inverse of equation set (3.1) to transform variables from { £(0,0,7)} to {P,pa, qa} and from to + hp + one has: (3.16) = 3f(2n)W\hp-P) + fa-?a + P+ii-fl) (2n)5(k'°-k°a + u + K-tJ,0) (27r)35(3)(i(P' - P ) _ I ( £ _ + (pfa _ pQ))(27r)a(A;,0 - fc°) ^ ^ ( J * - P) _ I ( £ _ &) _ _ P-Q))(27r)5(fc;0 - fcjj) ^ H ( ^ , % , * > H . ( * a , * / » , f e r ) « A f | || ^ ' ^ ' t y | d{P,pa,qQ) d{P\pa,^) f ^ f ^ f ^ d r ^ d ^ f p ^ dkl_ (2;r)3 (2TT)3 (2TT)3 (2TT)3 (2TT)3 (2TT)3 Ja^ (2TT) 6 One does this integration to get a relativistically invariant quantity so that a direct comparison of the same physical quantity obtained by the IA and the E P M is possible. 3. T h e impulse approx ima t ion 36 Again , using the multivariable version of one of the basic properties of 5 function: 5(n)(x) = | det(T) | 8{n\Tx) (3.19) where n is any positive integer and T is any non-singular linear transfor-mation and | det(T) | is the absolute value of the determinant of T . We have, (3.16) = 3 j"(2TT)3<5(3)(P' - P + v + K - /2)(2TT)<5(£' - E + v + K - /j°) ^ T T ) 3 ^ 3 ^ - pa)(27r)3<5(3>(& - qa + 2-(u + K - p)) (2n)5(k'» - k°p)(2n)5(k'f - k^t^PV«,O^He(Ap«,9a) - - - d3P' d3<f d3f d3P U(ka, k0, ky, k°0, k°y)M(k'a, ka, M p ) ( 2 7 r ) 3 ( 2 7 r) Q3 ( 2 7 r) Q3 ( 2 7 T ) 3 fq^ofp^dE^dE_ n d^_dk\l (27T)3 (27T)3 (2TT) (27T) . j ^ { 2 n ) ^  ^ U> The assumption of impulse approximation is that the two subsystem nucleons are both on-shell; therefore, u>(ka, kp,ky,kp, k®) can be writ ten as: u(ka, fy, £ 7 I Arj, fc?) = (2kp)(2k°)(2Tr)6(kj - M2)(2ir)5(k2 - M p 2 ) (3.21) Using the above functional form of ui and integrating equation (3.20), we have: (3.16) = 3l{2TT)3S{3\P'-P + u + K-fi)(2Tr)S(E'-E + u + K-fi°) (2ir)36Wtfa - pa)(2Tr)35Wtfa - qQ + \{V + K - fi) ^ H . ( ^ . P a . fa)1»H*(P,Pa, Qa)M(k'a, ka, Mp) ^ l ^ f ^ ^ l ^ o f ^ d E ^ d E _ (2TT)3 (2TT)3 (2TT) 3 (2TT)3 (2TT) 3 (2TT) 3 (2TT) (2TT) 1 ' It is tempting to associate the centre of mass momentum of the three nucleons P(P') of I A with P W B H ) of E P M and E = k°a + ifcjj + fc° wi th P ^ e o f E P M and E' = k'£ + k'p+k'® wi th P ^ o f E P M ; however, there are s t i l l two problems which need to be resolved: 1. What is the dispersion relation of the struck nucleon? 3. T h e impulse approx ima t ion 37 2. Do (E,P) and (E',P') satisfy the "on-shell" condition? It turns out that if one puts the struck nucleon on-shell, the answer for ques-tion (2) is negative; one could have adjusted the energy-momentum relation of both the struck incident and final nucleon in a way such that (E,P) and (E',P') are on-shell but the latter approach has the disadvantage that the off-shell nucleon wi l l no longer satisfy the Dirac equation, the equation that has been used extensively to obtain the Adler and Dothan terms in the I A amplitude. Let us see the extent the on-shell condition is not satisfied by (E,P) and (E',P') if we put the struck nucleon on-shell. One wants to know whether P 2 = = Ml and P ' 2 = P4 = M 2 . Now, P = "^"^  ki • kj = 3 M 2 + £ ki-k = 3 M 2 + £ (M2 + \(ki - fc,)2) + 0(fc 4 ) (3.23). Except for the difference in binding energy, the error in assuming that P 2 = M 2 is of the order of the square of the relative momenta of the in i t ia l nucleons. Even for the case of 3 H , the error is at most 0((j%-)2) which is about 1%. The approach that is taken is to regard al l the nucleons as free nucle-ons and assume (to a considerable accuracy) that (E, P) and (E',P') is a four-vector satisfying the on-shell particle criterion. Then, noting that ipme(P,fa,q'a) = V*He(&>&)> o n e c a n associate •M(P3H,P3He ,M t ) o 3 | ( 2 7 r ) 3 ^ 3 > ( £ ^ V'»H(&»OV'»He(Pa,«a) The right hand side of the above expression is just Mia defined previously. We have successfully separated the centre of mass coordinates from the inter-nal coordinates of the I A amplitude. In order to actually do the calculation, one also needs to expand those 6 functions in (3.24) in terms of angular mo-mentum eigenstates. To this end, one has (after setting /7 = 0 and denoting 3. T h e impulse approx imat ion 38 v + k = s): (2*)*8Mtfa-pa) = (2n)M ~ P a ) }2(-iyS^Yi0i0(p'a,Pa) (3.25) Pa 2 (47r)5(2/1 + l)(2i2 + l) / h (2/3+ 1) \ 0 ( 2 7 r ) » * < » > ( & - & + |s) = £ ^ »=l -3 • iix-i a+i» y,i.-™. (^ > g a ) y - ™ . ( S ) ' I^(^^(^ .^(l^dr (3.26) since {pa,qa | -0) = £ i c ^i c(p, <?)