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Radiative kaon capture processes : K⁻p --> Y [gamma] and K⁻d --> [Lambda] n [gamma] Workman, Ronald Lester 1987

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R A D I A T I V E K A O N C A P T U R E P R O C E S S E S : ' K~p —• Yj and K~d —> A«7 by R O N A L D L E S T E R W O R K M A N B.Sc , University of Victoria, 1981 M . S c , University of British Columbia, 1984 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L 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 D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E STUDIES Department of Physics 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 October 1987 © Ronald Lester Workman, 1987 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. Department The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6(3/81) A b s t r a c t The radiative capture reactions K~p —* A7, K~p —• E°7 and K~d —• Anj are inves-tigated. Branching fractions are calculated for the K~p reactions within a pole model alternatively utilizing both pseudoscalar and pseudovector coupling schemes. The pho-ton spectrum has been calculated for the K~d —> Anj within an impulse approximation model. The above calculations are particularly important as new and more precise data pertaining to the above reactions are expected from a recent Brookhaven experiment. .- Both the pseudoscalar and pseudovector coupling schemes give branching fractions within the current experimental bounds. Sensitivity to the A(1405) resonance is ex-plored and improved constraints are determined for its contribution. A transition op-erator is also derived for use in the K~d —* Anj calculation. Our methods and results are compared with those of other recent calculations of these reactions. The K~ d —• A717 photon spectrum is also calculated in order to explore its sensitivity to the A — n final state interaction. The high energy end of the photon spectrum is found to be sensitive to the choice of A — n scattering parameters, yet relatively insensitive to the contributions from EA?" —> A n conversion, the choice of deuteron wavefunctions and momentum dependence in the transition operator. 11 C o n t e n t s Abstract ii List of Tables v List of Figures vi Acknowledgements vii 1 I N T R O D U C T I O N A N D H I S T O R I C A L P E R S P E C T I V E 1 2 T H E K~p -> Y-y P R E L I M I N A R I E S 6 2.1 Introduction 6 2.2 Rate Relations and Conventions 8 3 T H E I N V A R I A N T A M P L I T U D E 12 3.1 Construction of the Amplitude . : 12 3.2 The Two-Component Reduction 16 3.3 The Momentum Dependent Transition Operator 21 4 D E T E R M I N A T I O N O F C O U P L I N G C O N S T A N T S 24 5 N U M E R I C A L R E S U L T S F O R K~p y 7 29 6 C O M P A R I S O N S W I T H O T H E R M O D E L S 41 7 G E N E R A L F O R M A L I S M F O R K~d -> A n 7 49 7.1 Introduction 49 ii i 7.2 The Differential Rate Relation 52 8 T H E M O M E N T U M D E P E N D E N T T R A N S I T I O N O P E R A T O R 62 8.1 Introduction 62 8.2 Matrix Elements of f 64 9 T H E R E Q U I R E D W A V E F U N C T I O N S A N D A M P L I T U D E S 68 9.1 Introduction 68 9.2 The Deuteron Wavefunctions 69 9.3 The K~N -> Yf Transition Amplitudes 73 • 9.4 The Y'N' -» YN Transition Amplitudes 76 10 N U M E R I C A L R E S U L T S F O R K~d -> A n 7 82 11 C O M P A R I S O N S W I T H P R E V I O U S C A L C U L A T I O N S 93 12 C O N C L U S I O N S 97 References 99 Bibliography 105 A The E° -> A Transition Moment 110 B Some Useful Integrals 114 IV L i s t o f T a b l e s 4.1 Summary of Coupling Constants 28 5.1 Branching Fractions for K~p —> A7 and K~p —> S°7 35 9.1 PS Transition Amplitudes for K~N -> Y-y 75 9.2 A n Scattering Parameter Sets 81 v L i s t o f F i g u r e s 2.1 Non-resonant diagrams for K~p —> A7 10 2.2 Resonant diagrams for K~p —• A7 11 5.1 PS branching fractions for K~p —> A7 36 5.2 P V branching fractions for K~p —• A7 37 5.3 PS branching fractions for K~p —• S°7 38 5.4 P V branching fractions for K~p —> E°7 39 5.5 K~p —* A7 dependence on PA(I405) 40 7.1 Impulse approximation contributions to K~d —• A«7 53 9.1 Deuteron S-state wavefunctions 71 9.2 Deuteron D-state wavefunctions 72 10.1 7-spectrum peak for the reaction K~d —• An7 85 10.2 7-spectrum dependence on A — n scattering parameters 86 10.3 7-spectrum sensitivity to the parameter sets of Nagels et al 87 10.4 7-spectrum sensitivity to the A — n effective range 88 10.5 7-spectrum sensitivity to the S-state deuteron wavefunction 89 10.6 7-spectrum contributions from the deuteron D-state 90 10.7 7-spectrum sensitivity to 0(k/m) terms 91 10.8 7-spectrum sensitivity to 0(pp/m) terms 92 VI A c k n o w l e d g e m e n t s I would like to thank Dr. H . W . Fearing for his invaluable guidance and patience over the past four years. I would also like to thank Drs. D. Beder, D . Measday, B . Jennings, E. Truhlik and A . Gal for helpful suggestions and comments. Credit should also be given to Dr. J . Lowe for sending his Fortran code. Finally, I would like to thank the Natural Sciences and Engineering Research Council and the University of British Columbia for financial assistance. vn C h a p t e r 1 I N T R O D U C T I O N A N D H I S T O R I C A L P E R S P E C T I V E The kaon-nucleon and kaon-nucleus systems have provided a wealth of information on meson-nucleon interactions. Both the pion and kaon are spin-zero mesons with negative intrinsic parity. The kaon, however, introduces an additional strangeness quantum number, a quantity which is conserved in strong and electromagnetic interactions. The spectrum of near-threshold and sub-threshold resonances is also much richer in the kaon-nucleon system than in the analogous pion-nucleon system. The properties of these kaon-nucleon resonances have put strong constraints on the various models of meson-nucleon interactions. In the following, we will study the reactions K~N —• Yj and K~d —* Anj in which a kaon with negligible energy is captured from an atomic s-state. This thesis has thus been divided into two parts. The first part describes the K~N reactions. The K~p —• A7 and K~p —> E°7 rates will be calculated from a set of Feynman diagrams, which will include, in addition to the usual Born terms, the most important resonant contributions in the s-,t- and u-channels. Both pseudoscalar (PS) and pseudovector (PV) coupling schemes will be employed. In addition, a two-component radiative kaon capture operator will be derived for use in the K~d reaction. The second part of this thesis will then be an investigation of the photon spectrum from K~d —> An7, calculated using the impulse approximation and the above mentioned radiative kaon 1 capture operator. Before describing our calculation in detail, however, we should first reflect on the importance of understanding these processes. The K~d —• Awy reaction is the strange analogue of n~d —> nn-f. The latter, non-strange reaction has been utilized[l] in an accurate determination of the s-wave n — n scattering length. Early theoretical calculations oiit~d —> nnj, for at-rest pion capture, found[2] the high energy end of the photon spectrum to be sensitive to the final state n — n interaction. This sensitivity is understood if one realizes that the high-energy limit of the photon spectrum corresponds to the geometry in which both neutrons are emitted anti-parallel to the photon and with zero relative momentum. A n added bonus in this reaction is that, in the final state, the 7 — n interaction is negligible in comparison to the n — n interaction. Thus, this calculation is relatively free of theoretical uncertainty when compared with calculations of reactions having three strongly interacting particles in the final state. The use of K~ d —> An7 in a determination of A — n scattering lengths is more beset by difficulty. Unlike the 7r~d —• nwy reaction[2], the radiative K~d capture is strongly suppressed[3] relative to final states containing three strongly interacting particles. This large background is a problem as existing kaon beams are not as refined as the pion beams at laboratories such as T R I U M F . Thus, sufficient statistics will be more difficult to obtain. In addition, background photons from the 7T° decay in the K~d —* Ann0 reaction may overwhelm a large part of the K~d —> An~j photon spectrum. However, preliminary indications[4] suggest that these extraneous photons can be experimentally vetoed in the region of the photon spectrum most sensitive to the A — n final state interaction. These experimental difficulties have been confronted by an experimental group work-ing at Brookhaven[3]. This group proposes to measure the photon spectrum from K~d —> A « 7 as well as the branching ratios for K~p —> A7 and K~p —> S° 7, which we shall shortly discuss. 2 As the K~d —»• Any reaction has not been studied theoretically in sufficient detail, we have begun this calculation in order to extract information from any forthcoming data. Apart from some early considerations^] of K~d —*• Any and yd —• AnK+, only two detailed calculations of the former reaction exist. The first[6] was carried out in analogy with the ir~d —• nny calculation of reference 7. The radiative capture operator was taken to be momentum independent and of the form a • <f. A Reid soft-core deuteron was used, and the influence of short range effects on the final state wave function was examined using the model of Picker, Redish and Stephenson[8]. The approximation of a frozen nucleus was made, and the kaonic wavefunction was assumed to be approximately constant over the range of strong interactions. From their investigation of the photon spectrum shape as a function of the scattering length combination (as + 2a t)/3, it was found that the theoretical uncertainty in a determination of the above combination of singlet (as) and triplet (at) scattering lengths would be less than 0.2 fm. A n interesting outcome of this calculation was the discovery of a region of the photon spectrum which was very sensitive to the A — n final state interaction. This region extended from approximately 285 M e V to 293 MeV, the endpoint of the photon energy spectrum. Although this calculation was not published, it was in some ways more revealing than a calculation which appeared five years later[9]. This second calculation considered the capture reactions K~d —> YNy for the YN states A n , S°n and E~.p, and allowed for conversion between the various YN states. Unfortunately, there were several inadequacies associated with this calculation. While relations were given for a full calculation of K~d —> YNy, only the Hulthen s-state deuteron was used in generating numerical results. More importantly, the calculated results concentrated on variations of the photon spectrum peak, which exists near 280 M e V , for different sets of scattering parameters describing the K~N —> Yy and YN —» Y'N' processes. As mentioned, the sensitive region is actually nearer to the endpoint of the spectrum. 3 In order to allow for conversion effects, explicit amplitudes of the form Fa • e are required for each of the reactions K~N —• Y~f. The relative magnitudes of these ampli-tudes are sensitive[10] to the method of calculation. For this reason, the K~N —> Yy amplitudes are interesting apart from their influence on the K~d —> Anj interaction. The K~p - * A 7 and K~p -» E ° 7 calculations are particularly interesting as they require knowledge of the poorly understood A(1405) resonance. Two recent calcula-tions of these processes, one in the cloudy bag model[ll] ( C B M ) and the other in the non-relativistic quark model[12] (NRQM), indicate how widely opinions vary on the treatment and importance of this resonance in the K~p —> Y7 reactions. In the C B M , the A(1405) is incorporated implicitly through K~N rescattering and is manifested mainly as a K~N bound state. The lowest mass \ three-quark state is the A(1630). Very different is the N R Q M treatment. Here the dominant contribution to K~p —• Yj is due to the A(1405) which, in this model, is a 3-quark state and also the lowest mass | resonance. Many diagrams, which are usually incorporated into a pole model calculation, have been neglected from the start in this treatment. A recent pole model calculation of K~p —* A7 has also been performed[13]. In this calculation, experimental data[14] for the (K~p —• Aj)/(K~p —• all) branching ratio were used to obtain a value for the A(1405) —• A7 transition moment. Unfortunately, this calculation appears to contain some errors. It has also been criticized by both the C B M and N R Q M papers[11,12] on other more fundamental grounds. Our calculation of the K~p —• 5^ 7 and K~d —> An7 reactions wil l be compared in detail with previous calculations. First, however, we will indicate how our results were obtained. In Chapter 2 we describe the methods and conventions followed in our K~p —• Y^f rate calculation. In Chapter 3, the included set of Feynman diagrams is described, the invariant amplitude is constructed and the overall gauge invariance of our calculation is demonstrated. The two-component reduction of our amplitude, required for the K~d —• An7 calculation, wil l also be performed here. Next, in Chapter 4, the 4 required coupling constants are determined. These couplings are then used to produce the K~p —> Yy rate estimates of Chapter 5. Having obtained results, in Chapter 6 we confront the methods and predictions of the CBM[11], NRQM[12] and the previous pole model calculation of reference 13. In Chapter 7 we present the general formalism for our calculation of the K~d —> Any reaction within a simple impulse approximation model, neglecting any momentum dependence of the K~N —> Yy transition operator. A general relation for the invariant amplitude will be given, including the effects of T,N —¥ An conversion. The differential rate relation will also be presented here. In Chapter 8, we will indicate what additional complications arise when a momentum dependent transition operator is utilized. Having obtained the general relations required in our calculation of the K~d —» Any 7-spectrum, specific wavefunctions and parameters will be chosen in Chapter 9. In Chapter 10, we will present the numerical results of our 7-spectrum calculations. Here we will explore the sensitivities of our K~d —> Any 7-spectrum calculation to the various parameters chosen in the previous chapter. In Chapter 11, our results and methods will be compared to those of previous calculations. Finally, in Chapter 12, we will summarize the conclusions that may be drawn from our calculations of the K~p —* Yy and K~d —> Any reactions. Appendix A addresses the problem of determining a sign for the S ° —• A transition moment. Appendix B describes in detail the methods we have used in order to evaluate the radial integrals arising in Chapters 7 and 8. 