p watts}: ill llllllllll W Hill 3 1293 00837 0805 mam U Mill lit/ll Michigan State University This is to certify that the dissertation entitled Radiative Decays in the Bethe-Salpeter Equation With 3 Static Kernel and Harmonic Oscillator Potential presented by Walter W. Becker has been accepted towards fulfillment of the requirements for Ph .D . degree in Physics fl Major profess Date October 29, 1982 MSU is an Affirmative Action/Equal Opportunity Institution 0-12771 MSU ’ LIBRARIES M » RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. FbADIATIVE DECAYS IN THE BETHE-SALPETER EQUATION WITH A STATIC KERNEL AND HARMONIC OSCILLATOR POTENTIAL By Walter H. Becker A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy l982 ABSTRACT RADIATIVE DECAYS IN THE BETHE-SALPETER EQUATION WITH A STATIC KERNEL AND HARMONIC OSCILLATOR POTENTIAL By Walter W. Becker Radiative decay widths are calculated for the radiative decay processes observed experimentally in the charmonium system. The model uses a Bethe-Salpeter equation with a static kernel and harmonic oscillator potentials to model the c-E system. Each decay width is calculated for 2l different choices of the c-quark mass. The potential used was a linear combination of a vector coupled and a scalar coupled harmonic oscillator potential. The quark mass and the scalar to vector coupling ratio were determined by trying to fit simultaneously the u’(3685) - w(3095) mass difference, the ¢(3095) + e+ + e' decay width and the 3PJ mass splittings. A single choice of the quark mass and scalar to vector coupling ratio could not simultaneously fit all these constraints. The “best fit" to these constraints occurred when the quark mass was 5.5 and the scalar to vector coupling ratio parameter was -0.l5. The decay width calculations are shown graphically for values of the quark mass from 0.00 to TS Gev. The decay widths were calculated two different ways: (i) using the matrix elements of the quark momentum; (ii) using the matrix elements of the quark position. Most of the published calculations use method (ii). The widths computed by methods (i) and’(ii) are quite different for all masses and all ‘transitions implying that the usual method (ii) gives incorrect results, and the fits with experimental data are fortuitous. ACKNOWLEDGMENTS I would like to thank Dr. Wayne N. Repko for initially posing this topic and for his continued assistance on various phases of the work. The use of a plotting routine supplied to me by Dr. Fred Kus was of great value in displaying the results in a coherent form. To my wife Elaine I owe much. Initially she assisted me in the learning of Fortran programming and helped in the debugging of various parts of the computer routines. Finally, the thankless task of typing all the miserable equations in the appendices, as well as the more standard text material, was also done by my wife Elaine on an experimental technical word processor in the Project Physnet office at Michigan State University. ii TABLE OF CONTENTS List of Tables List of Chapter Chapter (A) (B) (C) (D) Figures 1. Introduction 2. Outline of the Calculational Scheme The Calculation of the Airy Function and its Derivative The Transformation of the Trial Navefunctions The Integration Algorithm Reduction of the Differential Equation(s) to ~ Finite Dimensional Matrix Problems (E) (F) Chapter (A) (B) (C) Scaling and Fitting Parameters Radiative Decay Width Equations 3. Results of the Calculations and Conclusions Calculational Results Discussion and Conclusions Summary Appendix I. The Bethe-Salpeter Equation Reductions I.l Kinematic Constraints on the Singlet and Triplet Havefunctions 1.2 Schoonschip Decomposition of the Bethe-Salpeter Equation vii ix lO l4 l6 l6 I7 29 3l 35 35 ll7 130 l35 l35 l42 1.3 Derivation of the Radial Equations 146 (i) Derivation of the IJJ Equation for a 145 Coupled Vector-Scalar Interaction (ii) Derivation of the 3JJ Equation for a 149 Coupled Vector-Scalar Interaction (iiia) The Radial Equation for the 3(Ja1)J Bethe-i l52 Salpeter Equation with Vector Harmonic Potential (iiib) The Radial Equation for the 3(le)J Bethe- ‘53 Salpeter Equation with a Scalar Harmonic Potential (iiic) Combining the 3(Je1)J Scalar and Vector 169 Potential Forms 1.4 Integrating Factors and the Sturm-Liouville Form 172 1.5 Limiting Cases of the Radial Equations and the 177 Lagrangians for the Radial Equations (i) p +.m 177 (ii) Case of p + o, for finite m > o 179 (iii) Case of m + a 18l (iv) Case of m + 0 182 (v) Lagrangians of the Radial Equations ‘34 1.6 Solutions of Limiting Forms of the Radial Bethe- ‘87 Salpeter Equations and the Choice of the Variational Trial Nave Functions Appendix II. Decay Rate Calculations I92 II.l The Transition Rate for 3w1(3095) + e+ + e‘ 192 iv (i) The term = = I (ii) The Photon Propagator Term (iii) The Hadronic Current Term 11.2 Derivation of the Transition Rate for the Radiative Decays in the Bethe-Salpeter Equation (i) Conversion of T to we (ii) Conversion of 11 (iii) Combining Terms and the Bethe-Salpeter Equation (a) The (l-4) Initial State System (b) The (3-5) Final State System (c) An Aside on the Bethe-Salpeter Equation (d) Substitution into the Bethe-Salpeter Equation (iv) Final Reduction of the Current Matrix Element (v) Final Reduction by Using Center of Mass Coordinates 11.3 The Calculation of the Angular Momentum States of the Vector Particles arising from qq States 11.4 Radiative Decay Rate Formulae (i) Results of the Schoonschip Evaluation of the Current Matrix Element (ii) The P0 Integration \l 193 l94 196 204 206 207 208 209 209 2l0 213 213 2l5 2l9 228 228 228 (iii) The Transition Rate Formulae 11.5 Schoonschip Current Matrix Element Reduction Program, Case 2351 + l3Po Appendix III. The Program used in the Calculation of Wavefunctions and Decay Widths Bibliographical Essay vi 230 234 235 283 Table Table Table Table Table Table Table Table ' Table Table Table Table Table Table Table Tabie tooouaim 10 ll l2 l3 l4 IS 16 LIST OF TABLES Quark Properties Comparison of the Computed and Exact Values of the Zeros of the Airy Function Check on the First Three Singlet S Zero Wavefunctions X between .le and 38.0 Check on the First Three Singlet S Zero Wavefunctions X between zero and 7.75 Orthogonality Check, Case L = 0 Orthogonality