(p, Q \ ic) where | ic) is a particular LS coupled state; therefore, one can separate the angular and radial components and use angular momentum algebra to evaluate (3.24) by picking up terms which give non-zero results. The radial integral is a little bit difficult to deal with, as it involves a product of three (highly oscillating) spherical Bessel functions (see ref. [20], chapter 11). Analytical methods, if any, to handle those spherical Bessel functions are not feasible since the radial channel wavefunctions are in numerical form. One numerical method to get around this is to handle these three functions one by one (see appendix A) to avoid this highly singular integral (see also appendix D for definitions of expressions). 3.6.1 Operators Upon the expansion of those S functions into their respective angular momen-tum eigenfunctions, the next thing to do is to manipulate M(k'a,ka, Mp) in equation (3.24) into a form that is suitable to couple with those 8 functions, keeping in mind that the aim of doing all this is to let Mia have a similar format to M(PsR, Psne, Mt) in the E P M except that all the non-EPM degrees of freedom are encapsulated in some formfactors or reduced matrix elements. The method used is to expand the expression non-relativistically in powers of momentum of the struck nucleon ka (which equals q, upon setting the initial centre of mass momentum of the trinucleon zero) and then couple all the spin and angular momentum operators into tensors of rank 0 or 1. Since the total angular momentum of both initial and final states are \, there is no need to couple the operators into tensors of other ranks. There is also no need to couple operators into odd parity as both the initial and final states are of even parity. Recognizing the total angular momentum of the trinucleon system in the IA as that of spin in the EPM, one can easily see that any 3. T h e impulse app rox ima t ion 39 operators of rank 1 in the IA corresponds to (within a factor) a matrices in the E P M and operators of rank 0 in the I A corresponds to identity hadronic operators in the E P M . A l l the coefficients are expanded to O ( ^ ) except that of gp since one of the kinematic endpoints of R M C is quite close to the pole of gp and might make the value of gp large at those places. Therefore, coefficients of gp are expanded to C ( ( ^ ) 2 ) . The correspondence between the I A and the E P M for al l forms of op-erators up to first order in momentum are shown below. Please refer to appendix B for operators of second order of momentum. Please note that the [.. .]s are actually reduced matr ix elements between the ini t ia l and final states (i.e. results of integration of "internal" degrees of freedom) and the numbers inside denote some specific spin and angular momentum combina-tion. They wi l l be defined in equation (3.33). For now, it is sufficient to note that the first digit of [...] is about the nucleon momentum q and the second digit comes from the spherical harmonics of the 8 function. These two terms couple together to have angular momentum of value represented by the th i rd digit. The fourth digit is about the hadronic spin matrix and the subscript is the rank of the whole reduced mat r ix element. 1 (or sometimes denoted as lo) is defined as the hadronic identity matr ix element in both the I A and the E P M , please do not confuse it wi th [1]° although they are related. I A after coupling and reexpressing in E P M format a • v -H-a • q -H-q • v «->• 1 f + (3.29) (3.28) (3.27) (3.30) 3. T h e impulse approx ima t ion 40 * * 9 - f f <* { - ^ [ ( ( 1 , 1 ) 2 ® l ) i ] +^[((1,1)1 + ^ [ ( ( l , 1)0 ® l)x]}v xs-o + *^ | [ ( (1 ,1 )1 ® l)o]v • 5 (3.31) q.va-u o 1 ) 2 ® l ) i ] - ^ | [ ( ( 1 , 1 ) 0 0 -£ ^ [ ( ( 1 . 3 ) 2 ® l ) i ] } a { - ^ [ ( ( 1 . 1 ) 2 ® l ) i ] +^[((1,1)1 ® l ) i ] -^ [ ( ( 1 . 3 ) 2 ® l ) 1 ] } a . i m - s + { - ^ [ ( ( 1 , 1 ) 2 ® l ) i ] - ^ [ ( ( 1 , 1 ) 1 ® l ) i ] -^ y f [ ( ( i > 3 ) 2 ® i ) 1 ] } « ? - H ^ + ^ | [ ( ( 1 , 3)2 ® 1)!](7 • six • sv • § + i ^ | [ ( ( l , 1)1 ® l)o]« • v x 5 (3.32) Note that while a on the left hand side acts on the spin of the spectator nucleon, a on the right acts on the entire nucleus; u and v are mutually commuting vectors that are not concerned wi th the internal momentum (i.e. not p nor q) and commute with o. In this notation, [l]° = [((0,0)0 ® 0) 0]; [of- 1 = [ ( ( 0 , 0 ) 0 ® IK], [of- 1 = - [ ( ( 0 , 2 ) 2 ® IH], [<?]+ = [of- 1 + v ^ ] 2 ' 1 and [<?]" = [a] 0 ' 1 — ^ f ^ ] 2 ' 1 - The precise relationship between [iP]1,1 and the reduced matrix elements defined here is unclear but it has a magnitude to the order of [((1,1)1 ® 0)i] or [((1,1)1 ® l ) x ] . A s one w i l l see later, these two matrix elements are very small. The definition of [((a, b)c[a] ® d)e] (a function of s = + K | | ) is: [((a, b)c[a] ® d)e] = 32^<^l|Te||^-1(^|r1|i)-1£ £ ( - l ) N ' - ' ^ ic,i'c h,h,Li 3. T h e impulse approx ima t ion 41 / P'dp r2drjb(^sr){^,(p, q')jv(q'r)qadq'}{^c(p, q)j,2(Qr)q2+adq} b 0 h l A BV, , h •/ • > / ( 2 L 1 + l ) (2 / 2 + l ) (2 / 1 + l ) 0 0 J ^ ' ^ ' ^ ^ V ^ r j ^ * (^ (^ )ll{(n°1,L1(75'>75) ® y,f q)) ® ( 1 0 ® T d ) } e | | » c ( J ) ) « ( X ) | | ( l 0 ® f ) | | i c ( J ) ) ( 3 . 