5 C h a p t e r 2 T H E K~P - F 7 P R E L I M I N A R I E S 2.1 I n t r o d u c t i o n The K~p —> A7 and —> E°7 reactions are attracting renewed interest due to a new Brookhaven experiment[3], which proposes to measure the branching ratios of these reactions with greatly improved accuracy. The A7 reaction currently has an experimental branching fraction of (2.8 ± 0.8) x 10_3[14] while there exists only an upper limit[15] of 4 X 10~3 for the E°7 branching fraction. Since the C B M [ l l ] and NRQM[12] predict different structures for the sub-threshold A(1405) resonance as well as different rates for the K~p —> Yj reactions, these reactions provide an important test of the two models. In addition to these quark model calculations, it is important to have a detailed pole model calculation, as Born approximation calculations of pion-photoproduction[16] have been successful in reproducing the threshold behavior of the •yp —* mr+ and •jn —» p-K~ reactions. There, apart from small N* contributions, the Born terms are sufficient to describe such processes. In the present calculation, however, the separation of Born from resonance effects is complicated by the presence of sub-threshold resonances. While this is indeed a problem, the threshold radiative kaon capture reactions are best suited for a determination of the A(1405) coupling to A7 and E°7. At higher energies, one must contend with a large number of other contributing resonances. In order to extract the A(1405) couplings, all other contributions to the amplitude 6 must be known or estimated to be small. This method of determining the A(1405) radiative couplings was utilized recently by Burkhardt, Lowe and Rosenthal[13] for the A7 reaction. In that calculation a PS coupling scheme was employed. In addition to the usual Born diagrams, the E° —> A transition, as well as some N* and K* contributions, were included. The A(1405) contribution was then parametrized and adjusted to fit the experimental A7 branching fraction. We have repeated the K~p —> A7 branching fraction calculation and have extended it to include P V strong couplings as well as the K~p —• E°7 reaction. We have also considered the contribution of higher mass Y* resonances in addition to the A(1405). The N* contribution has been given more careful consideration and, couplings have been derived from experimental information whenever possible. Since the A(1405) is just below the K~p threshold and decays almost 100% of the time to E71-, there exists no direct experimental information on its coupling to K~p or F7. Thus, as in reference 13, the A(1405) coupling is undetermined and adjusted to fit the data after all other contributions have been considered. 7 2.2 R a t e R e l a t i o n s a n d C o n v e n t i o n s We consider the reaction in which a kaonic atom is formed prior to the strong K~p interaction. Here the kaon energy is approximately zero and the strong capture is expected from an S-state[17]. Unfortunately, however, the principal quantum number of the state is not well known. It appears, though, that the kaonic wave function may be approximated by a constant over the range of strong interactions[17,18]. Thus, it may be removed from integrals involved in the rate calculation for K~p —> Yj. The decay rate for this process is then given by myk TK-^y, = | 4>K(0) I2 A , 7 \ E lMl2> (2-1) 4ir(mK + mp)mK af£3 wherein <J>K(Q) is the kaonic wave function evaluated at the proton. The quantities my, rriKi mp a n d & a r e the hyperon, kaon and proton masses and the photon energy. The invariant amplitude, M, is given by the diagrams in figs. 1 and 2. S denotes a sum over final and average over initial spins. In order to obtain branching ratios, we also require the total rate for all K~p pro-cesses. We will use the pseudopotential method employed by Burkhardt et al.[13] in order to calculate this rate. The total rate is then given by 2WP | MO) I2, (2.2) wherein Wp is the imaginary part of the K~p pseudopotential and has the value[13] (560 ± 135) M e V / m 3 . One must assume, as seems reasonable, that | <^ A'(0) |2 averaged over the states from which capture occurs is the same for the K~p —• all processes as for K~p —>• Yy, so that it cancels in the branching ratio. It should be emphasized that all branching ratio results we obtain, except the ratio K~p —»• K^/K~p —> S°7 , scale inversely with Wp and will be uncertain by an amount corresponding to the uncertainty in Wp. In the following, we shall use pp, px and py to denote the proton, kaon and lambda or sigma hyperon 4-momentum. The corresponding 3-vector will always be represented 8 by an arrow over the 4-vector symbol. In addition, the unit vector in the direction of a 3-vector s will be represented by s. In the kaon-proton center-of-mass, we have the following simple energy and momen-tum conservation relations mp + mK = E A + k, ' (2.3) and PA + k = 0, (2.4) wherein E\ = \J(PK)2 + rn\. The Mandlestam variables for general momentum are given by the relations s = (PP+PK)2, (2.5) t = (pK-kf, (2.6) u = ( P p - k ) 2 . (2.7) In the particular case of at-rest kaon capture, these reduce to the relations s = (mp + mKf, (2.8) t = m\- 2mKk, (2.9) u = m2p-2mpk, (2.10) which we shall use often. The metric and 7-matrix conventions of Bjorken and Drell[19] have been followed throughout. 9 K P K A ( a ) y A P P ( b ) K P ( d ) Figure 2.1: Non-resonant diagrams for K~p —> A7 10 C h a p t e r 3 T H E I N V A R I A N T A M P L I T U D E 3.1 C o n s t r u c t i o n o f t h e A m p l i t u d e In Chapter 2 we found that the poorly known kaonic wavefunction could be eliminated from our calculation, if we restricted our attention to branching ratios. Thus, apart from the pseudopotential input to the total rate calculation, all of the model dependence in our calculation is contained in M, the invariant amplitude, which we shall now construct. We describe first the pseudoscalar(PS) K~p —* A7 calculation. To construct the amplitude we start with the simple Born terms involving the intermediate radiating p, A and K graphs as well as the S° —+ A transition graph, as given in fig. l(a-d). In addition to these graphs, we considered all Jp = ^ s-wave Y* resonances below 2 GeV. Of the possible u-channel N* resonances, we retained only those which had an appreciable branching fraction to the p~f and AK final states. The t-channel K*(892) contribution of fig. 2(c) was also included. The form of the amplitude, M , required in equation (2.1), is given below. Equa-tion (3.1) contains terms which are representative of each exchange considered above, using PS couplings. M = Y,Mi, (3.1) wherein E K A _ , [(PA + k) • 7 + mA] M i = gxpA-z—uAe-7fc-7 ~2 y5up z/nXp s — 777.^  12 " u — ml I ' ' 2mr ' * i ' 2gKPAeuAy5- ^up, (3 .2) t — mk \/r eKE\- , [(PA + *0 • 7 + "»s] / 0 o A M2 = gKpx-z u A e - 7 f c - 7 - ^ * — - 2 -y5up, (3 .3) ZiTTlp S — 77?. ^ e/cA ( 1 4 0 5 ) A_ (pA + fc) • 7 + mA(1405) ,„ M3 = gKpA(i405)—£ —uAe • yk • 7 7 5 - ^ ^ — 1 (3 .4) 2mp s - m A ( 1 4 0 5 ) + 2mA(1405)r A*4 = gKM+)A-Z ~uA"f5—-—Y- • jk • yup, (3 .5) 2mp u - m'NH) + imN(+)T M „ eKN(-)P-. [(Pp ~ k) • 7 + mM_)] -^ 5 = gKN(-)A—^ " A 5 —: • 7A; • 7 7 s w p , (3.6) TIT - r V a . T 1 \ a l J , ( P A — P p ) a ( P A ~ Pp)n M6 = uA[gK.pA~f +gIK.pJi(pA - pp) -77°] <-ga„ + m In the above, refers to N* resonances with Jp = | . Equation (3 .2) gives the p, A and K contributions, while equations (3 .3 ) and (3 .4) are due to the E ( 1 1 9 2 ) | + and the A ( 1 4 0 5 ) | " . Finally, equations (3 .5 ) , "(3.6) and (3 .7) give the N+, N~ and JK"*(892) contributions, respectively. The form of M for higher Y* s-channel Jp = ^ resonances can be easily obtained from equations (3 .3) and (3 .4) . The quantities T, g and K are respectively the width, strong coupling constant and anomalous magnetic or transition moment associated with each particle exchange. Note that, although the third (kaon radiation) term in equation (3 .2 ) is required in general for gauge invariance, it does not contribute for at-rest capture reactions calculated in the transverse gauge. The dominant contribution to M , in the transverse gauge and in the the absence of resonances, is due to the second (proton radiation) term, as should be recalled from pion-photoproduction[20]. 1 3 In order to calculate the P V form of M, we have used the following substitution for the coupling at the strong vertex 9KNY7S -» 9KNYq • 775- (3.8) The on-shell equivalence UY9KNY7SUN = uYgKNYQ • 775«iv ; q = PY - PN, (3.9) is used to relate CJKNY to gxNY, and gives (JKNY — QKNYlipiy + "Uv)> which is appro-priate when the N and Y have the same parity. In those cases where N and Y have opposite parities, 75 —• 1 in equations (3.8) and (3.9) and, (JKNY = 9KNY/(MY ~ m i v ) -The K*NY coupling was left unchanged. In addition to the above change, to ensure gauge invariance in the P V case, we must add to the amplitude in equation (3.1) the following contact term corresponding to fig. 1(e): Mcontact = gKPAeuAe • ^ 5 u p . (3.10) In the transverse gauge this is the dominant non-resonant contribution to M for the P V case. Once again, this is expected from pion-photoproduction[20]. The K~p —> S°7 amplitudes have the same basic form as described for K~p —> A7. Differences arise from the choice of appropriate couplings at the strong and electromag-netic vertices, corresponding to a S°7 final state which affects the momenta, propagators and phase space. The set of intermediate resonances considered for the A7 final state is also considered for K~p —> S°7, though now of course it is the A —> S° transition which enters, rather than S° —> A. The numerical branching ratios of Chapter 5 are a result of two independent numer-ical evaluations of equations (2.1) and (2.2). The amplitude M in equation (3.1) was converted into a matrix equation using the 7-matrix representations given in Bjorken and Drell[19]. Once in a matrix form, the amplitudes were obtained by explicit nu-merical evaluations of the matrix products. Checks were also made by hand wherever feasible. 14 The requirement of gauge invariance is clearly satisfied by the invariant amplitude. The PS and P V Born terms are gauge-invariant sets of diagrams. Since the JV* and Y* diagrams involve only magnetic couplings, they are individually gauge invariant. The K* diagram is also individually gauge invariant, as the corresponding amplitude is proportional to a term in which both eM and k*1 are contracted with the totally antisymmetric tensor. 15 3.2 T h e T w o - C o m p o n e n t R e d u c t i o n In this section the invariant amplitude of equation (3.1) wil l be reduced to a two-component form. This form of the amplitude will later be utilized in an impulse ap-proximation calculation of the K~d —>• Any process. Furthermore, in a two-component form, the requirement of gauge invariance can be checked more easily. The relative size of the various contributions to our amplitude will also be obvious. First we will write the amplitude of equation (3.1) in the form M = euK(A • e + Be • 7 + C • ek • 7 + De • 7 & • 7 ) 7 5 ^ . (3.11) By reducing the K~p —» A7 Born plus E° —» A plus A(1405) amplitudes, we find for the terms in equation (3.11) 2pp • e 2pK -_e m„ is — m\ \ mpj u — mi . (^A\ PA • k ^ ( 1 4 0 5 ^ PA-k + - { - ^ - ) .—W (3'13) A(1405) KP \ pp • e I K A \ pA- e y\mv) u-m^ \mpJs-m\ («EA\ PA • e (KA(I405)A\ PA • e . . gKpY. 2" + ^PA(1405) " — 2 , (3.14) p 5 - mi '\ mr J s - m j ( 1 < 0 5 ) [ \ mv ) s — m\ u — mlv «SA \ ( m A + m s ) | /«A(I405)AA (^A(I405) - raA) (315) m p y s - m | ^ M ( i 4 0 s ) ^ ^ j s - m A ( 1 4 0 5 ) ' wherein, curly brackets isolate contributions from the Born terms, and the replacement "^A(i405) —* "^ A(i405) — «r/2 is implicit. From the form of equation (3.11) and gauge invariance, we find the relations k-A = 0, (3.16) 16 and B + k-C = 0. (3.17) The term in equation (3.11) proportional to D is gauge invariant by itself and thus gauge invariance does not constrain it. A two-component form for equation (3.11) results if one uses the following relations, valid in the transverse gauge: JEA + mA lEp + mp t f a-pp a • p"A ) Ukl5Up = \— — X A i ^ — T r " ; }XP, (3-18) V 2m A V 2 M P {Ep + mp EA -f m A J / . E A + m A / £ p + m p f / _ _ <? • p A _ ^ a • pp \ V 2 m A V 2 m p I - ^ A + rn A Ep + mp) F - . IEA + mA IEP + mp . x t i _ t . k + k ( + - • fc-^M X p , (3.20) \ \EA + mA^ Ep + mp) EA + mA Ep + mpjXl" y ' , JEA + m A lEp~Tmp~ «A« • l k • 7 7 s u p = ^ • XA \ k(? • e + i— ——a - e x k — ia • e x k——— — ——ka • e ° ^p > YD, \ ^ A + m A £ p + m p £ A + m A £ p + m p J A p ' (3.21) wherein x represents a two-component spinor. The further imposition of the energy and momentum constraints given in equations (2.3) and (2.4), corresponding to an at-rest kaon capture process, significantly reduces the number of contributing terms. Both the A • e and C • e terms give zero contribution for an at-rest capture in the transverse gauge. In this simple case, the amplitude in equation (3.11) can be written in the form pPS M = ^A*-?XP, (3-22) 17 where Fps is the momentum dependent factor corresponding to a PS set of diagrams. For the Born plus S° —> A transition contribution, given in equations (3.2) and (3.3), we have the following result *r = - « I W { I + « A ( I — — ) _ —) (ka—~£—d+"*))} I V rnK + mpJ \EA + mA J \ mK + mp v>) J + gKpLKnk——r I (ma + mA) (1 + ——7 ) - 2(mK + mp) \. (3.23) s ~ m s I \ EA +mA 1 The term proportional to gxP\ in equation (3.23) contains the contribution from Mi , while the term proportional to gxp?. contains the contribution from M 2 . This reduction shows clearly the relative importance of these diagrams. Note that the 0(k/m) terms in equation (3.23) are much larger than the analogous terms from pion photoproduction and cannot be neglected. The P V amplitude can also be reduced to the form of equation (3.22). This is accomplished most easily by reducing the q • 7 factor of equation (3.8) using the Dirac equation. In this way, the following expression, appearing in the P V Born amplitude PK • 7 (Pp~ k) • 7_ + mp can be reduced to the form HA *\ 1 7 5 ^ P ' V \ (3.24) mp + mA u — mt. u ^ { — ^ - + ( P ' - k ) - \ + m ' \ . (3.25) 1 mp + mA u — 1 Similarly, the P V expression (PA + k) • 7 + mA PK-1 is equivalent to 2 , 7 s u P , (3.26) s - mA rnp + mA + { P A + k ) - \ + m * \ - l s « p . (3.27) ^ mp + mA s — mA Using the above relations, one finds that the PS and P V prescriptions give two-component results differing only by a small term: FT = F?S + k ( - ^ ( K A + «p) + - ^ - * £ A ) ( l + T H T - ) • (3-28) \mA + mp H m s + mp ) \ EA + mAJ 18 The Born contribution to FPS can be simplified to Fps(Born) = -gKpK {1 + — ^ (1 + KP + «A)1, (3.29) by regrouping terms in the first line of equation (3.23). In this form, we can easily compare the amplitude with the analogous pion-photoproduction amplitude for jn —> p7r~. Dressler[20] has derived explicit PS and P V two-component expressions which, when written in the notation of equation (3.22), pps(7n -> p T T " ) = -gwpn | ^ 1 + + | ^ - ( *P + «n) j , (3.30) FPV{in - * prr-) = - < ^ n + , (3.31) compare favorably with our expressions (3.28) and (3.29). Comparing equations (3.28) to (3.31), we can easily see why PS 0(k/m) effects are larger for K~p —• A7 than for the analogous n~p —> nj reaction. Apart from a larger photon momentum in the K~p —> A7 reaction, the proton and neutron anomalous magnetic moment contributions nearly cancel in equation (3.30), while the sum of proton and lambda contributions in equation (3.29) does not. By the same argument, Born PS and P V calculations will differ more for K~p —> A7 than for 7r~~p —> n 7 . Two-component reductions of the K~p —*• S°7 and K~n —• £""7 amplitudes will also be required in the K~d —» Kn^ impulse approximation calculation of Chapter 7. For the PS Born plus A —* S° transition K~p —> S°7 amplitude, a reduction to the form of equation (3.22) results in the momentum dependent factor '-,PS 1 - . / - m s \ ( k \ ( mj I V rnK + mpJ \EE + m E / V mK + mp + S K p E K E A — ^ — 5 - 1 ( m E + m A ) ( l + -=7—7 J - 2(mK + mp) 1, (3.32) .s - m A [ y £/ S + m E / J which can be obtained by simply making the replacement A S in equation (3.23). A simplified relation for Fps{Born) is also obtained by making the replacement A —* E 19 in equation (3.29). As K%O and KA have opposite signs, the difference between Born PS and P V calculations will be greater for E°7 than A7. The K~n —* S~7 PS Born amplitude is constructed as was the K~p —> A7 amplitude described in Section 3.1. Diagrams involving the intermediate radiating n, E _ and K are summed to give the total Born amplitude: M(iv n -+ S 7) = gKnve-UY, -e • 7 + e • 7 * • 7 5 7 5 u n \ ZjTTlrp J S — 777 ^ _ (Pn - &) • 7 + m n Kn + ^nse«E75 5 ~ — e - j k - j u n u — m„ 2m p - 2gKn^eu^5 P K 6 un. (3.33) i — mK Once again, reducing M(K~n —•»• £"7) to the form of equation (3.22) we find FP = - g K n J - ^ — + ^-( I m K + mn \ 1 m s + top—; «E- : ; «n J , (3.34) where, in equations (3.33) and (3.34), is the S~ mass. The coupling constants gxPT, and gKnE are