Check, Case L = l Orthogonality Check, Case L = 2 Orthogonality Check, Case L = 3 Polarization Program Corrections Schoonschip Evaluation of Current Matrix Elements Spin Average Results Electric Dipole Constants Magnetic Dipole Constants Variation of the Charmonium Masses as a Function of the Quark Mass Parameter and the Scalar-Vector Coupling Constant Ratio Variation of Magnetic Dipole Widths with RAT (Gev) Radial Integrals Key 00 \l“ 20 Zl 23 25 26 27 28 32 33 34 p 34 34 6l 92 93 Table Table Table Table Table Table Table Table Table Table Table Table Table Table 17 18 19 20 21 22 23 24 25 26 27 28 29 30 The Radial Integrals as a Function of the Quark Mass for RAT = 0.00 The Radial Integrals as a Function of the Quark Mass for RAT = -0.16 The Radial Integrals as a Function of the Quark Mass for RAT = -o.275 The Radial Integrals as a Function of the Quark Mass for RAT = —0.33 Accumulation of the Radial Integrals Two-Gamma Decay Widths for some Charmonium States (kev) Two-Gluon Decay Widths fbr some Charmonium States (mev) Key to Tables 25 to 30 Cheat Transition Widths, RAT = -0.16 Cheat Ratios, RAT = -0.l6 Cheat Transition Widths, RAT = -0.275 Cheat Ratios, RAT = -0.275 Cheat Transition Widths, RAT = -0.33 Cheat Ratios, RAT = -O.33 94 96 98 100 102 108 109 110 111 112 113 114 115 116 Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 LIST OF FIGURES Graphs of the Binding Energies of the Ground States of the 150, 1P1, 102, 3P0, 351, 3D1, 3P2, 3F2 con- figurations of the q a system for RAT = 0.00 as a function of the quark mass Graphs of the Binding Energies of the Ground State and First Excited State of the 1So, 1P1, 1D2, 3P0, 351, 301, 3P2, 3F2 Configurations of the q a system for RAT = 0.00 as a function of the quark mass Graphs of the Binding Energies of the First and Second Excited States of the 150, 1P1, 102, 3P0, 351, 301, 3P2, 3F2 Configurations of the q a system for RAT = 0.00 as a function of the quark mass I Graphs of the Binding Energies of the Ground State and First Excited State of the 150, 1P1, 1D2, 3P0, 351, 301, 3P2, 3F2 Configurations of the q a system for RAT = +0.33 as a function of the quark mass Determination of the Potential Parameters, RAT = 0.00 Plot of the Wavefunctions of the First Three States RAT = 0 of the 180 system, for m 37 39 4'1 43 46 49 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure F i gure F i gure 10 11 12 13 14 15 16 17 18 19 Transition Width for l3Poa+ l331 + y, RAT 20 Transition Width for l3Po-+ 1381 + y, RAT Plot of the Wavefunctions of the First Three of the 1P1 system, for m = RAT = 0 Plot of the Wavefunctions of the First Three of the 102 system, for m = Determination of Potential Determination of Potential Determination of Potential Determination of Potential Transition Widths for 2331 RAT = -0.16 Transition Widths for 2351 RAT = -0.275 Transition Widths for 2351 RAT = -0.33 Transition Widths for 13P2 states, RAT = -0.16 Transition Widths for l3P2 states, RAT = -0.275 Transition Widths fbr 13P2 states, RAT = -0.33 RAT = 0 Parameters, RAT Parameters, RAT Parameters, RAT Parameters, RAT + 3P + y states, + 3P + 7 states, 1+ 3P + y states, and 13P1 + 1331 + BHd 13p] + 1351 + States States 0.00 -0.16 -0.275 -0.33 Y Y and 13P1‘+ 1331 + v -0.16 -0.275 50 51 53 55 57 59 65 67 69 72 74 76 79 81 Figure Figure Figure Figure Figure Figure Figure Figure Figure 21 22 23 24 25 26 27 28 29 Transition Width for l3Po +.1351 + y, RAT = -0.33 Magnetic Dipole Transition Widths, RAT = -0.16 Magnetic Dipole Transition Widths, RAT = -0.275 Magnetic Dipole Transition Widths, RAT = -0.33 ' Radial Wavefunctions and Radial Integrals for Charmonium Transitions, RAT = -0.16 Radial Wavefunctions and Radial Integrals for Charmonium Transitions, RAT = -0.l6 Radial Wavefunctions and Radial Integrals for Charmonium Transitions, RAT = -0.16 Radial Wavefunctions and Radial Integrals for Charmonium Transitions, RAT = -0.16 Radiative Decay Widths in Charmonium xi 83 86 88 A 90 123 125 127 129 133 Chapter 1 Introduction With the discovery of the I~meson in 1947 a certain stage of "explanation" of atomic structure was reached. The electron, proton and neutron were the fundamental building blocks of nature. The photon was the instrument that was responsible for transmission of the electromagnetic forces from one charged particle to another charged particle. The I-meson did the analogous thing for the nuclear forces found to be acting between protons and neutrons. The neutrino was needed to conserve angular momentum in beta decay of nuclei. The only oddity was the u-meson (or muon) whose existence escaped reason. Certain other particles, like the positron, were known or strongly suspected to exist on the basis of a symmetry of the Dirac equation, which was believed to describe the behavior of massive spin 1/2 particles. The discovery of several additional particles ruined this nicely reasoned structure. These strange particles came in two varieties - called mesons (particles with masses between the n-meson and the proton) and hyperons (masses greater than the proton’s mass) which eventually i decayed into a proton or a neutron. These particles are now called K , 0 K9 and R0 mesons, and A , 2 and E- for the hyperons. If one o + o lodks at the eight particles p, n, A0, 2 , Z“, , 5- one finds that inany of their properties are identical e.g. spin, parity and their interaction strength with nuclei. In an attempt to explain this similarity of properties Cell-Mann and Ne’eman (independently) developed time 80(3) or unitary symmetry model of elementary particles in 1961. The group SU(3) has an eight dimensional representation which accomodated these eight heavy particles. This group also has representations of dimensions 3, 6, 10, 27, ... At this point the question arises, do there exist other, still to be found particles that could be described by these other allowed representations. The lO-dimensional representation was found to be an acceptable representation; it described certain resonances and predicted a new S = -3 heavy particle, the 9-. This 9- was subsequently found in an experiment conducted at Brookhaven in the 1960’s. So far no particles have been detected directly that do not belong to the representations of dimensions 1, 8, or 10. Fairbanks, et a1, [43] have