3 3 ) Notice a specifies the dimension of the matr ix element. When a is not shown explicitly on a reduced matrix element, a = a; that is [((a,b)c <g> d)e] = [((a, b)c[a] <S> d)e]. F(h, l2; a, b, c; i'c, ic) is defined as: F(li,l2;a,b,c;i'c,ic) c a b - l / Yff*(v, x Y Y ° r ( v , x j l f t^t ) , X)dvdx J_C_1 \h+h+b j h h b 1 ATTK ' \ a c I J Z 0 h a \ / (2/ 2 + l ) (2a+Tj 0 0 J V 26 + 1 6hi< ( 3 . 3 4 ) being some dummy angular variables. The notations used here are the same as Br ink and Satchler [28] (also see ref. [29]) except that the Clebsch-Gordon coefficients are denoted by J M J \ J 2 M i M 2 as opposed to ( J M | J x J 2 M i M 2 ) . Please note that the definition of reduced matrix ele-ment is: (-1) 2J" J M J ' J " M ' M ' „ (J||7>||J') = ( J M I T j n M „ | J ' M ' ) ( 3 . 3 5 ) <*c(^)IK(>i 1 ,£ 1 (p ' .P) ® g)) <8> (lo «» T d ) } e | | i c ( ^ ) > is the spin and angular momentum part of the reduced matr ix element between the helion and tr i ton channels. Its calculation is tedious but standard. (i(J)\\{(YLL1(P'>P) ® Y?A?,Q)) ® ( lo ® Td)}e\\ic(J)) = 3. T h e impulse approx ima t ion 42 4 7 r ( - l ) £ ' + 2<+c + L ' + S ' + i + S < + d { ( 2 £ + 1 ) ( 2 £ , + 1 ) ( 2 < S + 1 ) ( 2 < s , + 1 } < | l | r « , | | | > < J L ' 011^,^(^,^)110 Xr> (3.36) where Td = 1 0 for d = 0 and <? for d = 1; also note that £ ( - l ) L l V/2L 1TT<L' 0\\Y°iiLi(p>,p)\\0 L) (3.37) (^ cC^ ilio <S> T^ J|«c(X)> is the isospin contribution of the reduced matrix ele-ment, which equals B y doing all the procedures mentioned in this section, one can match the I A amplitude piece by piece wi th its E P M counterparts and thus a direct comparison between each piece would become possible. Upon doing all the appropriate couplings and integrating the internal coordi-nates, one would obtain the right hand side of equation (3.24) which is M{a. One then obtains ^ v ia equation (3.11). Figure (3.2) shows the statistical I A R M C photon spectra from wavefunc-tions of various model potentials. Figure (3.3) shows the photon spectra at different values of gp and figure (3.4) plots the statistical R M C rate vs. gp. Since the IA treatment of O M C is less complicated than R M C , all the nucleon momentum (q) terms in the I A O M C amplitude are expanded to 0(£j). Note the four-momentum transfer at the hadronic vertex Q2 for O M C is (3.38) 3.7 Results 3.7.1 Ord ina ry M u o n Captu re Q2 = - ( l ^ 1 ) 2 * -0 .964m 2 : M™(k'Q,kQ,Mp) = 9*v«(L {gi +g2v -i> + g3aL • a + —(g4o • q + 3. T h e impulse approx ima t ion 43 > s • a 0.014 0.012 0.01 0.008 0.006 0.004 0.002 X # + EPM calculation for comparison + * + EPM calculation with only GI terms x + IA calculation (Bonn-A) x + IA calculation (Bonn-B) x IA calculation (CD-Bonn) x * IA calculation (Nijmegen) x + IA calculation (Paris) ••7§|i:i;, x IA calculation (AV14) "•(Js^ .. x Klieb and Rood values • 20 X C X 40 60 photon momentum [MeV] 80 100 Fig. 3.2: R M C photon spectra from two E P M calculations (one with the full Adler and Dothan amplitude and the other with only gauge invariant terms) and from IA calculations using various model potentials. A l l wavefunctions used have 22 Faddeev components and the permutation is projected on the same set of states. The infrared divergent part is not shown. <75<?L • q + 9&i> • q + giOL • qo •£> + g8aL • aq- £> + g9a -i>q-i> + iglQD x a • q) + 7^ (<7n<? •qotL-q + 9u | q |2 SL • a + gna • qv • q + g14 \ q \2 a • u)} ( 3 . 39 ) where 92 9v(l v r) - 9M = 9P( 2MP 8 M p 2 v _ Zv3 v 2MP ~ l6My~9v2M~p mu AM] 9 A ^ k + 3. T h e impulse app rox ima t ion 44 gp=1.75of PC AC value 0 20 40 60 80 photon momentum [MeV] F i g . 3.3: Photon spectra from I A calculations for various values of gP. Wavefunction derived from B o n n - A potential is used for calcu-lation. u mu u2 2MP ' 4 M p 2 4 M p 2 ; *3 = - 9 v 2 W P + 9 A { 1 - M I ) + u mu u2 9m{~2M~p + 4M] ~ lM]} mu2 ,2m mu 9 i = 9p41^ + 9A{Wp + 2Ml) mu 2m 9 5 = -9A2M}+9VWP mu 96 = -9A 2M2 3. T h e impulse app rox ima t ion 4 5 u < 0.8 1 1.2 F ^ ( F P at PCAC) or gp/(gp al PCAC) F i g . 