related by isospin, which we use to find[21] gKriZ equal to y/2gxPs- In Chapter 7, the operator Ffsa • e*will act between the deuteron and Y N final state spin wavefunctions, where the factor Ffs is equal to (e/2mp)Ffs. 20 3.3 T h e M o m e n t u m D e p e n d e n t T r a n s i t i o n O p e r a -t o r In Chapter 8, an impulse approximation calculation, utilizing a momentum dependent K~p —> A7 transition operator, will be performed for the K~d —» Anj reaction. Here we will construct the operator to be used. The following derivation is similar to the two-component reduction of Section 3.2. However, we will now retain the proton momentum in order to account for its motion in the deuteron. As the kaon is captured from an atomic state with negligible momentum, in the deuteron rest frame, the following momentum conservation relation holds for the capture process: PP=PA + k. (3.35) We then have the following relations for the Mandlestam variables: s = m2K + m2p + 2mKEp, (3.36) t = m\ - 2mKk, (3.37) u = m2 -2{Epk-pp-k). (3.38) As the proton's fermi momentum distribution is peaked below 100 MeV, Ep can be replaced with mp to a good approximation in equations (3.36) and (3.38). Thus, s and t are equivalent to their at-rest capture values. The propagator (u — m2)-1 is modified and can be approximated using 1 ~ _ 1 ^ l + S L l * V (3.39) u — mp 2mpk where the pp term is a small correction to the at-rest capture propagator. In practice, the momenta appearing in our transition operator will be converted to gradient operators for the coordinate space impulse approximation calculation. From 21 equation (3.35) it is clear that we may eliminate the lambda momentum in favor of the proton and photon momenta. This we will do in order to restrict the required derivatives to the photon and deuteron wavefunctions. It is convenient to first eliminate the explicit dependence on pA in the Born ( M x ) plus E° —• A ( M 2 ) plus A(1405) ( M 3 ) amplitudes of equation (3.1): Mx — gxpA-—uKe • 7A; • 7 -2 7 5 u p - \{Pv - k) • 7 + mp] J KP . + 9KP\euA75 o ^ e ' 7 + ^ — e - 7 f c - 7 ^ u p u — m* \ 2mp J - 2gKpAeuAj5fK \up, (3.40) M2 = gKvY. -z uhe • jk • 7* F—2 Ly5up, (3.41) M 3 = fif/fPA(i405)—^ —uAe • jk • 77s * '- 5 u p , (3.42) ZMP S mA(1405) where as in Section 3.2, the A(1405) mass is complex, and the third ( kaon radiation ) amplitude in M i again gives zero contribution. We will retain terms of 0(pp/m) and 0(k/m) relative to the leading order term in our two-component amplitude. To this order, the following truncations are sufficient: —* + @ * Jc UAlsUp « XA^ Xp, (3.43) z m A uAe • 7 7 5 U p » -xAa • exP, (3.44) uAk • 7 7 5 U p « -x\o • kxP, (3.45) uAe • 7& • 77 5 u p » x U ^ • e ^1 + X»> (3-46) 22 when combined with the factors, defined in equation (3.11), IllpfX, \ ''tip J ( 3 - 4 8 ) \ m„ I 2mr,k \ m , , (3.49) 2mpA; y m p / \2mpJ (m# + m p) + m A ( «EA \ 1 , / «A(1405)A ^ 1 „ v -flfATpS 7: 7 ; r - r-fifxPA(i405) I —^ 7 ; : • (3.50) \2mp J (mK + rnp) + mx '\ 2mp J (mK + mp) - mA(i405) Combining terms, we have the following transition operator: egKpAf, f Pp-k _ _ pp • -\ — (1 + Kp) < — cr • e —cr • k ) 2mp { mp mp ) k iegKpT, ( 2 m A \ egKpA(1405) / 2mA \ 1 _ _ ~ o \ -7, -7=-. K2A 7T^ L -7= «A(1405)A > cr • e, (3.51) 2m A [ 2m p \ v s + m E / 2mp \ V 5 _ mA(i405)/ J which depends on the two momentum scales, A; and p"p. The terms involving pp have been simplified through the neglect of pieces proportional to (1 — m p / m A ) . The approx-imations made in deriving equation (3.51) will be justified in Chapter 10, where the effect of these momentum dependent terms is calculated. Having gained some insight into the structure of M , we next proceed to a discussion of the coupling constants used in our numerical calculations. 23 C h a p t e r 4 D E T E R M I N A T I O N O F C O U P L I N G C O N S T A N T S In order to evaluate numerically the amplitude in equation (3.1), we require several strong coupling constants as well as magnetic and transition moments for the electro-magnetic vertices. Since our results are sensitive to the choice of these couplings, this chapter will give a detailed outline of the methods we employed in their determination. The values used in our numerical calculation are given in Table 4.1. The most important of these parameters are those which determine the Born plus E° —»• A transition terms of fig. 1, but fortunately these are for the most part fairly well determined. However, two comments should be made about the values in Table 4.1. First, one should note that only the magnitude of KEA is experimentally known. Thus, we have chosen a sign from the SU(3) prediction[21]: K S A = —Y K n ~ +1-7' (4i1) which is in good agreement with the experimental magnitude. As this choice is not universally agreed upon, the «SA sign is discussed in Appendix A . The magnitude of KEA is taken from the most recent experimental determination[22]. In addition, since the E° magnetic moment is not experimentally determined, we have used the isospin relational] = |(^s~ + (4-2) 24 to estimate its value. Experimental values for the total magnetic moments, /iE±, have been taken from the most recent particle data compilation[23]. For the strong KpA coupling, we have taken the recent dispersion relation deter-mination of Martin[24]. In addition, we have taken Antolin's very recent value[25] for the KpH coupling, which is based on an analysis of charge exchange data and is consis-tent with the earlier estimate given by Martin. The relative sign of the gKpS. a n d gKPz couplings is determined by the SU(6) expectation. The radiative K* —> K~f coupling can be estimated from the K* total width and branching fraction to J<7[23]. Using our form of the K*Ky vertex factor, as given in equation (3.7), the radiative width and coupling constant KK*K are related up to a sign wherein a is the fine structure constant. The strong K*AN and J f *EiV couplings are not well known. Thus SU(3) relations have been employed in order to relate these couplings to the better known pNN couplings. The relations are[26] rfSv = + 2a«<»>)*33, (4-4) and 9KQN = C1 - 2ae(m))gVpNTm (4-5) wherein ae and independent constants appropriate for the vector and tensor couplings respectively. We have used the dispersion relation results[27] of Pietarinen for the pNN couplings. Using Pietarinen's values 9VpNN = 2-6 gTpNN = 16.0, (4.6) and the static SU(6) values[28,29] of 1 and | for ae and am respectively, we find K*AN and K*HN couplings 9K*KN = - 4 - 5 9K-AN = -16-7, (4.7) 25 9K*ZN = -2-6 gJc*xN = 3-2, (4.8) which are in basic agreement with Nagels' fitted values[29]. The tensor K*YlN coupling shows the poorest agreement with Nagels' value. However, we shall find that this discrepancy does not appreciably affect our final result. The relative sign of the KYN and K*YN couplings was chosen to conform with Nagels' value. The largest Jp = | resonant s-channel contributions, apart from that due to the A(1405), are due to the A(1670)i~, A(1800)i~ and £(1750)|~. Strong couplings have been estimated from the full widths and branching fractions of these resonances, using relations of the form r 9BB'M O B + MB-f -m2M i B ( i - ) - B ' ( i + ) M ( o - ) - g 7 r m | wherein Q is the decay 3-momentum of the outgoing kaon. Since there are no photon decays listed for these resonances in the most recent particle data compilation[23], we have used the N R Q M results as guidance. From the decay widths, generated using amplitudes from the Isgur/Karl model[30], tabulated in Table II of reference 12, we have calculated transition moments using ak3 R B(|HB'(|)7 = ~2KBB" (4-10) The A(1600)| + , A(1800) i + and E(1660)§ + can also contribute in the s-channel. However, as we have neither experimental nor N R Q M estimates of their photon decay widths, they have not been included. The effect of this omission is considered in the next chapter. There are two N* resonances with significant decay fractions to and AK final states, the JV(1650)§~ and the JV(1710)| . Here, equation (4.10) was used directly to obtain values of K^*P from experimental information. Similarly, equation (4.9) was employed to estimate the strong Af(1650)| coupling, and the following relation was used to give the iV(1710)| + value: QBB'M ( M B ~ mB,)2 - m2M B ( ! + ) - B < ( i + ) M(0-) - G 7 R M 2 26 Q. (4.11) One type of contribution, which we cannot easily determine, is the coupling to N* resonances which are below the AK threshold. In the kaon photoproduction calculation of Adelseck[31], two values were fit for the coupling constant product <7/CAT(I440)AKJV(I440)P-Using our conventions for magnetic couplings, these two values are 4.1 and 5.0. The signs associated with these various N* and Y* amplitudes are not experimentally determined. When numerical results are presented, however, it wil l become clear that these amplitudes are usually small, and thus do not significantly effect our results. 27 Table 4.1: Summary of Coupling Constants Electromagnetic Couplings «A -0.613 KN(1650)p + 0.320 Kp +1.793 KN(1710)p +0.030 «S° +0.619 KA(1800)E° +0.693 «E°A +1.60 KA(1670)E° +1.250 KA(1800)A +0.117 KE(1750)E° +0.242 A^(1670)A +0.094 KR*K +1.58 ^£(1750^ +0.102 Strong (pseudoscalar) couplings gK-pk -13.2 9KN{\Q50)k 0.810 gK-pY.° +6.0 gKN(l7W)A 6.5 V 9K*VA -4 .5 fi,A"AT(1710)E° 13.0 T 9K*PA -16.7 gKpA(1670) 0.42 v 9K*PI:0 -2.6 gKpA(1800) 1.40 T 9K*PT,° +3.2 9Kp?:(17S0) 0.72 C h a p t e r 5 N U M E R I C A L R E S U L T S F O R K~p — » Y7 Both PS and P V results are presented in Table 5.1 for A7 and E°7 final states using the couplings of Table 4.1. Results have been given for various combinations of the am-plitudes given in equations (3.2) to (3.7), so as to understand the relative contributions of the various terms. Consider first the A7 Born graphs of fig. la-c and, for P V couplings, le which give 3.61 x 1 0 - 3 and 2.65 x 10~3 for the PS and P V coupling schemes respectively. It is interesting to note that both coupling schemes give branching ratios in basic agreement with the published[14] experimental range of (2.8 ± 0 . 8 ) x 10~3 for this simplest possible calculation. Addition of the S° —• A transition diagram, fig. Id , moves both the PS and P V results closer to the central experimental value, but produces only a 13% and 4% effect respectively in the two cases. Further addition of the K* contribution, fig. 2c, lowers both the PS and P V branching ratios by about 10%. The S° —-> A transition diagram and the K* diagram contribute primarily through interference with the other diagrams. Thus the alternative choice of sign for K 2 A , as was taken in reference 13, or for K,K*KI where the sign is really not known, would change the sign but not the magnitude of these contributions. The N* contribution, fig. 2b, which combines the individual contributions of the 7V(1650) and ^(1710) is basically negligible in both coupling schemes. As mentioned in Chapter 4, this N* contribution neglects the effect 29 of TV* resonances below the AK threshold. The PS iV(1440) couplings fit by Adelseck produce a 10% change in our PS branching ratio. Thus, although our estimate of the iV(1440) contribution is model-dependent, it appears that this resonance plays a minor role in determining the A7 branching ratio. Likewise, the Y* contribution, fig. 2a, which combines the individual contributions of the A(1670), A(1800) and £(1750), gives only a few per cent correction to the Born plus £° —» A transition graph result. The omission of Jp — | + Y* resonances should therefore not seriously effect our results. The A(1116)| + itself gives only a small contribution to the K~p —» A7 rate. In order to determine the A(1405) contribution, we have added the amplitude in equation (3.4) and have allowed the composite <7A>A(1405)kA(1405)A coupling to vary freely. If we take the value of 3.2 for gKPA(i405), as was done in reference 13, the resulting dependence of the A7 branching ratio on A(1405) is illustrated in fig. 5.1. In fig. 5.1, the PS A(1405) amplitude has been added separately to the Born plus E° —> A transition and the Born plus E° —•> A transition plus K* amplitudes. For the latter combination, the >«A(1405)A values of 0.0 ± 0.1 and —0.8 ± 0.1 are compatible with the published experimental range of branching ratios. The first range of small «A(I405)A values should be expected, as the theoretical branching ratio lies within the published experimental bounds without any contribution from the A(1405). The actual value of KA(i405)A here will be sensitive to the choice of diagrams, since in this case the small A(1405) contribution will be comparable to the small contributions from, for example, the Y* and N* diagrams. The A(1405) —*• A7 radiative width corresponding to this solution is 0 ± 2 keV. The second range of values requires a large cancellation between the A(1405) and other diagrams. Now the radiative width is 93 ± 25 keV. The value of /«A(i405)A) a n d thus this width, is relatively insensitive to the other included diagrams as the A(1405) contribution is dominant. The above values of «A(I405)A a n < ^ * n e A(1405) radiative width depend very directly on the one published experimental branching ratio[14] which is soon to be supplemented 30 by a new value from the Brookhaven experiment [3]. Fig. 5.1 is thus valuable as it can be used to extract a value for KA(I405)A, and thus for the contribution of the A(1405), from any new branching ratio data. For example, the minimum branching ratio obtainable this way is about l x 1 0 - 3 , which corresponds to «A(I405)A ~ —0.4 and a A(1405) radiative width of 23 keV. Any preferred value of fifKpA(i405) c a n be accommodated by simply scaling «A(I405)A-The above investigation has been repeated for a P V coupling scheme as well. In fig. 5.2, the P V analogue of the A(1405) amplitude was added separately to the Born plus £ ° —> A and Born plus £° —• A plus K* amplitudes as in the PS case. For either of the combinations, any given branching ratio gives KA(I4OS)A values and A(1405) radiative widths which are essentially equivalent to the PS values. The E°7 calculation appears to be more sensitive to the addition of diagrams beyond the Born graphs. The A —»• E° transition diagram dramatically lowers the PS rate while having a 16% effect on the P V rate. The resulting rate is in the PS case about an order of magnitude below the Born plus £° —• A result for A7. The large difference between our A7 and E°7 Born results can be understood simply from equation (3.29) and the ratio of the strong KpA and KpH couplings we have used. Addition of the K* amplitude produces an essentially negligible change in the E°7 rate. The same is true of the ^(1710) contribution represented by N* in Table 5.1. The ^(1650) resonance has not been included, as it is below the UK threshold, and thus we have no experimental information on the -K"iV(1650)£ coupling. The Y* contribution, which represents the same set of resonances considered in the A7 case, is appreciable for a E°7 final state. The addition of this amplitude to the Born plus A —* E° terms increases the PS and P V rates by 44% and 22% respectively. Recall that in Chapter 4 we obtained the Y* couplings indirectly from the N R Q M ; thus this last contribution is model-dependent both as to the magnitude and the relative signs of the various contributions. The omitted Jp = \+ Y* resonances could also make a large contribution here. If the E°7 31 branching fraction were of the order of 10~4, the Y* resonances would severely limit our ability to determine the A(1405) contribution. However, we will see that the E°7 branching fraction is probably much larger than 10 - 4 . We can add to our calculation the S°7 analogue of the A(1405) amplitude of equa-tion (3.4) within both the PS and P V coupling schemes. Results are given for PS and P V couplings in figs. 5.3 and 5.4 respectively showing the E°7 branching ratio for a range of «A(i405)E values. The A(1405) amplitude has been added