reported results that seem to imply the existence of free quarks. These experiments still await verification by Other experimental groups. This peculiar behavior has puzzled particle physicists for a long time. If one ascribed to the fundamental, or three-dimensional, representation, hypothetical particles called quarks, then all of the observed "particles" corresponded to representations of SU(3) obtained by taking the product representations [SU(3)]3 G [SDTETTB - I 0 [SU(3)]3 or [SU(3)13 6 [SU(3)13 G [SU(3)13 - 1 9 [SU(3)13 9 [SU(3)110 9 [We- These hypothetical quark particles have the following (unusual) properties. TABLE 1 QUARK PROPERTIES Spin charge baryon number other quantum numbers up 1/2 2/3 1/3 none down 1/2 -1/3 1/3 none strange 1/2 -l/3 1/3 S a -1 charm 1/2 2/3 1/3 c = +1 bottom 1/2 -1/3 .1/3 b - +1 top 1/2 2/3 1/3 t - +1 This scheme could be looked upon as indicating the existence of three "elementary particles" belonging to the three-dimensional representation of SU(3), and that the observed particles, e.g. fi-mesons, protons, etc., were bound states of these quarks. The results from high energy electron~ proton scattering also seem to indicate the existence of a point-like substructure for the proton. The point-like substructure had the same properties that were being ascribed to the SU(3) quarks. One should note here that in the 1960’s only the first three quarks in TABLE 1 were believed "to exist." The only essential difference between then and now is the replacement of SU(3) by SU(6) as the underlying symmetry group. Eventually, as things usually do, this simple quark picture became somewhat more complicated. If quarks obeyed the usual Pauli spinrstatistics theorem, then the quark model of elementary particles caused problems. The n‘ and the n + p (1238) resonance were viewed as being composed of three quarks called (3,3,8) and (u,u,u) respectively, and all three quarks were to have zero orbital angular momentum. This condition violated the Pauli spin-statistics theorem, which stated that no two spin 1/2 particles could have the same set of quantum numbers. In the case of the 9- and I + p (1238) resonance, all three quarks did have the same set of quantum numbers. Another problem for the quark model surfaced in the analysis of high energy e+ + e- annihilation experiments. The ratio of e+ + e- * hadrons to that for e+ + e- + 0+ + u" is predictable on the basis of the quark model [44]. On the basis of the simple model above, the predicted and observed ratios differ by a factor of three. The predicted decay rate of ,0 + 27 is also off by this same factor when one uses this simple quark model. In an effort to eliminate these problems, it was then hypothesized that each of the quarks u, d, s (and also subsequently discovered or inferred quarks like c, b, t) come in three varieties, e.g. red, green, and blue. The quark model now predicts that 9- is the bound state of three 8 quarks, one red, one green, and one blue. Since these S quarks are now distinguishable (this new quantity called the "color" of the quark is the new quantum number), the spin-statistic problem goes away 7 immediately. The additional multiplicity of quarks also eliminates the problem of the factor of three in the no decay rate and the e+ + e- annihilation experiments mentioned above. The picture or model that emerges for those elementary particles that participate in strong V ~ 5 interactions is that they are composed of quarks or quark-antiquark combinations. Other considerations of nature seem to restrict this same set of elementary particles to be made up of either three quarks or of quark-antiquark pairs. No deviation from these conditions seems to exist. The particles to be subsequently considered here are a special class of heavy mesons and are composed of a quark and its associated antiquark. The simplest thing to do is to assume a certain potential energy function for the qJE interaction, insert this into the Schroedinger wave equation and solve for the energy levels of this q4fi system. This is what was done in [29] and [30]. The assumed potential was V(r) - or + B/r. There were reasons for this choice, aside from giving good ,agreement with the observed energy levels for the c4; system. The or term arises from the requirement that the quarks are permanently bound or confined, i.e. free quarks do not exist. As the separation increases between two quarks, the potential energy also increases. The only way the two initial quarks can have arbitrarily large separations is to produce a new q¥q pair, one of which escapes with the first quark and the other stays behind with the remaining quark. The B/r term comes from the known coulomb interaction occurring at short distances between the quarks. This treatment is completely non-relativistic in character. Since the discovery of the Y(309S) meson and other members of the Charmonium system, a number of other potential functions have been used t1) describe these new particles. A recent comparison of the predictions Of' several of these model calculations with experimental findings is 81ven in [41] . Because of the close agreement between these tunr-relativistic calculations and experiments, one assumes that the motion of the quarks in these charmonium states is non-relativistic. Most of these calculations have problems predicting the splittings of the various P states (13P0, 13P1, and I3P2) as well as the relative decay rates from the 2381 state to these afore-mentioned P states. The radiative decay rates of these triplet P states to 1381 are for the most part not well enough determined to be useful in making theoretical comparisons between various model potentials. In the case of the hydrogen atom the non-relativistic Schroedinger equation with a pure coulomb potential is unable to reproduce the fine structure observed. Initially additional terms were incorporated into the potential and treated perturbatively to produce the observed fine structure. With the development of the Dirac equation, which treated the electron’s motion relativistically, the fine structure of the hydrogen atom was correctly produced, the major defect of the Dirac equation being the failure to account for the (281/2 - 2Pl/2) splitting (Lamb effect) discovered about twenty