3.4: Same as figure (2.2) but for I A calculation. Using Bonn-A poten-t ia l , T ^ ( « > 60 M e V ) = 0.1387 s " 1 . = 9P 97 = 9M( 9s 99 §10 Su §12 9l3 914 rrr mu ) ~ 9 A mu 2M] .rn mu mu ~M{MJ ~ mf + 9AMI Zmu2 4Mp 3 mu m 9v2~MI + 9MMI = 9A 2 m 2 M 2 -9A -gp -gp-2 m 2 M l m2u M l m2u (3.40) 3. T h e impulse approx ima t ion 46 To perform the internal integration, one uses equation (3.28) to (3.32) and the formulae of appendix B to couple the internal operators with the ap-propriate spherical harmonics from the 8 functions. The final product after internal integrations wi l l then be a amplitude that, similar to the R M C one, depends on the external E P M degrees of freedom. The only traces left by internal integration are the "formfactors" that are functions of s. In the case of O M C , s = V and therefore s — Use equation (3.12) to integrate externally with the phase space and get the I A O M C rate. The statistical rates for both the ordinary and radiative muon capture are shown on table (3.1): Potentials O M C rate (statistical) ( s - 1 ) Bonn-A 1368.6 1368.1 1367.8 1357.9 0.6114 0.1387 Bonn-B 1341.3 1340.8 1340.4 1330.7 0.6033 0.1366 C D - B o n n 1336.1 1335.7 1335.3 1326.1 0.6022 0.1364 Nijmegen 1298.0 1297.5 1297.1 1288.2 0.5894 0.1334 Paris 1270.9 1270.4 1270.1 1260.3 0.5800 0.1313 A V 1 4 1271.0 1270.6 1270.2 1260.0 0.5796 0.1313 E P M 1503 0.8189 0.2113 T a b . 3 .1: Various O M C and R M C statistical rates when gP or FP is at the P C A C value. The numbers in the second column are the values obtained by using the " | " prescription up to 0{^-) terms. The numbers i n the th i rd column are obtained v ia the correct approach up to 0{-jtj-) terms. The fourth column has values of the O M C rates using " | " prescription up to 0((-£-)2) terms. Numbers in the fifth column are the values obtained by the correct approach up to 0((jjr-)2) terms; the sixth and rightmost columns contain T ^ ( « > 5 M e V ) and T^(K > 60 M e V ) respectively. 3.7.2 There are several points to note: Discussion 3. T h e impulse approx ima t ion 47 1. Using the correct approach to handle the momentum terms instead of the traditional " | " approach 7 decreases the O M C statistical rate by a mere 0.5 s _ 1 in al l I A calculations up to first order in nucleon momen-tum terms. In the prescription that is used here, there are 24 reduced matrix elements instead of 3 in the " | " prescription. The decrease in capture rate is due to the non-zero value of those extra "internal formfactors" which can be attributed to the non-vanishing amplitude of the non-S wave components of the tri-nucleon wavefunction since the " | " prescription is exact for S waves. The smallness of the effect is pr imari ly due to the minute contribution of the P-state wavefunction to the trinucleon wavefunction. Please note that the effect is a genuine effect and not a effect caused by numerical calculation. To prove this, a two channel Yamaguchi wavefunction consisting solely of S-waves is used to gauge the numerical uncertainty in wavefunction integration. There is an increase of the rate (due only to numerical integration) by 0.1 s _ 1 for the correct approach. This difference is much smaller than the difference (stemming from errors in both the numerical integration and the | approach) of the calculations of other regular 22-channel wavefunctions. Even though the " | " approach is exact for S waves up to first order, it is no longer valid at second order and thus a correct prescription for second order nucleon momentum terms is essential, as can be seen by a 0.6% decrease in capture rate when the correct pre-scription is used for second order nucleon momentum terms. 2. The determinable Adler and Dothan terms have a significant influence on the E P M calculation of R M C rates but their effect on the I A cal-culation is much smaller. They make the ratio of the I A R M C total capture rate to the E P M R M C total capture rate about 10% smaller than the O M C counterparts 8 . For O M C , the ratio of the I A calcula-tion to the E P M calculation ranges from 85 to 90% but in R M C the same ratio is just around 75%. If only the gauge terms 9 were added 7 The " j " prescription, first used by Peterson [30] in his O M C calculation, replaces all nucleon momentum terms q by | . 8 Notice that the influence of those extra Adler and Dothan terms other than those GI terms on the IA amplitude is minimal. One can barely distinguish the two IA photon spectra (one with the full Adler and Dothan amplitude and the other with only GI terms) visually when they are plotted on figure (3.2) 9 The gauge terms are the terms in A J f a d r o n i c R M C (equation (1.35)) that are explicitly 3. T h e impulse app rox ima t ion 4 8 but not the full Adler and Dothan terms, the ratio would be about 83% (see figure (3.2)). One might blame the discrepancy on the failure of the IA in describing R M C by 3 H e but that seems unlikely given its good predictions for O M C . Indeed, the Adler and Dothan procedure seems suspicious when it is applied to both the E P M and I A descrip-tion of the 3 H e —>• 3 H transition, in particular the E P M one when the derivatives