to both the Born plus A —> S° and Born plus A —• S° plus Y* calculations. The current experimental limit [15] of 4 x 1 0 - 3 puts a weak limit on the range of KA(1405)E values allowed. Assuming, as before, that <7KpA(i405) has a value 3.2, we have /«A(I405)E constrained to be between 1.0 and —0.6 in both coupling schemes. One should note that for S°7, unlike A7, since the Born and other contributions are relatively small, the A(1405) dominates over almost all of the allowed range of KA(I405)E-It is of interest to consider the ratio of the branching fraction for A7 to that for S°7, as this ratio is independent of the quantity Wp appearing in equation (2.2), and thus is perhaps better determined than either of the individual branching ratios. Both the NRQM[12] and the CBM[11] make a definite prediction of 1.3 for the A7/ S°7 ratio. In our model, without the A(1405), the A7 rate is much larger than the S°7 rate. When the A(1405) is included, however, the results change dramatically depending on the relative values of KA(1405)A a n d «A(i405)E as can be seen directly from figs. 5.1 to 5.4. In principle, the «A(I405)A and «A(1405)E transition moments are independent. Given a model for the A(1405), however, these two constants can be related. The simplest possible approach is to treat the A(1405) as a pure SU(3) singlet. W i t h this assumption, «A(i405)s is a factor of y/3 larger than «A(1405)A- Using this relation and the value of KA(I405)A w —0.8, the branching ratio for E°7 is greater than 10~2. Thus this value of /CA(1405)A is probably ruled out by the experimental upper limit of 4 x 10~3. For «A(1405)A ^ —0.4, corresponding to the minimum A7 branching ratio, the branching ratio 32 for E°7 would also be slightly above the experimental upper limit. Although the A(1405) is mainly a SU(3) singlet in the N R Q M , care must be taken in making comparisons. In the N R Q M , interference between singlet and octet components is particularly important to the A(1405) —» £°7 transition, so that the factor of \/3 may not be appropriate. As indicated in Section 2.2, the value of Wp used in our determination of the K~p —• Yy branching fractions has a 25% error. Thus, this factor introduces a 25% error in our branching fraction results. However, since this factor is not present in our A 7 / E ° 7 branching ratio calculation, the results of these calculations, for a small A(1405) contribution, should be relatively firm. If the A(1405) contribution is large, we will require a ( model-dependent ) relation between the couplings KA(I405)A a n d KA(1405)£ in order to calculate this ratio. Another problem is linked to the uncertainty in the A(1405) full decay width. The full decay width is quoted[23] as 40 ± 1 0 MeV. If the A(1405) is the dominant resonance contributing to the K~p —» Yj reactions, then the magnitude at the minima of the curves in figs. 5.1 to 5.4 is related to the A(1405) width. In order to display this effect, fig. 5.5 shows a calculation of the K~p —> A7 branching fraction using the Born plus E ° —* A transition plus A(1405) amplitudes. Fig. 5.5 illustrates the effect of choosing a width between 30 and 50 MeV. Hemingway[32] has fit a Breit-Wigner form to the ETT mass distribution and finds a best value of 32 MeV for the A(1405) width. Fig. 4 of this reference also illustrates that the C B M and K-matrix methods are more successful than the Breit-Wigner form in fitting this mass distribution. In principle, one could make the A(1405) full width energy dependent, evaluat-ing T(y/s) at y/s = 1432 M e V corresponding to the A(1405) energy[33]. However, as T(1432 MeV) is within the 40 ± 10 M e V range, if one chooses T(1405 MeV) to be 40 MeV, the additional complication of an energy dependent width is not useful. The preliminary[4] K~p —• F 7 branching fraction results from Brookhaven appear to indicate a E°7 branching fraction of the order of 1 0 - 3 and in excess of the A7 value. 33 If this is indeed true, the A(1405) contribution will dominate our K~p —v E°7 branching fraction calculation. In addition, as can be seen from figs. 5.3 and 5.4, the uncertainty induced by the Y* resonances will have a small effect. 34 Table 5.1: Branching Fractions for K p —> A7 and K p —> £°7 Branching Fractions for K p —* A7 Diagrams Included Pseudoscalar Pseudovector Born 3.61 x 10" -3 2.65 x 10--3 Born + E° -+ A 3.13 x 10" -3 2.75 x 10" -3 Born + E° -> A + K* 2.84 x 10" -3 2.48 x 10" -3 Born + E° -> A + Y* 3.03 x 10" -3 2.70 x 10" -3 Born + E° —> A + N* 3.09 x 10" -3 2.78 x 10--3 Branching Fractions for K p —• E°7 Diagrams Included Pseudoscalar Pseudovector Born 6.57 X 10- -4 4.14 X 10" -4 Born + A -> E° 3.67 X 10- -4 4.79 X 10- -4 Born + A —• E° + K* 3.65 X 10" -4 4.76 X 10- -4 Born + A -> E° + Y* 5.29 X 10- -4 5.89 X 10" -4 Born + A —»• E° + N* 3.72 X 10" -4 4.80 X 10" -4 35 6 I o 5 <j) Z3 O 4 3 D 2 CD c o sz O m o 1 h V B O R N + I I > A / / / \ B O R N + S ° - > A + K * A / / - \ / / -• \ / / \ / / \ / / \ / / \ / / / / / / / / // - " V I I i i -1.0 •0.8 -0.6 -0.4 K'A(1405)A •0.2 0.0 0.2 Figure 5.1: PS branching fractions for K p —> A7 36 Figure 5.2: P V branching fractions for K~p —> A7 37 Figure 5.3: PS branching fractions for K p —• E°7 38 Figure 5.4: P V branching fractions for K p —• E°7 39 to I O 4 — o 00 o ~o C D c: o c a L. 0Q 0 _L 1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 ^ ( 1 4 0 5 ^ Figure 5.5: K p —* A7 dependence on FA(I405) 40 C h a p t e r 6 C O M P A R I S O N S W I T H O T H E R M O D E L S First consider the calculation of Burkhardt, Lowe and Rosenthal[13]. This is a pole model calculation done in the same spirit as the the one described here. It uses PS couplings only, but contains many of the intermediate resonances we considered. Our calculations do not, however, reproduce their results. The rate relation quoted in ref-erence 13 has an extra, and erroneous, factor of m j f / m p as compared to that given in equation (2.1) which was obtained from the rules of Bjorken and Drell[19], and which seems to agree with the analogous expressions used in other capture processes, e.g. muon capture. This is apparently a typographical error. The process of comparison was complicated by the fact that explicit definitions of vertex factors or matrix ele-ments analogous to our equation (3.1) were not given in reference 13, so that we could not compare at an analytic level. Our comparisons were aided by J . Lowe who supplied a copy of the Fortran code used in the calculations of reference 13. The erroneous phase space factor was not contained in this code. From the relations in his code, it appears that the rate calculations of reference 13 utilized two-component amplitudes analogous to our equation (3.22). The major difference between our calculation and the calculation of reference 13 seems to be a consistent difference of sign for all of the magnetic couplings relative to the proton charge coupling. This sign difference can arise if one is not careful to 41 distinguish between incoming and outgoing photons. For outgoing photons, the vertex relation in equation 10.88 of reference 19 leads to the vertex factors in equations (3.2) to (3.6). When reduced to a two-component form, as in equation (3.29), the Born terms contain a factor 1 + KP + «A which agrees with the standard Blomqvist-Laget[34] operator for radiative pion capture and photoproduction. However, the code supplied by Lowe contains the factor 1 — KP — KA with similar sign changes for all of the other magnetic couplings. This conflicts with Blomqvist-Laget[34] and Dressler[20], and is erroneous. If these signs are reversed in Lowe's code and allowances are made for the magnitudes of our couplings, then Lowe's code can reproduce our results, thus giving added confidence in the correctness of the other aspects of both calculations. Unfortunately, the interference between charge and magnetic terms is quite large, and so this sign change produces a large change in the branching fractions. Consequently, the results of reference 13 are quite different from those we have obtained. In reference 13, the contribution from the iV(1690), which is equivalent to our iV(1650), and the iV(1710) contributions were considered. However, only crude esti-mates were taken for their couplings. The strong i^"iV(1650)A and KN(1710)A cou-plings were taken to be equal to QKPK- In addition, the transition moments were ap-proximated by the Dirac static proton magnetic moment scaled by the ratio mp/mKT*. In comparison, our strong and electromagnetic couplings are much smaller, leading to an essentially negligible contribution from the N* resonances above the AK threshold. Our calculation validates the neglect of higher Y* resonances in the K~p —• A7 calculation of reference 13. However, we find a small but non-negligible contribution from the K*(892), in contrast to the negligible contribution found in reference 13. The K* couplings quoted in Table I of reference 13 are denoted SU(S) values. As described in Chapter 4, though, SU(3) alone is not enough to determine these couplings. Two other calculations of the process K~p —> Y7, one in the NRQM[12] and one in the CBM[11], also obtain branching ratios in fair agreement with experiment. As 42 these involve rather different physics both from each other and from the calculation we have described here, it is of interest to compare and contrast these calculations to try to understand the important features which are responsible for the result in each case. Consider first the N R Q M calculation of reference 12, which is in principle a rather attractive approach. The N R Q M involves a number of assumptions but does correlate a large amount of data and does a good job of describing the spectroscopy of a very large number of states. For the present process, one can calculate the vertex functions purely in terms of parameters fixed from the fits to masses and decays. Thus the result for the branching ratio is firmly predicted and can be compared directly to data. When one looks in detail, however, there is quite a difference between the physics incorporated in this model and that in our calculation. This difference can be traced to a quite different choice of contributing diagrams, which we now discuss. The N R Q M calculation of reference 12 starts from the framework of time-ordered perturbation theory, which effectively separates the Feynman diagrams into separate parts involving positive and negative frequency propagation. The positive frequency parts correspond to the usual non-relativistic theory, whereas the negative frequency parts represent the so-called Z-graphs. In reference 12 all these Z-graphs are neglected, as is the full contribution of the Born graphs, figs, l(a-c) and the transition term fig. Id. The entire contribution then comes from the positive frequency parts of diagrams such as fig. 2a involving a | exchange in the s-channel, plus some rather small quark exchange corrections, which cannot easily by related to any of the diagrams we have used. The major exchange is the A(1405), but also included are the well established resonances A(1670), A(1800), and £(1750) plus two other resonances, £(1620) and £(1810), which have not been seen experimentally with much confidence. With in the N R Q M , the A(1405) contribution dominates and taken alone gives a branching ratio of 2.9 x 10 - 3 . In contrast, with the present model based on Feynman graphs, for «A (1405)A = 0.0 ± 0.1 the A(1405) contribution is quite small and the Born 43 graphs of fig. 1 dominate, whereas for KA(I405)A = —0.8 ± 0.1, the A(1405) contribution is very large and the result is a consequence of a strong destructive interference between the A(1405) and Born graphs. The arguments used in reference 12, for neglecting all but the positive frequency part of the A(1405) and similar exchanges, are several. Parity requires that s-channel resonances be | when the initial kaon and nucleon are at rest in an s-state. This rules out the positive frequency parts of figs, l c and Id which involve A and S exchange. However, the Feynman graph approach also conserves parity, and one can explicitly show that these positive frequency contributions do vanish in that approach also. Thus the parity argument provides physical insight, but does not distinguish between the two approaches. The remaining graphs are neglected in reference 12, based on arguments originating in the composite nature of the hadrons in the quark picture. For the negative frequency part or Z-graphs, it is argued that form factors will suppress their contribution. For the other graphs such as figs, l a and l b , the claim is that the interaction is short ranged involving overlapping hadrons, and thus that the description of hadrons as point particles in this region is not appropriate. These arguments have a certain attractiveness in the context of the N R Q M . Note, however, that the second argument really does not imply that contributions from the diagrams of figs, l a and l b are small and should be neglected, but instead that the calculation would have to be done in some more complicated way. These arguments however present a number of difficulties. In the first place there is a long history of describing medium energy reactions successfully via a series of Feynman diagrams which include both positive and negative frequency parts. Perhaps the most relevant example is n~p — > ro7, which is the direct analogue of the process considered here. In this case the Feynman graph approach gives reasonable agreement with data and, at threshold, the Born terms are dominant [34]. 