years later. To account for this, one needed a relativistic quantum field theory of electrons and the electromagnetic field. This construction was first attempted in the late 1920’s and early 1930’s. These initial attempts were able to handle some things, e.g. the calculation of the KleinrNishina formula for compton scattering. The Klein-Nishina formula, from a relativistic quantum field theory viewpoint, represented the effect of the leading term in a perturbative expansion. When one tried to compute the radiative corrections to this result, certain divergent integrals arose. It was unclear how to deal with these divergences in the theory. In the late 1940’s, Feynman, Dyson, Schwinger, and others developed a covariant renormalization scheme which 7 enabled them to calculate these radiative correction terms. This method is described in several papers in the collection [23]. Shortly after the introduction of the Dirac equation, 6. Breit [16] and others attempted to generalize this equation to describe the interaction of two or more relativistic particles, the immediate application being to the fine structure of the helium atom. The initial attempts presented problems. (See discussion in [17]). These problems were only finally resolved with the developments in relativistic quantum field theory in the late 1940’s. The initial paper making use of these methods is by Bethe and Salpeter [18]. Most of the work done on the Bethe-Salpeter equation is directed to applications involving radiative corrections to the hydrogen atom or positronium. A recent survey on the applications to positronium is given by Stroscio [22]. The starting point for our calculations makes use of equations 2.1 to 2.10 and equations 4.20 to 4.25 of [26]. The derivation of the Bethe-Salpeter equation from quantum field theory is given in appendix C of [22] and will not be reproduced here.' The derivation given in [22] as well as the original one given by Bethe and Salpeter in [18] is based upon Feynman’s approach to quantum field theory described in papers 21 and 22 of reference [23]. The method of reduction of the general form of the Bethe- Salpeter equation as given by MEWS) + ¢n<-5) - knoci) - - uH(S.S’)¢ (1) to get the radial equation(s) needed to determine the eigenvalues An is done in several steps. The function ¢(S) can be expanded as follows. 8 MS) - 3(S)c + iuvutfim + a (5m + isiuAut‘Sm + isP(3)c <2) UVTIW The functions 8(3), Vu(3), ..., P(p) are not all independent of one another. Relativistic kinematics imposes certain restrictions, and these relations are derived in Appendix 1.1. It is seen there that the function 4(3) splits up into a singlet and a triplet case. we choose as the independent function for the singlet case P(p) and for the triplet case 3(3). The next step is to insert these singlet and triplet wavefunctions back into equation (1) and carry out the necessary Dirac algebra to get equations'4.3, 4.8, and 4.19 of [26]. The program Schoonschip was used to perform this reduction. The program to do this for the singlet wavefunction is shown in Appendix 1.2. Similar but more complicated computer programs are needed for the triplet cases and are not illustrated here. The wavefunctions for the singlet and triplet states of the scalar potential are related to those of the vector potential by + v + 48(p) - B¢v(p)B (3) One consequence of (3) is that by making the substitutions 3’ + -S’ and 4(3’) + -¢(3’) in equations 4.20, 4.21, and 4.22 of [26] one could convert the results for the "coulomb potential" given there to the case of a scalar potential. For the singlet case this reduction was explicitly carried out and checked by Schoonschip. For the triplet cases a complete reduction was not done, but this substitution rule was used to obtain the equations analogous to those of 4.21 and 4.22 of [26] for the scalar potential. The remainder of Appendix I gives the details of other :reductions needed to put the Bethe-Salpeter equations for the singlet 1" f. (“a '1 I If) ‘Ll and triplet states into a form suitable for numerical solution. Appendix II gives the details of the calculations of formulae for the leptonic and radiative decay rates of various qeq systems. The results of the numerical calculations of these decay rates will be discussed in chapter 3. The point of the work described below is to calculate properties of the qeq system, with an admixture of a vector and scalar harmonic coupling, and the comparison of these calculations with the experimental results on the charmonium system. The spectrum of the q4q system using a purely vector coupling was carried out in [31] and it was found that the level structure in the 3(J11)J system was not the same as that found for this spin assignment in charmonium. In [31] the level scheme is 1381, 13D1, 2381, 23D1, etc. In charmonium the appropriate level structure is 1381, 2381, 13D1, etc. The addition of the second, scalar harmonic, coupling here was an attempt to obtain the correct ordering of these states. This attempt was not successful, as will be seen below. This scalar potential gives us another parameter to vary to see if one can obtain better agreement with either decay rates or the fine structure of the P-states within this model. Initially attempts were made to incorporate a 1/r type of term in the potential. This attempt produced problems described in chapter 3. we hope to look further into these problems with a view to incorporating this potential form into our work. It is possible, based upon work using the linear plus coulomb type potential described in [29], that the use of a I/r potential would <:orrect the above level structure problem. Chapter 2 Outline of the Calculational Scheme The Bethe-Salpeter equation is a relativistic two body wave equation which represents a generalization of the Dirac equation. The Dirac equation is a linear partial differential equation; The Bethe-Salpeter equation is an integro-differential equation in general. For special choices of the potential function, the Bethe-Salpeter equation can be reduced to a partial differential equation. One of these special potentials is the instantaneous harmonic oscillator potential. This potential was picked for this work for precisely this reason. Once we have this differential equation