of formfactors are large 1 0 . Since the magnitude of the Adler and Dothan terms are large in E P M due primarily to the large size of the derivatives of the nuclear form factors, it is important that al l the extra terms (to a sufficient high order) should be determined in order to have the full effect of those terms on the resulting E P M capture rate. However, this does not happen: the terms of O(KQ) are nor determinable (or at least cannot be determined uniquely) by the Adler and Dothan procedure. Some of those undetermined terms of O(KQ) carry coefficients of magnitude of ( / X 3 H E — n*ii)RM and these terms are actually first order terms disguised as second order and might have a considerable influence on the resulting spectrum. A s a result, a full meson exchange current ( M E C ) calculation seems necessary to tell whether the inadequacy, if there is any, lies wi th the I A or the E P M or both. Congleton and Truhl ik [31] calculated the M E C contribution to the simpler problem of O M C by 3 H e and found that I A + M E C predic-t ion of the exclusive O M C statistical rate agrees wi th both the E P M calculation and experiment [21]. 3. The E P M calculation of both the O M C and R M C rate and the I A calculation of O M C rate using wavefunctions of non-Bonn potentials agree wi th that of K l i eb and Rood . However, the I A R M C spectrum is about 4% lower 1 1 than theirs. A t first it looks a bit contradictory that a more or less the same I A O M C rate but a slightly lower R M C spectrum is obtained in this work but a closer look reveals that K l i eb and R o o d used a lot of approximations evaluating the reduced matrix elements for the R M C spectrum which they did not use for O M C . independent of K (this independence does not include the pion pole terms). They are the terms determined by gauge invariance alone. 1 0 In their original paper, Adler and Dothan applied only the procedure to the case of R M C by a free proton. 1 1 This already takes into account the fact that they took the value of C = 0.965 while C = 0.9788 is used in this work. 3. T h e impulse approx ima t ion 49 (a) Instead of evaluating the reduced matrix elements directly in terms of s, they expressed s in terms of an infinite sum of spherical har-monics of v and k and an artificial cutoff was imposed on them. (b) They did not fully square the resulting matrix element: only prod-ucts of any two most dominant terms and products of one domi-nant and one small term were considered. (c) In expanding the plane wave exp(is • r), they only included the term having jo(vr)jo(Kr) and they used this approximation as the premise in deriving several relationships between various reduced matrix elements for R M C . None of these approximations is used here. Some terms in A J ^ " a d l . o n i c R M C that are present here but not included by Kl i eb and R o o d also tend to decrease the resulting photon spectrum. 4. The Bonn type potentials seem to give a higher (and perhaps better) R M C and O M C results than the other potentials. To analyze this properly, let us take a look at the three dominant reduced matr ix ele-ments: [ l ]° , [a] 0 ' 1 and [a] 2 ' 1 . A l l the curves produced by the non-Bonn potentials seem to be a bit separated from the curves of the Bonn type potentials, especially i n the region when s is large. For [1]° the prob-lem may be partly associated with the numerical normalization since [1]°(0) should be one in principle. However, even though one takes this into account (say, for example, scale the Paris potential curve so that it agrees wi th al l others at s = 0) the value of [l]° is st i l l smaller than that of the Bonn type potentials, as can be easily seen from the fact that the fractional deviation is larger at large value of s. This seems to have to do wi th the binding energy differences by the Bonn and non-Bonn potentials: the non-Bonn potentials generally underbind the trinucleon by about 1 M e V . Congleton and Truhl ik [31] explained since [1]° ~ l — const(r2)s2 (To see this, expand j0(^sr) in polynomials, note const > 0) and ( r 2 ) scales like the inverse of binding energy so one can expect a lower value for [1]° when a wavefunction with a lower binding energy is used. For [<T]0,1, they argued that because of the weak tensor force of the Bonn type potentials, one-body currents w i l l be favoured [a] 0 ' 1 , which is already dominated by contributions from one-body cur-rents. If their analysis is correct, this would potentially explain why 3. T h e impulse approx ima t ion 50 the Bonn type potentials consistently give higher results in both O M C and R M C I A calculations. The quadratic like curve of {a}2'1 