44 The Z-graphs are also known to be important in other contexts, notably in 'yd —• pn, where the importance of such graphs at threshold was a classic manifestation of meson exchange corrections[35], and in a variety of other situations, particularly few body elec-tromagnetic reactions at the higher momentum transfers[36], which are more relevant to the situation here. There has also been at least one explicit calculation in a strong interaction process[37], pion-nucleon scattering, which finds that neglect of the Z-graphs gives incorrect results. One can also gain insight into this problem by comparing PS and P V couplings as we have done. As is well known, one effect of the P V coupling is to suppress the pair terms. Thus, although we get roughly similar branching ratios with PS and P V couplings, in the P V case a major contribution comes from the contact term necessary for gauge invariance and, unlike the PS case, relatively little from the Z-graphs. Thus, neglect of these negative frequency contributions will give quite different results depending on the choice of basic coupling. There is a further difficulty arising from the gauge invariance requirement, which enters into the choice of diagrams and the interaction appropriate for each diagram. In the Feynman graph approach, the | exchanges, such as the A(1405), involve a mag-netic type coupling and thus are gauge invariant by themselves separately for positive and negative frequency components. However, in the N R Q M of reference 12 this am-plitude is generated by taking the product of matrix elements of single quark operators corresponding to electromagnetic and strong interactions. For a gauge invariant result however, even at the level of free quarks in the intermediate state, one must include the operators which will generate all of the quark level Born graphs, i.e. the quark analog of the Blomqvist-Laget[34] operator for pion-photoproduction, which contains for example extra contact type terms. In addition, for bound quarks, further terms, depending on the binding, wil l be necessary for full gauge invariance. This latter point is a well known, but not completely solved, problem which arises for example in attempts to define a 45 conserved current for the two[38] or three[39] nucleon system. For the Born terms in the Feynman graph approach, if any are included, all must be, as they are not individually gauge invariant. Thus, neglecting the Z-graph parts of the Born terms leads to a non-gauge invariant result. This is in fact consistent with the non-relativistic approach with only positive frequency components, where to get gauge invariance one must add contact terms of the same general type as those which would be generated in a PS theory from the Z-graphs in a two component reduction of the Feynman graphs. It is also consistent with the discussion above of the current operator at the quark level. Aside from the question of gauge invariance, one would usually expect that the Born graphs should be included, as they are normally important and are graphs which are included in simple non-relativistic calculations. For example, the very successful Bonn nucleon-nucleon potential is based on a sum of the positive frequency parts of such simple graphs[40]. In our calculation, these Born graphs give the dominant contribution, apart from possibly the A(1405), though the positive frequency pieces are relatively small compared to the negative frequency ones, in the case of PS coupling or to the contact term for P V coupling. In the N R Q M , it should, in principle, be possible to calculate these extra Born terms, as has been done for pion photoproduction[41]. The calculation of reference 41 indicates that, for pion photoproduction, the Born terms alone reproduce the data fairly well. It would be very interesting to see a detailed calculation of this contribution in the N R Q M . It is an interesting question whether or not one can distinguish between these two approaches empirically, rather than via arguments as above. As long as there is only one piece of experimental information, the branching ratio, the answer is probably not. The N R Q M , together with the particular assumptions of reference 12, makes a specific prediction for K~p —» A7 and K~p —> E°7 branching ratios, which can be tested and in principle ruled out. The approach here, however, depends on the unknown A(1405) 46 couplings which can be adjusted to fit the data, and so cannot be ruled out without additional experimental information. It might also be possible to distinguish the two sets of assumptions by looking at the in-flight process, as one might expect the momentum dependence of the various diagrams to be different. Detailed comparisons with the C B M calculation of reference 11 are more difficult, as it is a very different approach to the problem. The most obvious difference is the inclusion of K~p rescattering and coupling to intermediate HIT channels. In this model, the A(1405) is manifested as a KN bound state rather than the three quark state considered in reference 12. The lowest mass | state is then taken to be the A(1630) and treated as an SU(3) singlet, as opposed to the more complicated structure suggested in the N R Q M calculation. No higher mass Y* resonances are included. For the A7 and £°7 branching ratios, the C B M predicts values of 1.9 x 1 0 - 3 and I. 5 x 1 0 - 3 respectively. While these branching ratios are lower than the N R Q M values, both models predict a AjfE°j ratio near 1.3. In addition, this A7/£°7 ratio is shown to be insensitive to reasonable changes in the C B M parameters and is thus a firm prediction of the model. Although the C B M uses a P V strong coupling, a comparison with our P V calculation is complicated by the fact that only the numerical C B M amplitudes are given in reference 11. The ratio of K~p —> A7 and K~p —»• E°7 contact term amplitudes in Table 1 of reference 11 is, up to a relative sign, just what one would expect for K~pA and K~pTi° strong couplings obeying SU(6) symmetry. The ratio of our contact term amplitudes is determined by the K~pA and K~pTi couplings of Martin and Antolin which are not related in magnitude by SU(6) symmetry. As described in Chapter 4, the relative sign of the above couplings was determined by SU(6) symmetry. However, unlike the amplitudes listed in table 1 of reference 11, our contact term amplitudes have a different sign for the A7 and E°7 reactions. This may, perhaps, be due to a difference in sign conventions. There also appears to be a typographical error in Table 1 of reference I I , as the individual contributions to the imaginary part of the A7 amplitude do not 47 sum to the total. Thus, some care is required in making a detailed comparison. It now appears[42] that reference 11 also contains a computational error. The prelim-inary indication is that the C B M calculation actually predicts a S°7 branching fraction in excess of the A7 value. Since only the dominant contact term is retained from the full set of P V Born dia-grams, this calculation is clearly not gauge invariant. Since we find that the remaining P V Born terms are not negligible in our calculation, perhaps they should also be esti-mated within the C B M . This has recently been done in a chiral bag model[43] for the analogous pion photoproduction reaction, in which these terms are much smaller. 48 C h a p t e r 7 G E N E R A L F O R M A L I S M F O R K~d —> A717 7.1 I n t r o d u c t i o n The K~d —> A n 7 reaction is also attracting renewed interest, mainly because of the possibility of deducing a value for the A — n scattering length from the 7-spectrum. As mentioned in the Introduction, this sensitivity of the 7-spectrum to final state interac-tions has been exploited in the analogous non-strange reaction, Tv~d —> n n 7 . There, the n — n scattering length was determined from the high energy end of the 7 -spectrum of the Tr~d —*• nny reaction. As the high energy endpoint of that 7 -spectrum corresponds to a geometry in which the two neutrons are emitted anti-parallel to the photon, the n — n final state interaction is enhanced. In addition, as the relative n — n momentum is small near the 7 -spectrum endpoint, an effective range approximation is possible, giving a simple relation between the final state wavefunction and scattering length. It is hoped that similar techniques will apply to the K~d —• A n 7 reaction. Infor-mation on the A — n scattering length is currently due mainly to bubble-chamber and hypernuclear data. Dover and Walker[44] have extensively reviewed the attempts to learn about the AN interaction through (K~,ir~) scattering off nuclei. Clearly this is not an unambiguous procedure as the AN interaction takes place in the nuclear medium. Nagels, Rijken and de Swart have carried out a long program[29,45,46] of fitting EJV -» S iV, EiV -» AN and Ap -» Ap bubble-chamber data[47-50] within a 49 boson-exchange-potential model. Although no A n —• A n bubble-chamber data exists, as both particles are neutral, A — n scattering information has been obtained indirectly from the boson exchange potential model and the couplings fit to the abovementioned reactions, for which there exist data. The K~d —• Any reaction thus promises to provide an independent and relatively direct check on previous determinations of the low energy A — n scattering parameters. The K~d —> Any reaction is of particular interest now, due to a recent measurement of its 7-spectrum in a Brookhaven experiment[3]. As this is the first detailed measure-ment of the K~d —• An7 7-spectrum, it is important to reconsider the K~d —> An7 calculation. In this chapter, we will present a general formalism for the K~d —> An7 impulse-approximation calculation. The K~d —> An7 amplitude and differential-rate relations will be given in terms of the K~N —+ Yy and Y'N' —* YN amplitudes. As has been done in the previous[6,9] K~d —• An7 calculations, we will factor the kaon atomic wavefunction from our amplitude. The assumption of a constant kaonic wavefunction is clearly only an approximation to the true kaonic wavefunction. The effects of a strong TT~ — d interaction on the pionic wavefunction has been estimated[51] in calculations of the analogous n~d —> nn7 reaction. There it was found to add an uncertainty of approximately 0.2 fm to the n — n scattering length determination. Estimates given in reference 6 suggest that the uncertainty introduced by the strong K~ — d interaction will add an uncertainty of about 0.2 fm to the A — n scattering length as well. It should be mentioned that the derivation of a precise kaonic-atom wavefunction is a difficult problem. The difficulty lies in the large number of channels to which K~p can couple, and is manifested in the notoriously-difficult-to-fit K~p atom Is level energy shift. Considerable progress has, however, recently[52] been made toward an understanding of this-problem. 50 The effect of a momentum dependent transition operator will be tested in Chap-ters 8 and 10, as this contribution has not yet been considered for the K~d —• Any process. A momentum-dependent operator has been used in a previous n~d —> nny rate calculation[53]. However, the effect of momentum dependence on a determination of the n — n scattering length was not discussed. 51 7.2 T h e D i f f e r e n t i a l R a t e R e l a t i o n As described in Section 7.1, in the following we will develop the general formalism for an impulse- approximation calculation of the K~d —» Any 7-spectrum, using a momentum-independent transition operator. Alterations due to the inclusion of mo-mentum dependence in the transition operator will be considered in the next chapter. In this section, we will derive the invariant amplitude and the general differential rate relation accounting for A n —• A n final state scattering, as well as contributions from final state UN —* A n conversion processes. The transition amplitude for K~d —• An7 is given by 3 M = < I Tj I >, (7.1) j=i where the subscript j in equation (7.1) labels the three contributing processes illustrated in fig. 7.1. The j — 1 label denotes the final state A n —• A n scattering mechanism of fig. 7.1a, while j = 2 , 3 denote respectively the S°n —> A n and E~p —»• A n conversion reactions depicted in figures 7.1b and 7.1c. The initial state wavefunction, contains the deuteron wavefunction and an S-state atomic wavefunction for the kaon: *W = M 0 ) $ d ( r ) . (7.2) As mentioned in Section 7.1 , the kaon atomic wavefunction has been approximated as a constant over the range of strong interactions. The deuteron state[54] is represented by < ? \ $ d > = ^ F 0 o ( O r ) | l M > r + ^^J2Y2m{ttr) I lfJt >< 2mlfi I 211M >, (7.3) where u(r) and u>(r) are respectively the S- and D-state deuteron radial wavefunctions, I lfj, > is a spin 1 state constructed from the spin 1/2 proton and neutron spinors ^ ,1 , 1 1 1 , 1 1 , , I 1^ >= 2^ I 2 m i >l 2 m 2 > < 2 m i 2 m 2 ' 22 ^ > ' ^ ^ 52 53 and the Yim functions are spherical harmonics. The variable r is the proton - neutron relative coordinate. The center of mass wavefunction is not explicitly included here as our calculation will be performed in the deuteron rest frame. In equations (7.3) and (7.4), < 5 i m 1 3 2 m 2 | S1S2SM > is a Clebsch-Gordon coeffi-cient coupling the spins sx and s2 to a total spin S. The usual Condon and Shortley sign convention sign convention has been followed in assigning numerical values to the Clebsch-Gordon coefficients. The transition operators, Tj, in equation (7.1) are f\ = F i e - ' * V * £ ( r p - rK)8(rni - fnf )ax • e, (7.5) f 2 = F2e-rhif'6(rp - rA)6(fni - f n /)a x • e, (7.6) T 3 = F 3e-« f c >".«5(rn, - fh)8{fv - rn/)<72 • e, (7.7) where 3\ (CT2) is the proton (neutron) spin and nt- (« / ) labels the initial (final) neutron coordinate. The factors F{ can be deduced from the two-component reduction of the K~N —y Yy amplitudes in Section 3.2 with ^ Fx = F(K~p - A 7 ) , (7.8) F2 = F(K~p - E ° 7 ) , (7.9) F3 = F(K~n -> S - 7 ) , (7.10) taken as momentum-independent and derived from the at-rest capture reactions. The final state Y N wavefunction is = J ^ & i { P h + ? n ) ' % ^ - ( 7 - n ) 54 In the absence of spin, we have the usual relation for an asymptotic scattering wave-function [55]: = ( 2 ^ {* J ' i e * ' + M ™ 0 ^ } ' (7'12) where cos 8 — f • p, p is the magnitude of p, and the following separation of relative and center of mass variables is implicit: g = m A r A + m wF mA + mn r = r A - r n , (7.14) P=PA+Pn, (7.15) rnnpA - mApn P = : , (7.16) mn + mA where the subscript (n) now refers to the final neutron variable. The scattering ampli-tude fj is then the amplitude for the j t h type of final state interaction h = / i ( A n - » A n ) , (7.17) f2 = / 2 ( £ ° n - > A n ) , (7.18) /a = / 3 ( E - p - A n ) . (7.19) Including spin, we have U?) = 7^372 { ^ e ! p ~ \Sm> + (// | 00 > +/j | l m >) ^ J , (7.20) where the superscript s (t) labels a spin singlet (triplet) scattering amplitude. 55 It is useful to write equation (7.20) in the form ^(r) = j ^ e ' ^ (//&. + //fit) ^ } Xif)xif\ (7.21) where and X 2 ^ a r e respectively the final state lambda and neutron spinors, and we have used the spin singlet (a s) and triplet (a<) projection operators[56] aa = 1(1 - a, • a2), (7.22) and at = 1(3 + a i • a2), (7.23) in order to project out the desired spin states. The K~d —• Ara 7 amplitude in equation (7.1) can be split into pieces depending on the S- and D-state deuteron wavefunctions M = MS + MD. (7.24) The spin part of the S-state amplitude is then Ms = X{mX{2})] j i l i l t o • e + B1 £ (fta. + / » * Ftf • e),- J atX?X{2i], (7.25) where A\ and B\ are spin independent factors, and (<? • e)j is equal to 3\ • e*for j = 1,2 and is equal to B2 • e for j = 3. We have written the deuteron spin wavefunction in the form atX^x2\ with X i ^ ( X 2 ' ) being the initial state proton (neutron) spinor. Equation (7.25) can be reduced to a more manageable form if one commutes the final state spin projection operators past the transition operator using (<7i • <r2) <?i • e = — <?i • e (<7i • <J2) + 2<72 • e, (7.26) and the properties of projection operators atat = at, (7.27) and asat = 0. (7.28) 56 The result is Ms = X[f)txim {Atftffr • ?