form for the Bethe- Salpeter equation, next arises the problem of obtaining solutions of this equation. One of the tried and true methods for solving partial differential equations is to use the method of separation of variables to reduce the problem to a set of ordinary differential equations. we follow this time honored method. The next problem is the solution of the resulting ordinary differential equations. .If we use spherical coordinates in our separation of variables technique, then the two equations involving the polar angles are solvable and yield the well known spherical harmonic functions as solutions. The radial equation tihich results from this separation procedure appears not to be a well lcnown equation whose solution is easily recognized. In such cases one time honored method of solution says to express the (radial) function as a. power series in terms of the independent variable, substitute this Berries into the differential equation, collect terms of the same power 10 11 in the independent variable and set the coefficients of these terms equal to zero. This method yields a recurrence relation for the coefficients of the assumed power series solution. In the workable cases one has a two term recurrence relation. Three or more term recurrence relations have historically been very hard to deal with. One wants to be able to express the nth coefficient of the power series as a function of n and the first term in the recurrence relation. To,be able to solve the recurrence relation when it involves three or more terms is in general very difficult. It is only within the last couple of years that the 3-term recurrence relation arising from the non-relativistic Schroedinger equation with a linear potential has been solved using non-standard techniques. In the radial equation resulting from our Bethe-Salpeter equation, an eleven term recurrence relation results - so much for this time honored method. An alternate time honored method of obtaining approximate solutions of differential equations is to recast them as a variational calculus problem. This is the procedure to be used here on the radial equation to obtain an approximate solution to the overall Bethe-Salpeter equation. One of the basic types of problems in the calculus of variations is: Given a function £(x,y(x),y’(x)), find that function y(x) such that the integral . I - It.y’>dx is a minimum, subject to the constraint condition that {i It. Ea 12 Ipiy12dx = 1. 'For particular choices of £(x,y(x),y’(x)) and p(x), this problem requires, as a necessary condition for its solution, that y(x) obey a linear second order ordinary differential equation. Since any second order ordinary differential equation can be cast into a self-adjoint or Sturm-Liouville form, one restricts oneself to those choices of t(x,y(x),y’(x)) which give rise to such equations. The reduction of the ordinary differential equations obtained by the introduction of spherical coordinates is given in appendix 1.3. The further reduction to the Sturm-Liouville form and the obtaining of the associated Lagrangian is described in appendix I.4. we note here that the Lagrangian associated with the differential equation d/dx(p(x) df/dx) + (q(x) + A r(x)) f(x) = O is t - (1/2)p(x)[df/dx]2 - (1/2)q(x)[f(x)12 with the subsidiary condition (1/2) fr(x)[f(x)]2dx - 1. That this Lagrangian leads to the given differential equation is easily checked by inserting i into the Euler-Lagrange equations. (d/dx(3£/3(df/dx)) - ai/af - 0) The method we use to attack this variational problem is called the direct method. we do not rely on the associated differential equation beyond using it to obtain the Lagrangian for the variational Irroblem. This method says to select a set of functions {Ai(x)}:_l and to 13 write the function f(x) that minimizes the integral f£(x,y(x),y’(x))dx, subject to the constraint (1/2)fr(x)f2(x)dx = 1, as f(x) 3 avo(x) + a1A1(x) + ... + anAn(x)+ ... The usefulness of such a method depends upon how fast the above series for f(x) converges to the correct (or exact) solution. The choice made here was to set An(x) = (xkAi(x)xn)/(a + xk+1) The ideas behind making such a choice are the following. (i) we wanted to approximate as nearly as possible the correct asymptotic behavior (x + a). (ii) As x + 0, the exact solution must go to zero as xk where k is the orbital angular momentum of the two particle system. (iii) The correct asymptotic behavior (x + “0 is given by (xkAi(x))/(a + bxk+1) and this is also compatible with the choice (ii). It is shown in Appendix 1.6 that Ai(x)/x is the appropriate solution in the x + a limit. The third term is one that should not affect either of these limiting behaviors. This means that f(x) should reduce to one as. x + O and as x + ~. Several such terms were tried, none successfully. The one finally chosen was just x“. If one recalls that the correct solution is a linear combination of functions of the form Ah(x), then one sees that the x + 0 limit is still valid provided that a0 # O. The at + 0 limit however is not maintained for any choice of ai # 0, i > I. 'The x + N'limit is still dominated by the Airy function since as x + w it goes like exp[-2x3/2/3]. The choice of xn for the third part of our trial functions was 14 tested for the case in which the quark mass m a 0. In this case the known solution is just a "displaced Airy function," i.e. f(x) - Ai(x-xo), times a normalization constant, where x0 is one of the zeros of the Airy function. By choosing b = 1 and a a 2.385, the correct function was obtained to five decimal places over the interval from x.