is obvious if one notices j2(x) ~ fg for x « 1. Since [a] 2 , 1 is the result of S-D mixing, a tri-nucleon wavefunction with a larger component of D wave would probably have a larger magnitude of [a] 2 ' 1 which seems to be the case for the generally higher D wave component of the non-Bonn po-tentials. Recently Lahiff and Afnan [32] suggested that the Bonn-type potentials might be a better choice than the Nijmegen potential be-cause the energy dependence of propagators is treated exactly (during the evaluation of the potential v ia time-ordered perturbation theory) in the Bonn type potential but the Nijmegen group removes this energy dependence. In a nutshell, the reasons that the Bonn type potentials give a higher result for both R M C and O M C are pr imari ly due to its higher binding energy and possibly its weak tensor force. The higher D wave com-ponents of those non-Bonn potentials may give a boost to [cr]2'1 but the smallness of [cr]2'1 as compared to the other two would make its effect on the I A capture rates small . Figures (3.5) to (3.7) show the variation of [1]°, [a] 0 , 1 and [a]2'1 for various model potentials. Ordinary muon capture occurs at s = 0.52574 fm""1 and the kinematic range for radiative muon capture is 0 < s < 0.52574 f m - 1 . 3.8 Summary and conclusion A n impulse approximation calculation of R M C by 3 H e has been done using various trinucleon wavefunctions. Unlike most, if not al l , other calculations, the calculation is done wi th the momentum space wavefunctions instead of the tradit ional position space ones and the nucleon momentum terms are treated exactly without using the common " | " approach. Moreover, the non-relativistic reduction of the Hamil tonian is better in the present work than that of K l i eb and Rood's [4, 5] since al l terms having coefficient gp are expanded to second order in nucleon momentum terms here. There are slight variations in the resulting photon spectra from the IA calculations for different wavefunctions and these variations can possibly be accounted for by some general characteristics of tri-nucleon wavefunctions such as binding energy and par t ia l wave probabilities. The difference between the impulse approximation and the E P M description of R M C by 3 H e is larger than in the 3. T h e impulse approx ima t ion 51 case of O M C since the extra Adler and Dothan terms contribute a lot more in the E P M calculation. It is not possible to say whether the E P M or the IA is a better description of the nuclear reaction as a l l the experimental results are preliminary [2, 3]. A M E C calculation seems necessary because it might remedy the inadequacy of the Adler and Dothan procedure by providing terms of O(KQ), thus leading to a satisfactory resolution of the discrepancy between the I A and E P M R M C calculation. 0.95 0.85 0.75 F i g . 3.5: Plot of [l]° vs. s for different nuclear potentials. 3. T h e impulse approx ima t ion 5 2 -0.75 -0.8 -0.85 -0.9 h -0.95 F i g . 3.6: P lo t of [c?]0-1 vs. s. -0.02 F i g . 3.7: Plot of [of- 1 vs. s. A. Handling of radial wavefunction integrations in the IA 53 A. Handling of radial wavefunction integrations in the IA A . l Introduction Since the radial trinucleon wavefunction employed in this calculation is nu-merical and the spherical Bessel functions are highly oscillating, this ap-pendix explains the special procedures needed when doing evaluations like that in equation (3.33). These special procedures are indeed "brute force" approaches: parti t ion each zero of the Bessel function and perform a gaussian integration on each of the partitions. There is no sure method of determin-ing the accuracy of this procedure but the smoothness of I A form factors as a function of s obtained after integrating out the internal degrees of free-dom, among others, suggests that a high level of accuracy is achieved by this procedure. A .2 Integration of type: CO, r) = fip(p, q)ji(qr)q2+adq A set of about 70 mesh points on r is chosen in advance. To increase accuracy, this set of mesh points contains more points when C(p, r) has more structure. 1 O f course, one does not know where C has more structure unt i l after doing the integration but it is almost always the case that for r small , C would tend to have more structure. 