+ \Bx (F+a, • e- F_a2 • g)} atX?X%\ (7.29) where we have denned F± = F i ( / X s ± fly + F2{f° ± /<)*- F 3 ( / | =F /<)*• (7.30) The plane wave factors in M, coming from equations (7.5) to (7.7), (7.11) and (7.12), can be regrouped using the relations in equations (7.13) to (7.16). Writing these products in terms of the relative and center of mass coordinates, we have and - - - p—tpr -, -, p—ipr ,-iPR p - i k - r A ~ _ e-i(pA+Pn+k)-H e ~ i f r ~ r where we have defined - mnk q = T r. (7.33) (m„ + m A ) —* —+ Integration over the center of mass variable R then leads to a S(p\+pn + k) factor, enforc-ing momentum conservation. The remaining plane wave functions of r are decomposed using [5 7] o o I c** = 4rr ^ £ i[Ji{pr)YCm{np)Ylm{nT). (7.34) Z=0 m=-l The angular integral corresponding to the amplitude Ms is J dtlr y ^ ( f i r ) r o o ( f i r ) = WmO, (7.35) where the loo factor has come from the S-state deuteron wavefunction. The Si06m0 factor selects the 1 = 0 spherical Bessel function from the sum in equation (7.34) and converts the numerical factor A-JvilYim(Qp) to \/4~7r. We are then left with the radial integrals roo U0= / dr ru(r)j0(pnr), (7.36) Jo 57 and roo V0= dr u(r)j0(qr)e-ipT, (7.37) Jo which have been evaluated in Appendix B , for common parametrizations of the deuteron wavefunction. Thus, the spin independent factors of equation (7.25) are given by A1 = V^<f>K(Q)Uo, (7.38) and Bx = V^<t>K(0)Vo, (7.39) and the final result for Ms is then Ms = v / 4 ^ ^ ( 0 ) X ( 1 / ) t X ^ ) t {UoFJr • e + ±V0 (F^ • e- F_a2 • e)} atXfx{?. (7.40) Next, consider the amplitude MB due to the D-state component of the deuteron wavefunction. The spin matrix elements can be simplified, as was done for Ms, to yield MD =< S'm's | A2F\d\ • e + ]-B2 {F+8x • e- F_a2 • e) \ lft >, (7.41) where A2 and B2 are spin independent factors. Once again, the relations in equations (7.31), (7.32) and (7.34) are used to separate the radial and angular integrals. Here the angular integral is J dnr y;m,(oJ.)y2m(nr) = 8l28m,m, (7.42) which selects the 1 = 2 spherical Bessel function. The required radial integrals are then roo W2 = dr rw(r)j2(pnr), (7.43) Jo and roo V2 = dr w(r)j2(qr)e-tpr, (7.44) Jo which are evaluated in Appendix B. 58 The remaining factors, coming from the plane wave decomposition and the Clebsch-Gordon coefficient in equation (7.3), are of the form - 4 7 r ^ r 2 m ( ^ p ) | lfi >< 2ml// | 211M >, (7.45) mfj, and can be written as - S 1 2(p) | V >, (7.46) if we use the relation [54] Si2(r)Y00(£lr) | In >= V8Y,Y2m(SlT) I lfi >< 2mlfi \ 211M >, (7.47) wherein 5 i 2 ( r ) is the usual tensor operator[58] defined by 5i 2 (r ) = Zoi • f a2 • r - 6\ • a2. (7-48) Using the above relation, the spin independent factors defined in equation (7.41) are A2 = - ^ f M 0 ) W 2 S 1 2 ( i U (7.49) and B2 = -\JlM0)V2S12(q). (7.50) Combining the above relations with equation (7.39), we have MD = -^4>K{Q)X\1)]X(2)] {W2F1a1 • eS12(Pn) + \v2{F+ax.e-F.a2-e)S12{q^yatX^\2. (7.51) The relations for Ms and MD in equations (7.40) and (7.51) agree with those quoted in reference 6 thus giving added confidence in these results. In the calculation of \ M |2 we calculate J2 I Ms + MD |2 in order to isolate the S- and D-state contributions. A representative example of the calculational technique is given by the term M, = x¥)]Xl2)*W2F1ff1 • eS^aaPx?, (7.52) 59 in the amplitude MD- The sum over final and average over initial spins leads to "2" | M i |2= | Yl Wl I ^ |2 T n T r 2 • e^ 1 2(p B)a t5' 1 2(p n)5 f 1 • e) , (7.53) spins A where Tr ; denotes a trace over a matrices labeled i. Summing over photon polarizations (A) and factoring out non-spin factors, we have I T l M j 2 = §W? | Fx\2 TriTr2 ( ( T W ^ W 1 ) spins • [3Q3n)j(pn)fc - ^fe][3(Pn)/(pn)m ~ ( 7 - 5 4 ) Through use of the relation A w = 6'm + i e ' m V ( n ) , (7.55) we can eliminate the a factors in equation (7.54), utilizing the fact that the trace of a single cr-matrix is equal to zero. The final result, after some algebra, is then ~£ I M i |2= 1 2 W 2 \Fi\2. (7.56) spins For the total amplitude we have £~ | MS + MD |2= ~ £ | M s |2 +2i2e £ ~ M S M D + "]T | M D |2, (7.57) spins spins spins spins where l T | M 5 | 2 = ^4>2K(0){3U^\F1\2+UoRe[VoF*(3F+-F.)} spins + | | y 0 |2 (3 I F+ |2 - 2 i ? e ( F + F : ) + 3 | F_ |2)} , (7.58) 2Re~J^MSM*D = -2y/2ic</>2K(0)Re { (u0V2 + ^W 2Vo(3(p n • £ ) 2 - 1)) F*F_ spins + V0V*(F+F*_)}, (7.59) "S" | MD I2 = 2TT^(0) J3W 2 2 | Fx |2 + ^ 2 ( 3 ( p „ • £ ) 2 - l ) J?eF 2 F*(3F + - 2F_) spins + \ | F 2 |2 (3 | F+ |2 - 4 i 2 e ( F + F : ) + 3 | F_ |2)} . (7.60) 60 In order to calculate the differential-rate relation, we start with the general rela-tion [59] rfr = 7^ i M to N 3 / 0 N 3 / 0 N 3 0 F F ,( 2 7 R) * (P* + Pi> - PA ~Pn ~ k). 2{mK + mD){£n (2TT)3 (2TT)3 (2TT)3 2EnEAk (7.61) Integrating over d 3 p n we have d T = | M I2 v (7-62) ^ A S P - T S ' ' 2(27r)3(m* + m D ) The 7-spectrum is then found by integrating the result in equation (7.62) dT mkmn rE^max) dk 2(27r)3(mK + mD) JEA(min) s p m over the allowed range of EA values / d E K Y , I M | 2, (7.63) W l anc th JP (max^ (mK + mD- k){Emax - k) ± A rriK + m-D — £k rp _ (mK + m D ) 2 + m A - m 2 2(mK + mD) A = k. . (£roo* - k)2 - ^ -{mK + mD- 2k). (7.66) \ \mK + mD) It is dT/dk which we will plot versus the photon energy in Chapter 10, using the wavefunctions and scattering parameter sets of Chapter 9. In the next chapter, however, we will first indicate what changes are required in order to accommodate a momentum dependent transition operator. 61 C h a p t e r 8 T H E M O M E N T U M D E P E N D E N T T R A N S I T I O N O P E R A T O R 8.1 I n t r o d u c t i o n In this chapter, we will evaluate the required matrix elements of our K~p —> A7 tran-sition operator, given in equation (3.51), in order to account for proton momentum within the deuteron. As we will soon find that YIN —• A n conversion effects are very small near the 7-spectrum endpoint, it is sufficient to consider momentum dependence in only the above transition operator. As the momentum-dependent terms are of 0(pp/m) and 0(k/m) , relative to the leading 0(1) momentum-independent term in equation (3.51), we do not expect a large deviation from the results derived in Chapter 7. This expectation will be verified when numerical results are presented in Chapter 10. Clearly, as the factor k/2mA in equa-tion (3.51) changes by only 3% over the 285 MeV to 293 MeV energy range, such 0(k/m) terms would normally be considered negligible. However, if there is a large cancellation between the 0(k/m) A(1405) contribution and the 0(1) terms in T, as occurs for the KA(i405)A value of —0.8 considered in Chapter 5, 0(k/m) terms can have a larger effect on the 7-spectrum shape. The possible effect of 0(pp/m) terms is less obvious from the form of T, but will become more apparent in the next section. Matrix elements of the proton momentum operator will mix in new angular momen-tum states and require some additional radial integrals to be evaluated. The required 62 radial integrals can, however, be evaluated analytically and thus introduce no additional computational difficulties. 63 8.2 M a t r i x E l e m e n t s o f f From the form of our transition operator, given in Section 3.3, we require matrix ele-ments of the photon and proton momentum operators. As the photon is represented by a plane wavefunction, an eigenfunction of the momentum operator, terms involving the photon momentum present no additional difficulty. The photon momentum operator can simply be replaced by the photon momentum. The pp-operator can be written as p PP = Pd+Y (8.1) —• where Pj_ is the deuteron center of mass momentum and pj_ is the constituent proton-neutron relative momentum. Since we are in the deuteron rest frame, gives zero contribution so that pp is, in coordinate space, replaced by — iS7r where r*is the deuteron relative coordinate. Matrix elements of the gradient operator corresponding to the proton momentum are not as trivial as^those described for the photon momentum since the deuteron wavefunction is not a momentum eigenstate. The proton momentum operator couples initial states with orbital angular momentum I to final states with orbital angular momentum / ± 1. From the form of our transition operator, given in equation (3.51), we must evaluate matrix elements of the operators pp • k and pp • e, in order to include the effects of the constituent proton momentum. We will first examine the contributions from these operators to the amplitude Ms which, as in Chapter 7, is the amplitude involving the deuteron S-state wavefunction. Consider first the matrix elements of the operator pp • k. It is convenient to write the scalar product pp • k in the following way [60]: PP • k = J2(-mPpUkU, (8-2) where we have represented the vectors in terms of spherical coordinates. In this system, a Cartesian vector, V, written in spherical coordinates, has components[61] V± = T^(Vx±iVy) , V0 = VZ. (8.3) 64 Making the replacement pp —> —zV r in equation (8.2), the matrix element of this term will have an angular integral of the form £ J <mrYfm(nr) (-ivr)Y00(nr), (8.4) lm which is analogous to the integral in equation (7.35). As in equation (7.35), the Y00(Q,r) factor comes from the deuteron S-state wavefunction and Y/ m ( f i r ) is one term in the plane wave decomposition given in equation (7.34). Only the 1 = 1 term in equation (8.4) is non-zero and we have[62] £ / dfUrjftr) (V P) F 0 0 ( 0 r ) = < 1, - m , 1, n | 1,1,0,0 > | - , (8.5) with the derivative acting on the S-state deuteron radial wavefunction. The resulting radial integral, corresponding to the plane wave factor in equation (7.31) and analogous to UQ in equation (7.36), is given below: Ui = rdr*iM£{*r)> (8-6) and evaluated in Appendix B. The remaining terms in equations (7.34), (8.2) and (8.5) combine to give an overall factor Ci = 4 7 r£(-ir+' i + 1 < l , - m , l , M | 1100 > (k)^Ylm(SlPn). (8.7) This factor can be simplified through the use of equation (8.2) and the relations yio(p) = \Jl(p)0, (8.8) and Yi±i(P) = yi(P)±, (8.9) to obtain C i = V4^pn • k. (8.10) Similarly, the radial integral corresponding to the plane wave factor in equation (7.32), and analogous to VQ in equation (7.37), is given by V l = f d r r j ^ l ^ e - ^ , (8.11) 65 In addition, in Chapter 10 we will also find that the D-state contributions are quite small and, therefore, the 0(pp/m) corrections to our D-state contribution will produce an entirely negligible change in our 7-spectrum. These correction terms have therefore been dropped. 67 with the overall factor C 2 = -V^q • k. (8.12) Again, V\ has been evaluated in Appendix B , for common parametrizations of the deuteron S-state wavefunction. For the momentum dependent transition operator terms proportional to pp • e, we obtain the same radial integrals as given in equations (8.6) and (8.11). However, the overall factors analogous to C\ and C 2 are in this case equal to \/47r pn • e and — \/47r q • —* e respectively. Note that, since q is proportional to k, the second factor gives zero contribution in the transverse gauge. The first factor is also suppressed in the region of interest, as pn and k are anti-parallel at the high-energy endpoint of the photon spectrum. For contributions to the D-state amplitude, M p , we have an angular integral of the form £ / dnrY;m,(Qr) ( V P ) Y2m(Qr). (8.13) H I " ft I'm' —* Here, the V r operator couples both the 1 = 1 and / = 3 states to the deuteron D-state and we require the angular integrals jdnrY3*m, (vr)^y2m(or) = ( - l ) m + m ' + 1 ^ | < 3, - m ' , 1, n | 3,1,2, m > d_ _ 2N dr r, (8.14) and j dnrY?m.(yr)Y2m(sir) - ( - l ) m + m ' y f < 1, - m ' , 1,H I 1,1,2, m > ^ + fj , (8.15) where the derivative acts on the D-state deuteron radial wavefunction. The D-state matrix elements of our momentum dependent transition operator can be evaluated as described above for the S-state. We will see in Chapter 10, however, that these 0(pp/m) corrections to the S-state calculation alter the 7-spectrum only slightly. 66 C h a p t e r 9 T H E R E Q U I R E D W A V E F U N C T I O N S A N D A M P L I T U D E S 9.1 I n t r o d u c t i o n In Chapters 7 and 8, we have given expressions for the K~d —» Any amplitude and differential rate in terms of several transition amplitudes and integrals over the deuteron S- and D-state wavefunctions. Here, the required amplitudes and wavefunctions will be discussed in more detail. In Section 9.2, we will briefly indicate our choice of deuteron wavefunctions. Sec-tion 9.3 will deal with the momentum independent K~N —• Yy transition amplitudes. Finally, in Section 9.4, we will consider the Y'N' —* YN scattering and conversion ampli-tudes, indicate why conversion effects will be small near the endpoint of the K~d —• Any 7-spectrum, and give the relations connecting the A — n scattering parameters to our final state wavefunctions. 68 9.2 T h e D e u t e r o n W a v e f u n c t i o n s In order to calculate the initial-state wavefunction in equation (7.2), we must choose a deuteron wavefunction. Several such wavefunctions have been plotted in figs. 9.1 and 9.2. Both the S- and D-state radial wavefunctions show a reasonable agreement between the various models beyond 2 fm. The variation between models, as one would expect, is greatest near r = 0. The various S-state wavefunctions plotted in fig. 9.1 show, in general, closer agree-ment than the D-state wavefunctions. The Hulthen S-state wavefunction is an exception; constructed from only two exponential terms, one of which is fixed by the required asymptotic behavior. The S-state wavefunctions of McGee[63] and the Bonn group[40] are fitted to five and eleven terms respectively. The most widely used modern N — N potentials are those of the Paris[64] and Bonn[40] groups. These potentials have similar on-shell and off-shell behavior. In addition, the Paris and Bonn S-state deuteron wavefunctions show agreement, at the 10% level, down to 0.7 fm. We have chosen to use the Bonn deuteron, as a convenient parametrized form has recently been given[40] for the radial wavefunctions. The two Bonn D-state wavefunctions in fig. 9.2 are fits to the full and one-boson exchange potential( O B E P ) deuterons which we shall refer to as B O N N 1 and B O N N 2 respectively. Both agree reasonably well down to about 1 fm. The older wavefunction of McGee, by comparison, is quite different. The behavior of this wavefunction near r = 0 is due to a fault in McGee's parametrization, which gives an analytic D-state that is formally divergent at the origin. As tables of parameters sometimes contain errors, we have tested the deuteron parametrizations of reference 40 by checking the normalization (9.1) (9.2) 69 and the deuteron radius value r2D = i - / drr2 \u{r)2 + w(r)2} . (9.3) 4. Jo *~ J Both tests are satisfied by the parametrizations, with the exception that the version of equation (9.3), given in reference 40, is missing the factor of \. The value of ro quoted in Table 6 of reference 40 requires this factor, which appears in the standard definition of rD[65]. 70 0 1 2 3 4 5 r (fm) Figure 9.1: Deuteron S-state wavefunctions 71 0.3 0.2 0.1 0.0 -.1 I I BONN1 B0NN2 McGEE // •: / • / / • / / • y / I I I I 0 2 3 r (fm) 4 5 Figure 9.2: Deuteron D-state wavefunctions 72 9.3 T h e K ~ N Y 7 T r a n s i t i o n A m p l i t u d e s The momentum dependent factors, Ffs, in our two-component K~N —> Yy transition operators of Section 3.2 are given in Table 9.1. Recall that, from the form of equa-tion (3.22), the factors Ff5 are dimensionless and are to first order just equal to gKNY if one neglects 0 (k/m) terms. The expressions for Ffs obtained in Section 3.2 have been multiplied by e/2mp, in order to obtain the Ftps values appearing in the table. As only PS values have been considered, the PS superscript has been dropped. Values of the Fi have been given for factors derived from the Born, Born plus £° —*• A and, Born plus £° —> A plus A(1405) amplitudes. Notice the very different character of the Born plus E° - * A plus A(1405) results for F^K'p A7) and F2(K~p -> E°7). In the calculation of Fi we have taken the value «A(I405)A = —0.8. The A(1405) amplitude corresponding to this choice of KA(I405)A results in a contribution of (—0.617 + i0.458) fm to Fi. This destructively interferes with the Born plus £ ° —* A contribution, leaving a predominantly imaginary result. The choice of «A(1405)A ^ —0.4, corresponding to the minimum A7 branching fraction, is intermediate to the Born plus E° —• A and Born plus E° -> A plus A(1405) results listed in Table 9.1. Recall, from the discussion of Chapter 5, that «A(1405)E is only weakly constrained by the current experimental upper limit to the E°7 branching fraction. The value of 0.8 for KA(I405)S w a s chosen in the calculation of F2, as this coupling produces a K~p —• E°7 branching fraction near to the experimental upper limit, as well as the current experimental value for the K~p —»• A7 branching fraction. In this case there is less cancellation and the amplitude is mainly due to the A(1405) contribution. The corresponding factors Ti used in the calculation of Akhiezer et al.[9] have also been included in Table 9.1. These transition amplitudes were extracted from the K~N —> Y-y calculation of reference 10, which used SU(3) relations to relate the cou-plings in their Born plus E° <-> A plus A(1405) pole-model calculation. Here the A(1405) decay width to E7r final states was used to determine the A(1405)E7r coupling and SU(3) 73 symmetry, assuming a singlet A(1405), was used to determine a K~pA(1405) coupling. The A(1405)Y7 coupling was taken from dispersion relations[66]. Unfortunately, the factors T{ used in reference 9 have not been taken directly from the calculation of ref-erence 10, but rather have been modified in some unstated manner. A quantitative comparison between the transition amplitudes of reference 9 and our results has not been made due to the fact that a choice of units and conventions has not been clearly stated in references 9 and 10. 