= O to x 8 38. This corresponds to a range of values in f(x) from f(x) = 64 1.00 to f(x) 8 10- . we next discuss the principle steps in the calculational scheme. (A) The Calculation 2f the Airy function and its Derivative The Airy function and its first derivative are related to Bessel functions of fractional order through the following identities: 1/2 Ai(x) = moon/3) x1 ”(2) - (1/3)/;’{I_1/3(Z) - Il/3(z)} Ai’(x)': <-x/i)<1//5)x2,3(z) ' (‘X/3){I-2/3(Z) - Iz/3(z)} where z = 2x3/2/3 and x > 0. The determination of Ai(x) and Ai’(x) is done by making use of series expansions of 111/3(x)’ 112/3(x), and K1/3(x) and K2/3(x). The expressions used are the following: x-vIv(X) - n§0Aa(")T2a(X/8) where v - 11/3 and -8 < x < +8. The series for v a iI/3 was terminated 15 at 17 terms. The constants An(v) are taken from [32]. The quantities T2n(y) are Chebyshev polynomials. (See below.) The expansion for x > 5 makes use of the following approximation: K1/3(x) - «(t/2x) e"x 112.0 CnTn*(S/x), x > _s The values of the constants Cn are also from [32]. The quantities Tn*(y) are again Chebyshev polynomials. (See below.) The expansions for Ii2/3(x) and K2/3(x) have the same form as those for the case of Ii1/3(x) and K1/3(x), and the relevant constants for these expansions appear again in [32]. The Chebyshev polynomials which appear in the above series expansions are evaluated by the use of the following recurrence relations: I H To(X) T1(x) - xTo(x) - x T2(X) ' (2X2-1)To(x) - 1(2x2-I) T3(X) ' 2(2x2-1)T1(x) - T1(x) . 4x3-3x or in general sztx) - 2(2x2-1)T2n(x) - T2n_2(x), n>0 T2n+3(x) - 2(2x2-1)T2n+1(x) - T2n_1(x), n>0 16 ' a Note that Tn (x) a Tn(2x-1) so that * a * Tn+l(x) - 2(2x-1)Tn (x) - Tn_1(x), n>l The source for these recurrence relations is [32]. (B) The Transformation g: the Trial wavefunctions The next step is the transformation of the set of trial k+l 9 )ln functions (Ai(x)xkxn/(a + bx to a second set of orthonormal -0 functions. These new orthonormal functions are orthogonal with respect to a weight function x2. The important quantity here is the matrix Cum that transforms the functions {Ai(x)xkxn/(a+bxk+l)}:.6o the orthonormal set ¢m(x). The determination of these matrices Cmn is done separately for each value of k and is stored in the program for future reference. These matrices were used when some of the wavefunctions were plotted for values of x which differed from those used in the integration algorithm. One should note here that the value of RFC, when used in the 180 functions, yields zero, but this is incorrect. This is due to the fact that the above expansions for Ai(x) fail at xFO. To eliminate this problem the value of Ai(x), for x90, is explicitly specified in the main body of the program. (C) The Integration Algorithm In the calculation of the matrix Cmn mentioned in (B), several integrations were needed. The method of integration that was used throughout the program was one of the gaussian quadrature algorithms. ffhree different gaussian methods were tried. The first was a standard 17 32 point Gauss-Laguerre quadrature. The second was a truncated 68 point Gauss-Laguerre quadrature. The third one tried was a 28 point guassian quadrature over the interval from x = O to x - 20. In the Gauss-Laguerre routines, the values of x are prescribed, and in the standard 32 point case several values of x were sufficiently large to cause the value of Ai(x) to be less than 10.300, which the machine sets equal to zero. The truncated Gauss- Laguerre routine used only the first 32 points of the 68 points available. The xrvalue of the last point used in this routine was approximately x - 38.094, and the value of the Airy function in this region of the xraxis is about 10-68. The truncated Gauss-Laguerre also gives the best fit to the computed eigenvalues when compared to the known results for the 180 case. For all cases, the lowest eigenvalue is reasonably accurate, but for the 2 1So, 3 180, etc., the truncated 68 point routine proved to be the best. (D) Reduction 2: the Differential Equation(s) 22 Finite Dimensional Matrix Problems The final step in the solution of the differential equations is to write f(x), the solution of the differential equation, as a linear' cambination of the orthonormal functions ¢n(x) obtained in (B). This formal series 9 f(x) -n20 an¢n(x) Was substituted into the Lagrangian giving rise to the original d1ffeirential equation. Note that for terms like [f(x)]z in the Lagrangian, we substitute in Z aman¢m(x)¢n(x). we then integrate over m,n x. The resulting terms 18 are of the form fa a g (x)dx, a and a being constants independent of m n mu m n x, and where gmn(x) is a product of the form q(x) ¢m(x) 4110:) + r(X) 41:10:) 4:10!) and where q(x) and r(x) are the coefficients of [f(x)]2 and (df/dx)2 in the appropriate Lagrangian function. If we set an - fgmn(x)dx and view an now as just a matrix, we have the typical linear algebra problem of finding the relative minimums of the quadratic form 2 a M a m mn n m,n subject to the constraint that the vector I have unit length. The minimum value(s) of this bilinear form occurs when the vector 3 - (ao,a1,a2,...,ag) is an eigenvector of the matrix an’ the minimum being associated with the lowest eigenvalue ll, the next lowest relative minimum being associated with the next lowest eigenvalue A2, etc. The sought after approximate solution to the original differential equation is then given by 9 f(x) -1 a i (x) 1-0 i.i where the ai’s are the components of the eigenvector associated with the eigenvalue l1, l2, etc. This approach is expected to produce good results only for the lowest few eigenvalues and eigenfunctions for the solution of the original differential equations. To show that this expectation is verified, TABLE 2 lists the eigenvalues for the 150 case. The known eigenvalues for this equation are the zeros of the Airy function. The zeros of the Airy function are "well known" and are also listed in TABLE 2 for purposes of comparison. One can see from TABLE 2 ICA 19 that the lowest two eigenvalues are indeed very accurately calculated, but as we go to the larger eigenvalues the agreement becomes progressively worse, and in the last several cases, one can safely say there is no agreement whatsoever. Also shown in TABLE 2 are the corresponding results for including only 6 or 8 terms in the expansion for f(x). In TABLES 3 and 4 we compare the known solutions for the lowest three eigenvalues of the 180 equation. The case of the lowest or ground state solution is indeed very good, that of the next lowest one not quite so good. In principle one can increase the accuracy of the calculations by increasing the number of orthonormal functions used. In practice this presents two different types of problems: the first involves the amount of storage needed to do the calculations; the second is how bad the round off errors become in the process of constructing the orthonormal functions ¢n(x), or what is the same thing, the matrix C . To get some idea of the round off error problem, the program mn calculates numerically the quantity M a f” x26 (x)¢ (x)dx nm 0 n m This matrix, if there is no round off error, should be just the unit matrix. In practice it is not of course. Some examples of what Mnm comes out to be in a typical run are shown in TABLES 5,6,7, and 8. From these examples we see that the round off errors differ by a fem'parts in 10-.19 from the exact value of zero. One should also point out that the initial data used to start the calculation had an accuracy of one part in 10-29. So the resulting round off error due to machine computations is amplified by a factor of 1010. 