2 1 A loose definition of structure can be related to second derivatives: the smaller the second partial derivatives of a function, the less the structure of it. 2 One "reason" for the above argument is C(p, r), with all its partial derivatives to all orders, is expected to go to zero "sufficiently fast"when r —¥ oo that would make the r integral finite. A . H a n d l i n g of rad ia l wavefunction integrations in the I A 5 4 For each of the mesh points in r space, the zeros of ji(qr) (in q space) are evaluated. A gaussian integration on q is then performed on the integral for each interval between two adjacent zeros. Usually, 25-35 gaussian points per interval would suffice but there are cases (when the interval is large) which need as high as 70 gaussian points. Since the number of integration points used in q space integration is much larger than the number of mesh points, some techniques should be used for interpolation when the integration points do not coincide with the mesh points in q space on which the wavefunction is defined. A one-dimensional cubic spline (see ref. [33], chapter 3) wi th the "natural" boundary conditions 3 is used, this spline interpolation is seemingly good on inspection. This procedure w i l l minimize the error of the integral due to the highly oscillatory nature of the spherical Bessel functions. A .3 Integration of type: K(p, r)C{P, r)ji(sr)p2dpr2dr Since the p integration is not affected by spherical Bessel functions, it is straightforward. The method used is to integrate using the coefficients pro-vided wi th the wavefunction (remember that nothing has been done on p space yet). The technique used in the r integration is essentially the same as that in the previously discussed q integration, except that one needs two one-dimensional cubic splines (for £ and £') to interpolate the r space for each integration point in p space (the number of integration points in p space varies from 40 to 50). There are also other possible methods, say using a two-dimensional spline for C, but the current method is seemingly more economical in terms of both programming and computing resources. A . 4 "Proofs" of the accuracy of spline interpolation There is no solid proof that the spline used here is accurate but one can estimate the accuracy of the interpolation by some indirect methods. There are at least three hints showing that the spline interpolation in this work is fairly accurate. They are: 3 It means setting the second derivative of both endpoints to zero. A . H a n d l i n g of r ad i a l wavefunction integrat ions i n the I A 55 1. Calculat ion of normalization of the trinucleon wavefunctions using spline interpolation and regular gaussian mesh points agrees to the 0.01% level with that of the true normalization calculated using weighting coeffi-cients provided 4 , i f one uses at least the same number of integration points for integration using interpolation as there are mesh points. 2. [1]°(0) is in principle the numerical normalization of the wavefuction, the agreement between that and the numerical normalization is at the worst 0.15% for al l wavefunctions. 3. The smoothness of the internal "formfactors" suggests that the inte-gration is numerically stable wi th respect to the variation in s, at least in the region of interest. A.5 In tegrat ion cutoffs The integration cutoffs for various spaces are as follows: 1. p space: 45 f m - 1 . 2. q space: 30 f m - 1 . 3. r space: 75 fm. The cutoffs for the p and q space are determined by the domain of the wavefunction provided. The 75 fm cutoff of the r space seems to be a good one since this distance is over 35 times the nuclear radius. A s a test, almost no changes in the "formfactors" are observed when the cutoff is reduced to 70 fm. The set of "natural" mesh points (i.e. mesh points with weighting coefficients pro-vided) for the wavefunction is a totally different set than the set of mesh points used for testing the normalization. B. Second order terms for IA <-> E P M translation 5 6 B. Second order terms for IA <-> E P M translation The second order terms of the I A amplitude after coupling wi th the S func-tions are shown below. Note rf1^ is defined as V2a + 1 I 0 11) (B1) I A after coupling and reexpressing in E P M format qxa-vq-w <-> { - ^ ^ [ ( ( 2 , 2)2 <g> l ) x ] - ^ ^ [ ( f r 2)1 ® l ) x ] -^r 7 ( V ) [((2, 2)0 ® 1):] + ^ ^ [ ( ( O , 2)2[2] ® l)x] + 1 5 ' 7 ( 1 2 1 ) [ ( ( 2 , 0 ) 2 ® 1 ) 1 ] -9 ^ W W , 0)0[2] ® l ) ^ x w • a + ^ / l 7 7 ^ ^ 2 ' 2 ) 2 ® ~ 7 6 ^ 1 ) [ ( ( 2 ' 2 ) 1 ® + yf^V^CCO, 2)2(2] ® l)i]}w; x 5 • «™ • 5 + {\/ |(^ ( 12 1 )[((2 ) 2)1 ® l)x] + 7?(V )[((2, 2)0 ® 1):] -^ [ ( ( O , 2)2[2] ® l)x])}v xs-aw-s-l-V(\\(2,2)l®l)0}w-v + ffi\(2,2)l®l)0]v-sw-s (B.2) b. Second order terms for I A <-> E P M translation 57 q-vq-wa-u {-^=^[((2, 4)2 ® 1)^ + ^1^1,[((2, 2)2 ® 1):] --J=r7(121)[((2, 2)1 ® • wu • sv • s + { - ^ » 7 ( V ) [ ( ( 2 , 4 ) 2 ® l ) 1 ] - ^ T ? ( , i 1 ) [ ( ( 2 l 2 ) 2 ® l ) 1 ] -1 r? ( 121 )[((2,0)2® l)1]}o :-zZJir-T7+. 15 {- v 7=^ ( 1 2 1 ) [ ( (2 ) 4)2® 1),] - 2 ^ T ? ( V ) [ ( ( 2 , 2 ) 2 ® + y|r?