74 Table 9.1: PS Transition Amplitudes for K~N Transition Amplitudes (fm) Diagrams Included Fr F2 * 3 Born +0.535 -0.228 -0.119 Born + £ ° <-> A +0.497 -0.170 — Born + £ ° A + A(1405) -0.120 + ^0.458 +0.297 - iO.347 — Akhiezer's Transition Amplitudes F/(F+D) Ratio Fx ^2 ^3 1/4 0.99 - z2.47 - 7 . 0 3 - i 3 . 3 3 -1.82 2/5 4.07-z'1.10 - 2 . 5 2 - i l . 4 9 -0.73 75 9.4 T h e Y'N' YN T r a n s i t i o n A m p l i t u d e s The effect of final state Y'N' —> YN interactions on the K~d —> Any amplitude is contained in the final state wavefunction defined in equation (7.12). More specifically, it is the scattering amplitude, /j(cos#), which contains this information. There exist three distinct regions of the 7-spectrum for the reaction K~d —» Any, each of which requires special consideration in the calculation of Y'N' —* YN scattering amplitudes. Although we require knowledge of only that part of the spectrum near the high-energy endpoint, we will discuss the general behavior of the scattering amplitudes over the entire photon energy range. For photon energies below 222 MeV, there exists sufficient energy in the K~d inter-action to produce both YPny and Y~py final states as well as the Any final state. The amplitudes for A n —• A n scattering and YN —• A n conversion can be described by a two-channel S-matrix / Ve2iReSl ^ ?V1 - ^2et-fle(«i+fia) w. here The form of equation (9.4) follows[67] from unitarity and time-reversal invariance. The phase shifts Si and 82 in equation (9.4) describe the isospin \ An —• A n and YN —> YN interactions respectively. The corresponding T-matrices are [68] TEA„ = ^  (ve*™1 ~ l) , (9-5) n(§) 2ip and L Ue2iR^ - l) , (9.6) T ( | ) 1 /I ~ V2 JRe(6i+S2) / Q 7 N 76 where p and p' are the Y N relative momenta in the A n —• A n and UN —• I]N channels respectively. The isospin space relations in equations (9.5) to (9.7) can be transformed into a particle basis through the relations I £~J> > | £ ° n > in which | I, I3 > is an isospin state. The scattering amplitudes required in equation (7.12) are then / 1 ( A n - A n ) = T i £ A B , (9.11) / 2 ( £ ° n - An) = -^4HAn, (9.12) / 3 ( E - p -> An) = - ^ | r ^ l A n . (9.13) One should also note[69] that p' is the relative momentum in the isospin 1/2 channel. Using equations (9.8) and (9.9) we have , 2 1 t P = gPE°n^E°n + gP£-p-*£-p, (9.14) a relation between the isospin channel momentum and the particle momenta. Relations similar to those in equations (9.11) to (9.13) have been given in reference 9 for the scattering amplitudes required in a calculation of the K~d —» £~.P7 reaction. It should be noted that the above approach is valid over the whole 7-spectrum of the reaction K~d —> H~py. In the K~D —• A n 7 calculation, however, there are additional difficulties near the 222-MeV to 225-MeV photon-energy range. Since the threshold photon energies for E~p and E°n production are at 222 M e V and 225 M e V respectively, there are corresponding threshold cusps in the 7-spectrum at these energies. I 3 /2 , -1 /2 > - y | I 1/2 , -1/2 >, (9.8) 'I I 3 /2 , -1 /2 > + - ^ I 1/2 , -1/2 >, I A n >=| 1/2,-1/2 >, (9.9) (9.10) 77 Luckily these cusps are far from the endpoint of the 7-spectrum for the reaction K~d —»• Any. In the K~d —* Y°ny reaction, however, there is a cusp very near the 7-spectrum endpoint and thus, even in the absence of the existing very large background of extraneous photons, this reaction is less suited to an accurate determination of the E°n scattering length. Dalitz[70] has shown that the T-matrix of equation (9.7) may be expressed in terms of the A n —> A n scattering length in the following way: where A = a + ib is a complex A n scattering length. The imaginary part of A thus determines the strength of transitions between the YN and A n channels. For energies slightly below the YN threshold, a naive calculation of p' yields a complex value, as the corresponding energy is negative. Dalitz has further shown[71] that equation (9.15) can be extended below threshold, assuming that the corresponding K-matrix elements are approximately constant, by simply replacing p' with i | p' |. Clearly this is not a valid extrapolation near the 7-spectrum endpoint, where | p' | is about 280 M e V and our A n scattering length is real. A n analogous situation occurs in the K~d —• A p x - reaction[72]. Fig. 5 of refer-ence 72 illustrates the separation of regions in which Ap —» Ap and YN —• Ap are important [73]. Thus, near the 7-spectrum endpoint, we require only / i ( A n —>• An) given in equa-tion (9.11). In addition, the above simplification removes the need for the transition amplitudes F2ps(K~p —* Y°y) and Fps(K~n —> Y~y). In this case, for a momentum independent transition operator, Fps(K~p —> Ay) becomes an overall constant and may be factored out of the amplitude, leaving no dependence on the K~ N Yy amplitude other than its a • e*form. As our numerical calculations will concentrate on the high-energy end of the photon 78 spectrum, the effective range approximation[74] should be valid: pcotS = -- + \rp2, (9.16) a 2 since p, the relative A—n momentum, is small near the 7-spectrum endpoint. In equation (9.16), a and r are the usual s-wave scattering length and effective range. Using the relation e2iS - 1 1 2i ~cot6-V ( 9 ' 1 7 ) we can write the scattering amplitude in the form fi = jz x " ° , • • (9.18) (1 — ^arp') + tap Notice that for small values of p, equation (9.18) is basically dependent on only the scattering length and, in the limit of zero A — n relative momentum, the scattering amplitude is simply equal to —a. We can estimate the effect of higher partial waves by expanding the scattering am-plitude of equation (7.12) using 1 00 /i(cos0) = - ^ ( 2 / + l )P , (cos^)e i 5 ' s in^, (9.19) P 1=0 where P/(cos#) is a Legendre polynomial. Using the zero range relation[74] p2 / + 1cot<5, = - - , (9.20) ai the 1 = 1 contribution to our scattering amplitude is then ^ = 1> = -TTI£F< 3 C O S*>' ( 9 - 2 1 ) where a\ is the p-wave A — n scattering volume. The p-wave contribution is suppressed here for two reasons, the most obvious being that this amplitude is proportional to p2 and thus goes to zero for small values of p. In addition to this, the p-wave scattering volume is quite small. Nagels[29] has determined that the s-wave A — n scattering length is 79 approximately —2 fm, while the values[29] determined for the p-wave scattering lengths are all less than 0.2 fm3 in magnitude. We have chosen 5 sets of s-wave scattering lengths and effective ranges for use in these calculations. These have been listed in table 9.2. Set 1 is that which corresponds to no final state scattering. Sets 2, 3 and 4 are those fitted by Nagels, Rijken and de Swart in their potential model calculations of references 45, 29 and 46 respectively. Set 5 consists of scattering lengths, which are much larger in magnitude than those of sets 1 to 4 but are within the one standard deviation bounds of reference 47. Thus, set 5 produces, for comparison purposes, a final state scattering enhancement in excess of that which is expected, based on the potential model results. If charge independence holds for the AN interaction, we should also be able to estimate the AN scattering parameters from Ap values. The recently reviewed[75] Ap scattering parameters appear to justify a choice of A n scattering lengths near —2 fm and effective ranges near 3 fm. In the next chapter we will examine the effect of variations in the A n scattering pa-rameters on the 7-spectrum shape. We will also consider the sensitivity of the resulting 7-spectra to differences in the deuteron wavefunctions and momentum dependence in the K~p —* A7 transition operator. These results will then be compared to those of references 6 and 9. 80 Table 9.2: A n Scattering Parameter Sets A n Scattering Lengths and Effective Ranges - Singlet and Triplet Set Scattering Length (fm) Effective Range (fm) 1 as = 0 at = 0 r. = 0 rt = 0 2 as = -2.67 ± 0 . 3 5 at = -1.02 ± 0 . 0 5 rs =2.04 ± 0 . 1 r t = 2.55 ± 0 . 1 3 as = -2.03 ± 0.32 tH = -1.84 ± 0 . 1 rs = 3.66 ± 0.32 rt = 3.32 ± 0 . 1 1 4 as = -2.40 at = -1.84 rs = 3.15 rt = 3.37 5 as - —5.0 at = -3 .0 rs — 3.5 rt = 3.5 81 C h a p t e r 10 N U M E R I C A L R E S U L T S F O R K~d —> Arry Figs. 10.1 to 10.8 are the results of our numerical calculations, illustrating the sensitiv-ities and insensitivities of our 7-spectra to various modifications of the calculation. Fig. 10.1 shows the possible variations of our 7-spectrum shape, using the B O N N 1 S-state deuteron and the scattering parameter sets of Table 9.2. This figure also illustrates the insensitivity of the 7-spectrum peak position to reasonable changes in the final state scattering parameters. A l l three spectra peak near 280 M e V in photon energy. Clearly a A — n scattering length near the n — n scattering length value would shift the spectrum peak towards the high-energy endpoint. However, for scattering length values near to the prediction of reference 46, the peak position and region most sensitive to final state interactions are disconnected. The spectra displayed in fig. 10.1 have been normalized to equal area between 265 M e V and 293 M e V in photon energy. Recall that, as in our K~p —* Yj calculation, an absolute rate calculation would require knowledge of the distribution of atomic states from which the kaon is captured. Thus, as we are only exploring the 7-spectrum shape and each curve is normalized to equal area, our dT/dk values are given in arbitrary units. Fig. 10.2 displays the region between 280 MeV and 293 M e V and illustrates more clearly the difference in 7-spectrum shapes nearer to the high-energy endpoint. As in 82 fig. 10.1, and henceforth, the 7-spectra have been normalized to equal area over the displayed energy range. Fig. 10.3 focuses on the sensitive region between 285 M e V and 293 MeV, where final state effects are most prominent. Here 7-spectra have been calculated from the three parameter sets of Nagels et al. using the B O N N 1 S-state deuteron. Note that the 7-spectrum derived from the earliest parameter set ( Set 2 ) is separated from sets 3 and 4, which give very similar results. The result of variations in the effective range is shown in fig. 10.4, using the param-eters of set 4. Fig. 10.4 demonstrates that the spectrum shape will be insensitive to the errors quoted in Table 9.2 for the effective ranges determined by Nagels et al. In fact, a change of 1 fm in the effective ranges has a negligible effect on the 7-spectrum in the range 292 M e V to 293 MeV. In fig. 10.5 we have plotted the 7-spectra arising from the. scattering parameters of set 4 and three different S-state deuteron wavefunctions. While the B O N N 1 and B O N N 2 S-state wavefunctions produce very similar results over most of the range 285 M e V to 293 MeV, the Hulthen S-state wavefunction deviates only slightly from the two Bonn results over much of this range. Fig. 10.6 reveals that D-state contributions alter the 7-spectrum shape very little near its endpoint. Variations among the different D-state wavefunctions can then be expected to have a negligible effect on our results. The effect of momentum dependence in our K~p —> A7 transition operator is demon-strated in figs. 10.7 and 10.8. Fig. 10.7 indicates the effect of 0(k/m) terms, where k is the photon momentum. The solid curve in fig. 10.7 gives, for comparison purposes, the 7-spectrum resulting from a momentum-independent transition operator. The curve corresponding to a /«A(I405)A value of zero gives the 7-spectrum due to a transition oper-ator derived from the Born and E° —> A contributions alone. The curve resulting from a choice of —0.8 for «A(I405)A displays the effect of a large cancellation between 0(1) 83 and 0(k/m) terms in the transition operator. Clearly, in the case of a small A(1405) contribution, the 0(k/m) terms are negligible, as k/2mA varies only slightly over the displayed photon energy range. Fig. 10.8 indicates that 0(pp/m) terms are very small as well. Thus our expansion of T to first order in pp/m and k/m was justified. Clearly, although we are relatively insensitive to errors in the effective range, differ-ences in the deuteron wavefunction, and momentum dependence in our transition oper-ator, it will be difficult to differentiate between the scattering parameter sets of Nagels et al.[29,45,46]. Unfortunately, statistics will limit a measurement of the 7-spectrum in the last few MeV, where final state interactions are most important. Figs. 10.2 and 10.3, however, indicate that, for reasonable choices of the A n scattering parameters, the 7-spectrum is nearly linear over most of the region between the peak and high-energy endpoint. A measurement over this region is probably feasible since, for each photon in the range 265 M e V to 293 MeV, about 18% will have energies in the range 285 MeV to 293 M e V , given that the singlet and triplet A n scattering lengths are near —2 MeV. Twice that number will have energies in excess of 280 MeV. A determination of the 7-spectrum slope in this region could easily be compared to the results in fig. 10.3. Fig. 10.2 also suggests the possibility of comparing the number of counts near the 7-spectrum peak and endpoint. This method, however, would be more limited by statistics at the high-energy endpoint. 84 Figure 10.1: 7-spectrum peak for the reaction K d 85 2.5 2.0 1.5 h 1.0 0.5 0.0 I [ SET 1 SET 4 '* •- ^ \ — '' - \ \ SET 5 -\ . \ '. \ . V. V. 280 285 290 Photon Energy (MeV) 295 Figure 10.2: 7-spectrum dependence on A — n scattering parameters 86 285 290 295 Photon Energy (MeV) Figure 10.3: 7-spectrum sensitivity to the parameter sets of Nagels et al. 87 Figure 10.4: 7-spectrum sensitivity to the A — n effective range 88 285 290 295 Pho ton Energy (MeV) Figure 10.5: 7-spectrum sensitivity to the S-state deuteron wavefunction 89 285 290 295 Photon Energy (MeV) Figure 10.6: 7-spectrum contributions from the deuteron D-state 90 285 290 295 Photon Energy (MeV) Figure 10.7: 7-spectrum sensitivity to 0(k/m) terms 91 285 290 295 P h o t o n E n e r g y ( M e V ) Figure 10.8: 7-spectrum sensitivity to 0(pp/m) terms 92 C h a p t e r 11 C O M P A R I S O N S W I T H P R E V I O U S C A L C U L A T I O N S The calculation of Gibson et al.[6] is also an impulse-approximation calculation of the 7-spectrum for the reaction K~d —* Any. The initial state is constructed from a Reid soft-core deuteron wavefunction and a kaon atomic wavefunction. A momentum-independent transition operator is used and UN —> An conversion is neglected. As in our calculation, the kaon wavefunction is taken to be approximately equal to constant and is factored from the amplitude. The final-state A n radial wavefunction is constructed from a zero-range wavefunction for values of r greater than 1.4 fm, and a fifth-order polynomial in r is used for values of r from the origin out to the 1.4 fm matching radius. The polynomial and its first two derivatives are matched at r 0 = 1.4 fm and set equal to zero at the origin. This polynomial model is due to Picker, Redish and Stephenson[8] and has been utilized in the ir~d —• nn7 calculation of Gibbs et al.