(V)[((2, 2)0 ® l) x]}a • • S T J • i + { ~ V ^ O ^ 1 ^ 2 ' 4 ) 2 ® + h ^ ™ ^ 2 ' 2 ) 2 0 " ^r7(V )[((0,2)2[2] <g> 1),] + ^ r 7 ( V ) [ ( (2 ,0 )2 ® 1):] - . ^v /3r?(V )[((0,0)0[2]®l)1]-^r 7 (V )[((2,2)0(g)l) 1]}<7-7ju;.7j-{"7!io^ 1 ) [ ( ( 2' 4 ) 2 0 + { Y I 7 1 ^ ^ 2 ) 2 0 + 1 r7(12a)[((2,0)2<8> \)\]}8 • vu • w + 15 5 ^ f / r t l ^ M l V U . / l j U l r { 2 [ ( ( 2 ' 4 ) 2 0 + V n77 2 [ ( (2>2)2 ® + -^=r?(121)[((2, 2)1 ® l)!]}a • su • wv • s + { ; J u f ( ' 1 > l ( ( 2 , 4 ) 2 0 + /n T ? ( » 1 ) [ ( ( 2 , 2 ) 2 0 + -^r ? (V )[((2 ! 2)1 ® l)!]}a • su • vw • s+ {-y2=^ 12 1 )[((2,4)2 ® ljx] - ^ r ? ( 1 2 ' 1 ) [ ( (2 ) 2)2 ® l)x] + B. Second order terms for IA <-> E P M translation 58 ^-r/V ' tttO, 2)2 ® l ) ! ]}a • sw •vu • s + { V 7 f l 0 ^ 1 ) [ ( ( 2 ' 4 ) 2 ® + {f/^2' 2 ) 2 ® ^ " ^ » r r [ ( ( 2 , 2 ) 1 ® i)i]}<7 • • • s + r? ( 12 1 )[((2, 2)1 <g> l)o]u x u • sw • s + ?7 (121)[((2, 2)1 <g> l)0]w xu- sv-s -fc —77 ( 121 )[((2, 4)2 ® 1),]<? • sw • su • sv • s (B.3) f 1 7 in) 3 y gW[((2, 2)2 ® 1),] - ^VY ' [ ( (2, 2)1 ® 1)!] ^r7 (V )[((2, 2)0 ® - ^ ^ [ ( ( O , 2)2[2] ® 1),] -15r/ ( 12 1 )[((2, 0)2 0 1 ) , ] -^ ^ [ ( ( O ^ O ^ ] ® ! ) ^ - * ; + { \ / | ^ 1 ) [ ( ( 2 ) 2)2 ® 1),] + y | i 7 ( ¥ ) [ ( ( 2 , 2)1 <8) 1) :] + ' | f 7 ( 1 i 1 ) [ ( ( 2 , 2 ) 0 ® l ) 1 ] + y | r ? ( 1 6 1 ) [ ( ( 0 , 2 ) 2 [ 2 ] ® l ) 1 ] } 5 : . s i ; . s q-vq-w f-> ^r7 ( 1 2 ' 1 )[((2,2)l(8i0) 1]t;x s • a w • S + (B.4) "W^KC2, 2 ) 1 ® °)i]^  x s • • s + v 3 { - ^ ( 1 2 1 ) [ ( ( 2 , 2 ) 0 ® 0 ) 0 ] -2 V / 3 r 7 ( 1 o 1 ) [ ( ( 0 ) 0 ) 0 [ 2 ] ® 0 ) 0 ] } t ; - u J + Second order terms for I A <-> E P M translation /fi C. Wavefunction characteristics 60 C. Wavefunction characteristics C . l Channel specifications A l l the 22-channel wavefunctions [25] have the following channel specifica-tions. These 22 channels consist of a l l possible states of the trinucleon system up to and including J = 2, where J is the total angular momentum of the subsystem particle. Channel L I C S s / Channel L I C S S I 1 0 0 0 0 1/2 1 12 2 0 2 1 3/2 0 2 1 1 0 0 1/2 0 13 2 2 0 1 1/2 0 3 1 1 1 0 1/2 0 14 2 2 1 1 1/2 0 4 2 2 0 0 1/2 1 15 2 2 1 1 3/2 0 5 2 2 1 0 1/2 1 16 2 2 2 1 3/2 0 6 1 1 0 1 1/2 1 17 1 3 2 1 3/2 1 7 1 1 1 1 1/2 1 18 3 1 2 1 3/2 1 8 1 1 1 1 3/2 1 19 3 3 0 1 1/2 1 9 1 1 2 1 3/2 1 20 3 3 1 1 1/2 1 10 0 0 0 1 1/2 0 21 3 3 1 1 3/2 1 11 0 2 2 1 3/2 0 22 3 3 2 1 3/2 1 Tab. C.l: Specifications for 22-channel wavefunctions C.2 Model dependent quantities of wavefunctions Table (C.2) lists several important quantities that are dependent on the po-tential of the wavefunction. They are respectively the binding energy given C . Wavefunction characteristics 61 by the wavefunctions, various partial wave probabilities and the numerical normalization (ip | V)num of the wavefunctions. The experimental binding energy Eb of 3 H e is 7.72 M e V and 3 H is 8.48 M e V [34]. Potential E„ P(S) P(S') P(P) P(D) {1> 1 ^ ) n u m Bonn A 8.29 92.59% 1.23% 0.030% 6.14% 0.994 Bonn B 8.10 91.61% 1.19% 0.044% 7.16% 0.993 C D Bonn ( 3 He) 7.91 91.61% 1.35% 0.041% 7.01% 0.993 C D Bonn ( 3 H) 7.93 91.63% 1.31% 0.041% 7.01% 0.993 Nijmegen 7.66 90.31% 1.29% 0.065% 8.34% 0.990 Paris 7.38 90.11% 1.40% 0.069% 8.42% 0.988 A V 1 4 7.58 89.86% 1.15% 0.082% 8.90% 0.987 Tab. C.2: Some important quantities of trinucleon wavefunctions. Binding energy Eb in M e V . P(S) denotes the probability of S-wave and so on. D . Nota t ions and conventions 62 D. Notations and conventions Below are some notations and conventions that are often used in this work: 1. Units: Natural units are used. That is, h = c = 1. 2. Four vectors: Let U and V be two four vectors, their four "dot product" is defined as Note that U° = U0. Sometimes, U is used to denote \U\ but the context w i l l always clear the confusion. 3. 7 matrices: The Dirac representation of 7 matrices (the representation of ref. [35]) is used whenever an explicit representation is required. 4. Spherical harmonics: The bipolar spherical harmonic Y^'^3(x, y) is defined as 5. e: e = +\/4wa where a is the fine structure constant whenever e is used to denote electric charge quantities. 6. G F : G F is the Fermi coupling constant. G F = 1.16639 x 1 0 - u M e V - 2 . 7. V u d : V u d is the C K M matr ix element connecting up and down quarks. |Vud| = 0.9735 ± 0.0008 [27]. 8. m: m is used to denote the mass of a muon = 105.6583568 ± 5.2 x I O " 6 M e V [27] throughout. U -V = UaVa = U°V0 -U-V ( D . l ) YC(mr(y) (D.2) D. No ta t ions and conventions 63 9. m»: m* is the mass of a charged pion = 139.57018 ± 0 . 0 0 0 3 5 M e V [27]. 10. M t : M t is the average value of masses of 3 H e and 3 H . M t = 2808.66 M e V . 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