[7] as well. The resulting 7-specfcra of reference 6 have been normalized to equal etrCclj clS have the spectra of Section 10.1. The qualitative behavior displayed in these spectra is also in basic agreement with the findings of Section 10.1. For scattering lengths near —2 fm, the spectra rise rapidly in the photon-energy range between 293 M e V and 290 MeV, and are then nearly linear between 290 MeV and 285 MeV. Although we have verified that momentum dependence is unimportant in the K~p —* 93 Ay transition operator, except possibly for KA(I405)A values near —0.8, the reasoning put forth for its neglect in reference 6 is faulty. In reference 6, it is stated that the momentum-independent term dominates the transition operator. Clearly, from our two-component reductions of Chapter 3, this is not necessarily true. The 0(k/m) terms have only a small effect on the spectrum shape, due mainly to the fact that the overall factor k/2m\ changes by only 3% over the 285 M e V to 293 M e V energy range. The addition of a polynomial dependence in the final-state radial wavefunction of reference 6 is intended to account for short-range suppression in the A — n interaction. In the corresponding ir~d nnj calculation of reference 7, the matching radius was also taken to be 1.4 fm, the pion Compton wavelength. Outside this radius, a Reid soft-core potential was used to generate the n — n wavefunction to be matched at ro = 1.4 fm. The exterior wavefunction was found to be essentially independent of the potential used, as only the l7r exchange tail is important for large r values. Note, however, that in the A — n interaction the 27r exchange gives the longest range contribution to the A — n potential. Thus, a matching radius of 1.4 fm may not be appropriate. Since the radial integrals in equations (7.36) and (7.37) depend on an overlap of deuteron and final-state wavefunctions, the proper choice of r 0 is important. The effect of altering this matching radius has not been addressed in references 6 and 7. Clearly, a value of ro less than 1 fm would greatly reduce the effect of this polynomial extrapo-lation, as the S-state deuteron wavefunction drops off very steeply below 1 fm. A more appropriate method of evaluating the effect of short-range suppression would involve matching to an external wavefunction generated, for example, from the long-range tail of a A — n boson-exchange potential. We have used an asymptotic A n wavefunction for two reasons. The effects of short-range variations, introduced into the A n wavefunction in reference 6, are suppressed by the sharp decline of the deuteron S-state wavefunction below 1 fm. This short-range variation thus has a small effect[6,7] of the scattering length determination. In addition, 94 as we are evaluating overlap integrals with the deuteron wavefunction, an inappropriate choice of r 0 could itself introduce error into the scattering length determination. Although explicit matrix elements are not given in reference 6, it is stated that the 7-spectra can be parametrized in terms of a single effective scattering length equal to (2at + as)/3. The 7-spectra presented in this reference are then labelled by this constant and an effective range, which is presumably some combination of the singlet and triplet effective ranges. The amplitudes given in equations (7.58) and (7.60) and in reference 9, however, indicate that the 7-spectrum of the reaction K~d —> An7 depends on the singlet and triplet scattering parameters independently and cannot be expressed in terms of the above scattering length combination. Next consider the calculation of Akhiezer et al.[9], which allows for EA7' —> A n conversion effects and follows the formalism outlined in Chapter 7. A momentum-independent transition operator of the form F{(K~N —> Yj)a • e is also used here with the factors Fi as discussed in Section 9.3. Unfortunately, it appears that many important features of the K~d —* An7 reaction have been ignored in this calculation. Akhiezer et al. claim to have calculated total rates for the K~d —• An7 and K~d —> E~p7 reactions. The calculation of a total rate, however, requires knowledge of the distribution of atomic states from which the kaon is captured. The calculation of Akhiezer et al. has been performed only for a Is atomic capture. In addition, the ir~d —» nn7 results of reference 16 appear to indicate that the assumption of a constant kaonic wavefunction and the neglect of the deuteron D-state will add a large error to the total rate calculation. As mentioned earlier, the 7-spectra presented in reference 9 have not been normalized and concentrate on the rather insensitive 7-spectrum peak, instead of the sensitive region near the 7-spectrum endpoint. Thus reference 9 reveals very little about the one region of the 7-spectrum which is sensitive to the A n scattering parameters. 95 Relations for the photon and hyperon polarization calculations have also been given in Akhiezer et al. , though no numerical values have been calculated. As it may be possible[4] to measure the A polarization in the K~d —• Any reaction, this would place an additional constraint on the An —> An and K~p —• A7 amplitudes. 96 C h a p t e r 12 C O N C L U S I O N S In the following, we will summarize what has been learned through our calculations of the at-rest capture K~p —> A7, K~p —> E°7 and K~d — > An-y reactions. Our detailed findings from the K~p —* A7 and K~p —• E°7 branching-fraction calcu-lations are presented in Table 5.1 and figs. 5.1 to 5.5. Although conclusions concerning the A(1405) contribution to these reactions will depend upon the precise branching-fraction results expected from Brookhaven[3], several comments can now be made. The published[14] A7 branching fraction of (2.8 ± 0.8) x 1 0 - 3 results in two allowed ranges of KA(I405)A values, 0.0 ± 0 . 1 and —0.8 ± 0 . 1 , corresponding to a choice of 3.2 for 9K-pA(i405)- The second range of /«A(1405)A values requires a large cancellation between the Born and A(1405) contributions. If, however, the preliminary[4] indications of a A7 branching fraction close to 1 x 1 0 - 3 are correct, «A(i405)A is actually near —0.4 and definitely negative. If the preliminary[4] value of (2 — 3) x 10~3 holds for the E°7 branching fraction, the A(1405) contribution is dominant in this reaction. Our calculations of the E°7 branching fractions yield values an order of magnitude lower if the A(1405) contribution is neglected. In contrast with the calculations of references 11 and 12, we are able to obtain a E°7 branching fraction in excess of the branching fraction for A7 As no data are yet available, such quantitative conclusions are not yet possible in our 7-spectrum calculations for the reaction K~d —*• Any. However, as indicated in 97 Chapter 10, we have found our calculations to be relatively insensitive to changes in the effective range, differences in the deuteron wavefunction, and momentum dependence in the transition operator. Thus, an experiment with very good photon-energy resolution and a large number of counts in the peak region of the 7-spectrum could test the predictions of Chapter 10. It appears[4], however, that sufficient data is currently difficult to obtain. It is useful to compare with the ir~d —> nnj experiment of reference 1. In that experiment, 2500 events were obtained per 50 keV photon energy bin near the 7-spectrum peak. Preliminary[4] data from Brookhaven[3] indicates that, for the present K~d —» Any experiment, there will be two orders of magnitude less data in the peak region. Thus, this experiment may require a Kaon Factory in order to extract quantitative information on the A — n interaction. Throughout this thesis, necessarily crude estimates have been used for the kaonic wavefunctions. However, as there has recently been considerable progress[52] in the study of kaonic atoms, realistic kaonic wavefunctions may soon be available. The pos-sibility of using a A n wavefunction constructed from the model of Picker, Redish and Stephenson[8] is also being considered. In conclusion, several important results have come from this work. The influence of the A(1405) on the K~p —> Yy reactions has been constrained. The structure of this resonance, however, has yet to be determined. The low energy, in-flight K~p —• Yy reaction may help to narrow our choices. 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Physics Letters B171, 471 (1986). 109 A p p e n d i x A T h e E ° A T r a n s i t i o n M o m e n t The magnitude of the E° —» A transition moment can, in principle, be derived from the S° lifetime, since the E° decays essentially 100% of the time to A7. However, the E° lifetime is very short[23], about 10~ 2 0 sec, and thus this is not presently experimentally feasible. In practice, the Primakoff effect[76] is used to determine KEA-In this process, a beam of A particles is converted to E° particles through its interaction with a nuclear coulomb field. Here too, though, it is only the magnitude of KEA which is determined. In Chapter 4, we have chosen the sign of K^A using the SU(3) prediction «SA = - (A.l) relating K^A to the neutron magnetic moment. In the calculation of reference 13, however, the opposite sign was chosen for KEA- There the naive quark model prediction [77] KEA = (fid ~ Mu) , (A.2) was used to determine a sign. In the above, [xu and {is. are respectively the up and down quark magnetic moments. Recent values for the up and down quark magnetic moments are[78] 1.85 and —0.972 nuclear magnetons respectively. 110 Both of the above relations give a K^A magnitude near the experimental value of 1.60 quoted in Table 4.1. However, equation (A.2) gives a sign which is opposite to the one we have used. If one reviews the compilations of quark wavefunctions, the reason for this sign difference becomes apparent. Depending on the choice of quark wavefunctions, either sign is obtainable for «EA- It is thus important to use a set of quark wavefunctions which is consistent with the same SU(3) conventions as were used in the determination of our other couplings. Since we have used the SU(3) relations of de Swart[26], we will calculate K^A using the quark wavefunctions of Koniuk and Isgur[30], as these wavefunctions have also been constructed to conform with the SU(3) conventions of de Swart. The Y = A , S° flavor and spin wavefunctions are then given by where (f>y is the A or E° flavor wavefunction and x+ 13 a spin 1/2 wavefunction constructed from the three quark spins. The superscript A ( p ) indicates that the baryon wavefunction is symmetric ( anti-symmetric ) in the first two quark wavefunctions. The spin wavefunctions are given by XP+ = -j^aP<* ~ P<*a), (A.4) and xl = ^ (2aaP - aP<* - Paa)' (A-5) where the a (8) quark spin wavefunction has a spin projection value of 1/2 (—1/2 ). The A and S° flavor wavefunctions are given by <f>\ = —(usd + sud — dsu — sdu), (A. 6) = ^—(2uds — 2dus + usd — dsu — sud + sdu), (A.7) 111 and <^>2o = .—(sdu + sud + dsu + usd — 2uds — 2dus), (A.8) V12 <^ so = 7:(sdu — dsu + sud — usd), (A.9) where u, d and 5 are the up, down and strange quark wavefunctions. In the quark model, the stable baryon magnetic moments are given by a vector sum of the constituent quark magnetic moments. The baryon magnetic moment operator is then the sum of magnetic moment operators for the constituent quarks /r=X>9(0*(0, ( A . I O ) i=i where a{i) is a Pauli spin operator for the ith quark and fJ,q(i) is the ith quark magnetic moment defined by the relations [79] figU = /j,uu, (A.11) fiqd = fj,dd, (A.12) fiqS = nas. (A.13) The baryon magnetic and transition moments are then given by the matrix elements of p,z for baryon wavefunctions, which are maximally polarized along the z-axis: 3 l^B'B =< B' | 5>,(*M») | B > . (A.14) i=l Using the above relation for fiB'B a n d the quark wavefunctions in equations (A.4) to (A.9), we have KEA = - ^ ( ^ u _ (A.15) which has the opposite sign to equation (A.2) and agrees with equation ( A . l ) . If one were to use the wavefunction of Lichtenberg[79], the result in equation (A.2) would have been obtained. The wavefunctions given by Lichtenberg conform to the 112 conventions of Baird and Biedenharn[80] rather than those of de Swart. Unfortunately, however, the choice of conventions is not always clearly stated. Thus, the utilization of couplings from different compilations tends to be a perilous procedure. 113 A p p e n d i x B S o m e U s e f u l I n tegra ls Analytic relations can be obtained for the integrals Uo, Vo, W2 and V2 introduced in Chapter 7 and required in the K~d —> Any calculation. This is possible if simple analytic forms are used for the deuteron S- and D-state wave functions. Many N - N potentials, including the Bonn[40] potential, have associated with them deuteron wave functions of the form «(r) = E A j c - i ' , (B . l ) for the S-state radial wave function, and w ( r ) =S B '{ i +^ +(^} e"""' ( B 2 ) for the D-state radial wave function. The integrals Uo and Vo are easily evaluated: roo U0= dr ru(r)jQ(pnr), (B.3) Jo roo V0 = / dr e-ipru(r)j0(qr), (B.4) Jo and one may readily verify the relations j t i PI + <*j ^» = E ^ - ( l n A 2 - l n A 1 ) , (B.6) 114 where Ai = otj + (p - q)i A 2 = ctj + (p + q)i. (B.7) Recall that p is the relative A — n momentum, q is the photon momentum multiplied by mn/(mn + ms) and, pn is the outgoing neutron momentum. In order to calculate numerically Vo, the result in equation (B.6) is split into its real and imaginary parts using \nZ = lnR + id , Z = Reie, (B.8) to find ImVo = ± ^ { i + ( i q - P l ) . (B.10) Somewhat more involved is the integral ' dr rw(r)j2(pnr). ( B . H ) 0 Consider just one term of the sum in equation (B.2). Then we have Wi(r) = f°° drr{l + — + ) e~^T Jo { oijr {oLjr)2) 1(^ -^ )sinp"r-(^ C03p"r}' (R12) Term by term integration is difficult here as some of the individual terms diverge at r = 0. The infinities do, however, cancel to give a finite result. The following method[71] is useful in handling these problems. First define ^ o ) = / dr — , (B.13) 115 and /r=Um/J(r„) = r l i m ^ l , (B.14) where En(\r0) is defined by the expression /oo g - A r o i dt —pr. (B.15) The integral En(\r0) is well known and has the value[72] En(Xr0)=\ °{ (-lnA-lnr 0 + ¥(w)) + £ ^ f r r - r , (B.16) (n — 1)! (m — ra + l ) m ! m^n— 1 where \I/(n) has the value ^(1) = —7 and m = l m and 7 is Euler's constant. It is clear that the integrals in equation (B.14) would diverge if applied term by term to the integrand of equation (B.12). Note, however, that ji(qr) is of order r 2 near r = 0 so that the overall result in equation (B.12) is finite. Decomposing equation (B.12) into a sum of integrals of the form in equation (B.14), all terms proportional to 7*0 cancel - a useful check on the final result - and, after some algebra, one has the result Finally, we must calculate J roo ! dr w(r)j2(qr)e-'pr. (B.19) 0 In terms of the integral in equation (B.14) and for a single term in w(r), this may be written as 2iq3 2iq 116 ~ ^ ( ^ + 4 ) + 2 ^ ( 4 + 4 ) + 2 ^ ( 4 + * ) , (B.20) wherein, Ai = OLJ + (p + q)i , A 2 = aj + (p- q)i. (B.21) Once again the divergences cancel leaving a finite but very unwieldy answer - 2 ^ ( W 2 ) 9 (\\ A? A 2 - A 2 l l \ 3 2ior5 (A 2 .L 2 — ^I-^I + A 2 — Ai) 9 / A j _ Af A 2 - Aj 50N  + 2 ia 2g 3 \24 1 24 2 + 24 ' 24, 3 / ^ 1 R 2^ r . 2^ ~ 1^ 3\ ~ 2ic7q~\2Ll-^L2 + ~ 2 ~ ' 2 j 3_ 2<72 — ( A 2 ( L 2 + 1) + A i ( L i + 1)) A | ^ A| A 2 + A 2 3 2aq2 \ 2 ' 2 9 / Ao _ A? An — A? 11 i _ _. ' ^ 2 + - ^ - - V ^ - - 5 - > ( B - 2 2 ) 2 a V V 6 6 where we have used the abbreviations L1=\n.\1 , Z 2 = l n A 2 . (B.23) In order to calculate the matrix elements of our momentum dependent transition operator in Chapter 8, we must evaluate j f * ' V . ( f c r ) ! f ! £ l ) . (B.24) 117 The term involving a radial derivative of the deuteron S-state wavefunction d (u(r)\ ldu(r) 1 . . , n x is separated into the above two terms where, for the usual parametrization of u(r) given in equation (B . l ) , I M 0 . t ( _ a , A ) £ Z l , ( B , 6 ) we must evaluate integrals of the form roo h = dr r e - a r j x { p n r \ (B.27) Jo and J roo 1 dr e~arji(p„r). (B.28) o Using the above described methods, we have anc J 2 = — ( l - — tan" 1 ( ^ ) \ . (B.30) Pn [ Pn \a J ) We must also evaluate those integrals containing the e'pr jr term of equation (7.32). The results are roo / dr j1(qr)e-are-%pr = Jo ± - (In A 2 + In A x + 2) + (A 2 In A 2 - X1 In A x ) , (B.31) Iq Ziq and Jroo e-ipT ' d r i 1 ( 9 r ) e - a p = o r 4 ^ 5 {A?(ln Ai - |) - A»(ln A 2 - |)} - ^ {A 2(ln A 2 - 1) + A,(ln A, - 1)} , (B.32) with Xi and A 2 again given in equation (B.21). 118 

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