20 TABLE 2 COMPARISON OF THE COMPUTED AND EXACT VALUES OF THE ZEROS OF THE AIRY FUNCTION COMPUTED VALUE EXACT VALUE NEXP=6 n- ] .2338]07t]25+0] .2338]07u] 5+0] N- 2 .40883]22]]5+0] .408794944 5+0] u- 3 .56]]404421£+01 .552055983 5+0] w- u .8096949395E+01 .678670809 5+0] w- 5 .1543180161E+02 .7944l3359 5+0] w- 6 .55665696035+02 .902265085 5+0] NEXP=8 u- l .2338]07411£+0] .233810741 5+0] u- 2 .40879496565+0] .408794944 5+0] N- 3 .552]6322905+0] .552055983 5+0] N- 4 .68833526025+0] .678670809 5+0] u- 5 .896248]8375+0] .794413359 5+0] u- 6 .]3609233095+02 .902265085 5+0] Na 7 .27]279]0805+02 .1004017434E+02 w- 8 .1004682793E+03 .]]008524305+02 N5xP-io n- l .2338]07410£+01 .23381074] 5+0] n- 2 .4087949444£+01 .408794944 5+0] w- 3 155205620285+0i .552055983 5+0] u- 4 .6788694482E+01 .678670809 5+0] u- 5 .80u26376635+0] .794413359 5+0] u- 6 .9863035]9]5+0] .902265085 5+0] w- 7 .]3330]78525+02 .1004017434E+02 N- 8 .20953365285+02 .I]00852430£+02 u- 9 .42825007505+02 .]]9360]5565+02 N-lO .]6]008933u5+03 .]2828776755+02 21 TABLE 3 CHECK ON THE FIRST THREE SINGLET S ZERO WAVEFUNCTIONS X BETWEEN .211 AND 38.0 X(l) N81 st u-3 .2]]0695-0] - 9998275+00 .9997045+00 .9979565+00 9998275+00 .9996975+00 .99959]5+00 .111223E+00 -.99530]5+00 .99]7065+00 .9889h55+00 .99530]5+00 .99]7075+00 .98877]5+00 .2733995+00 -.9728005+00 .95]h975+00 .9343755+00 .972800£+00 .951498E+00 .9342665+00 .5077555+00 -.9]28]65+00 .8432055+00 .78843]5+00 .9]28]65+00 .843205E+00 .7885705+00 .814421E+00 -.7997785+00 .6399575+00 .5232955+00 .7997785+00 .639958£+00 .523]8]5+00 .i]93565+0] ~.634660£+oo .3542045+00 .]784955+00 .634660E+00 .354204E+00 .178549E+00 .164537E+01 -.uh]9845+00 .55]9565-0] .116453E+00 .hu]9845+00 .5519565-0] .116456E+00 .2]70]05+0] -.26]6905+00- -.]586935+00 .2]93]95+00 .26]6905+00 -.]586935+00 .2]93495+00 .2768035+0] -.]276335+00 -.2288375+00 .]]27395+00 .]276335+00 -.2288375+00 .]]26905+00 .343948E+01 -.496930E-OI -.]82]275+00 .59]5065-0] .496930E-01 -.]82]275+00 .591014E-Ol .u]848]5+0] -.]hg69&5-0] -.98]9]5£-0] .]396385+00 .1496945-0] -.98]9]65-0] .]396675+00 .500445E+01 -.338]555-02 -.3708655-0] .]]062]5+00 .338]555-02 -.3708655-0] .]]06245+00 .5898835+0] -.555]7]5-03 -.9772525-02 .5096225-0] .555]7]5-03 -.9772405-02 .5092235-0] .6868h65+0] -.6h]8625-04 -.]765]25-02 .147425E-Ol .6h]8625-04 —.]765225-02 .1478305-01 .79]3885+0] -.50622]5—05 -.2]33]75-03 .2687555-02 .50622]5-05 -.213457E-03 .2739905-02 .9035705+0] -.2637]55-06 -.167705E-04 .3027755-03 X(l) .102346E+02 .115112E+02 .128663E+02 .143007E+02 .158153E+02 .174111E+02 .190890E+02 .208501E+02 .2269S7E+02 .246269E+02 .266451E+02 .287517E+02 .309482E+02 .332363E+02 .356176E+02 .3809425+02 TABLE 3(CONT'D) N31 .878275E-08 .878258E-08 .lBOBBOE-09 .180869E-09 .222694E-11 .222655E-11 .153329E-13 .158266E-13 .627448E-16 .626952E~16 .133659E-13 .133469E-18 .147441E-21 .147098E-21 .810499E-25 .807633E-25 .213384E-28 .212323E-28 .258229E-32 .256545E-32 .137651E-36 .1355595'35 .309212E-41 .306472E-41 .279523E-46 .277056E-46 .969233E-52 .962143E-52 .122602E-57 .122157E-57 .536897E-64 .53847OE-64 22 N-Z .831496E-06 .8383895-06 .251494E-07 .256u295-07 .448093E-09 -465977E-09 .1535915-]] .486603E-ll -2513735-13 .282]285-]3 .7343965-]6 .876490E-16 .]088505-]8 .140650E-18 .787113E-22 .]]22555-2] .266822E-25 .4283695-25 .4070185-29 .7507]85-29 .2678725—33 ~579290E-33 .7282h55-38 .188422E-37 .781792E-43 .2468855-42 .3162h65—48 .124303E-47 .4590645-54 .2289325-53 .2272385-60 .146493E-59 N-3 .2050h45-04 .23303]5-04 .8098595-06 .]022455-05 .]808695-07 .2635525-07 .2213545-09 .3870h65-09 .]437785-]] .3]3h795-]] .479642E-14 .l35328E-l3 «794358E-l7 .3005005-16 .630u935-20 .330744E-l9 .23]]905-23 .1736285-22 .376933E-27 .4177OOE-26 .2625525-3] .hh]7605-30 .7493585-36 .1967075-34 .8389305-4] .352555E-39 .35]9h25-46 .242684E-h4 .527375E-52 .6]09375-50 .2684325-58 .53h38h5-56 COMPUTED SOLUTION IS ON SAME LINE AS XII) VALUES 23 TABLE 4 CHECK ON THE FIRST THREE SINGLET S ZERO WAVEFUNCTIONS X BETWEEN ZERO AND 7.75 X(I) N31 N-Z N-3 . - IOOOOOE+OI .100001E+Ol .997442E+OO .250000E+OO -.977106E+00 .959227E+OO .944917E+00 .977106E+OO .959227E+OO .944736E+00 .SOOOOOE+OO -.915237E+OO .847583E+OO .794269E+OO .915237E+OO .847583E+OO .794414E+OO .7SOOOOE+OO -.825840E+OO .686489E+OO .582872E+OO .825840E+OO .686489E+OO .532778E+OO .IOOOOOE+01 -.720513E+OO .500483E+OO .349834E+OO .720513E+OO .500483E+OO .349780E+OO .125000E+01 -.609593E+00 .312709E+OO .132405E+OO .609593E+OO .312709E+OO .132476E+OO .ISOOOOE+OI -.501320E+00 .1h1730E+00 .416246E-OI .501320E+OO .141730E+00 .415743E-Ol .17SOOOE+OI -.401525E+OO -.789459E-04 .157424E+OO .401525E+OO “.788153E-04 .157461E+OO .200000E+01 -.313722E+OO -.106706E+00 .212720E+OO .313722E+OO -.106705E+00 . .212780E+OO .225000E+01 -.239453E+OO -.177627E+OO .215059E+OO .239453E+00 -.177627E+OO .215068E+OO .250000E+OI -.178756E+OO -.216346E+OO .I73057E+OO .178756E+OO -.216346E+OO $178011E+OO .27SOOOE+OI -.130654E+OO -.228775E+00 .117592E+OO .130654E+OO -.228775E+OO .117541E+00 .300000E+01 -.935876E-01 -.221772E+00 .486397E-01 .935876E-01 -.221772E+00 .486283E-OI .325000E+01 -.6S7508E-01 -.202015E+00 .168483E-01 .657508E-01 -.202015E+00 .168149E-01 .3SOOOOE+OI -.453415E-01 -.175279E+00 ..709170E-01 ' .ASBAISE-OI -.175279E+00 .708673E-01 .37SOOOE+OI -.307110E-01 -.146077E+OO .109603E+00 .307110E-01 -.146077E+OO .109572E+OO X(I) .400000E+01 .425000E+01 .450000E+01 .47SOOOE+OI .500000E+OI .525000E+01 .550000E+01 .575000E+Ol .600000E+Ol .625000E+01 .6SOOOOE+01 .675000E+01 .7000OOE+01 .725000E+01 .7SOOOOE+01 .77SOOOE+OI N-l .204436E-01 .204436E-01 .133823E-01 .133823E-01 .861347E-02 .861847E-02 .546340E-02 .546340E-02 .341048E-02 .341048E-02 .209730E-02 .209730E-02 .127105E-02 .127105E-02 .759404E-O3 .759404E-O3 .447436E-03 .447436E-03 .260058E-03 .260058E-03 .149148E-O3 .149148E-O3 .844280E-04 .844280E-04 .471837E-04 .471837E-04 .260398E-04 .260398E-04 .141946E-04 .141946E-04 .764442E-05 .764442E-05 24 TABLE 4(CONT'D) N'2_ .117589E+00 .117589E+OO .917967E-01 .917969E-01 .697032E-Ol .697033E-01 .515996E-OI .515997E-OI .373086E-Ol .373086E-01 .263872E-Ol .263872E-01 .182788E-01 .182786E-01 .124145E-01 .124144E-Ol .827453E-02 .827442E-02 .541667E-02 .541661E-02 .348499E-02 .348500E-02 .220506E-02 .220514E-02 .137288E-02 .137301E-02 .841504E-O3 .841655E-O3 .508026E-03 .508184E-03 .BOZZOSE-OB -3CZ3SSE'03 N83 .i322625+00 .]322675+00 .140577E+00 .1406IZE+00 .]375355+00 .]375805+00 .i265675+00 .]265975+00 .]]09245+00 .]]09275+00 .9331805-0] .9329325-0] .7576805-0] .757255E-OI .5960395-0] .5955865-0] .455610E-01 .455272E-OI .339]7]5-0] ~339°39E‘01 .246333E-01 .246439E-Ol .l74797E-01 .]75]]95-0] .]2]3325-0] .]2]8]25-0] .8246755-02 .8303655-02 -.549336E-02 ~555253E‘02 .3588975-02 .364520E-02 COMPUTED SOLUTION IS ON SAME LINE AS X(I) VALUES 25 —O+woo.. mpim—wh. m—uw—m—. ONimMsv. Owimwow. .Niwhnu. NNithw. mwimmmm. Nuiwwuu. NNimpmp. mpimvwh. .O+moo.. ONinvw. Owiwuwv.i —N-wwmm. wwimmoh.i wuiwwm—. mfliw¢VN.i nfiimh—n. vniwvom. muiw—m—.i Onimnbv. ONINOVD. ONImN0—.i .O+w00w. .Nimvom. ¢Niwv0m. «0+wocw. «Niwn—..i Nuimmmm. Nfliwnvn. mwimpmh.i nNiw—Om.i nuimp¢w. nu:w®0—. vwiwmwv.i Quiwmmn.i mNiwnOw. vwiummm. VNimNON.] Ouiwwop.i wwiwhmu. NN-th®.i «Nimwmm. NNiwwwh.i «Niwmmp. «Niwmw—.i wwimmwn. nniw—Ow.i uniwmnm. mniwumh.i naiwpvfl. w0+m00v. mwiw—hn. eniw.0m.i mwiw¢hw. .O+mOO.. wwimmmn. vnim.0m.i vwimmmw. ¢O+w00—. «wiwhmw. muiwmhw.i mNin—N. mwiwnv..i DNimOmv. mniwmw—.i maimmvm. mniwmop.i buiwowh. nuimmmn. mwiw—vu.i mnimmop. vmiwmwv.i vwiwhm—. mwiwohw.i mwin—N. .O+wOOw. hNimva. hNimva. o I A mmh.i «Niwwvm. mpimmvp.i Ouimwmw. ON-mON..: puiwnvfl. .NNiwp.N.i Oniwmnm. Omimvhw.i .Niwvmw. «Niwwmh.i nwiwbmh. p0+wOO—. ¢Niwwmm. .Ninv..i NNimvmm. mNinNw.i puiwmwm. .O+uoow. «Nine—v. nwiwbmh.i emiwmmm. pNimOvp.i «Niwopw. —O+u00.. nuiuVB.. Qwiupvp.i «Niwvmfl. muiwhwh.i nu:wvn—. .O+w00.. mniwwpn. oninNN.i vuimmmm. vuiwwvo.i mfliwwwn. .O+wOO.. vmimmww.i vuinNp. muinmN.i Ofliwnmm. wNinN..i mwiuowu. maimumm.i mniwphu. wNivaO.i oniwmow. N I A Mm. wwimmum. muiwmum.i buiw—vw.i m.l A mma|31|Ce12R2(p)P and (ii) for the magnetic dipole cases r(n'mm = 8aISI3IchZR2

P. The quantities P are the spin sums and are given in TABLE 11. The constants Ce and Cm are given in TABLES 12 and 13 respectively. These equations along with the final steps of their derivations are given in appendix II.4. 32 TABLE 9 POLARIZATION PROGRAM CORRECTIONS State Correct Polarization Programmed Needed Quantity Quantity Factor 3s1 (ix/ii); 2; up; 3n, -3/<2/5T)i(e~p>p -e/3] (2°p)p - e/a -3/<2/EF> 3p, {3mm - . ;; um: 3?, -s/3/(2i>(EXP) Exp . -a/3/(2t) 352 k/3ln t-S, E°p t¢3ln 1s0 1/JZT 1 1//ZT Ex- NH\H 2.0x- 2e. w\m\i 33 Nfl\mx a~\m\i ~H\w acuumm wouuweo onuv. xnmi aux + oa_A~m\Ae] imvvuwfi mam onuV. ammia up + oQHA~m\Ae~+mvvm u mam w A.wxuv.wmm I Hmv + onam m MHMW o.uHm i emu + ea.m u may ch] 8 u.m.ofim i a» + oaammau we Auxov..mflm . aux + ea_m Luau ] o.unm . aux + oa_m C:Mi uaamom nanomcoonom msszNAN ”Hma<= HZflMfiDO ho onHm mHmomzoomom OH mam<9 :oauaaaaue 34 TABLE 11 SPIN AVERAGE RESULTS 351 + 3P0 2/3 3s, + 351 4/3 331 + 3P2 10/9 331 + 130 2/3 3P0 + 3s, 2 351 + 331 4/3 352 + 351 2/3 150 + 3s, 2 TABLE 12 ELECTRIC DIPOLE CONSTANTS 331 + 3P0 4WI9 3D1 + 3P0 4D/E/3 331 + 3P1 -2n1/6/9 3D1 + 3P1 -2n1/§/3 381 + 3P2 4n/3/9 3D1 + 3P2 ~2fl/6/5 TABLE 13 MAGNETIC DIPOLE CONSTANTS 331 + 1$0 “/9 3D1 + 180 Z/Efl/g Chapter 3 Calculational Results and Conclusions (A) Calculational Results The calculational results of this work will be presented in a series of graphs. In most cases the quantity being varied is the quark mass and usually the variation in mass is from zero to 20, in dimensionless units. The conversion to energy units is made by requiring the calculated splitting in the 2381 - 1381 to be equal to the y’(3685) - ¢(3095) splitting. This is one of the quantities plotted in the graphs presented below, and since this calculated splitting varies with the mass parameter, so does this conversion factor. To date most of the published calculations relating to the charmonium system give only the results for the parameters that result in the best fit to the system. I have tried to show how the different calculated quantities vary as a function of the parameters in this model. My hope is that by presenting the results in this form this will give some feeling as to how sensitive the calculated results are to the fitting parameters. It was pointed out in Chapter 2 that one of the programming checks was to see if the computed results were consistent with the known ‘behavior of these equations in various limiting cases. The case in vihich RAT-O and n+0 is known to yield the Schroedinger equation with a I lxinear potential. In this case, systems with the same total angular momentum, .1, will have the same eigenvalue, or binding energy, for this mass. In the case in which the mass becomes very large, our equations go over into the Schroedinger equation for the spherical harmonic oscillator. In this case we know that the difference in the binding 35 36 .mde xwmsv osu mo coauucsu m on o u Bax pom aoumhm v c may we meowumuamamnoo «hm .Nmm aanm .amm .omm .Nnu .~m~ .omH 0:0 mo mOuMum museum 0:» mo mufimuocm mcwoafim ocu we mammuo .H ousmam 37 ON H shaman L OH we om mm om mm as me am 38 .mmms xumsv osu mo coauoaSN m on o u HHSU HOW meUm HGUMHH®> warm. .>oo me e was .m .N .H mo>h=u pom onum HmUMHeo> one moumum cam ocm Nam use now use m o>asu How we oEmm oucoeomwfio mmmE oopsaaou Adam .Nmmv Q u mucouommwo mmme oo>pomoo Aaam .Nmmv upon: o\mo-uv cause mmwomv,s as ooomfimoe we Ammomvs Heap umouxo m o>psu how we oEmm map A-: + +1 + ammomvsve pom some: swoop oopsmsou m A-a + .a + flmaomVeea now seen: snooe eo>nonoo a upon: m\< mo poo» mono may moumum Hmma - Namfi use pom mommma wousnsou one a“ mucouommwo Amufics mmoH:0wm:oEmo :wv Emewoam one an mousmsou oucoaomwwo mmmE ago an poom>wo m>oo ewm.ov oucohowwfio mmmE s -.s oo>eomoo one nee age 11 .54 oo.o n epsu o>szu o>usu o>uso o>csu o>msu 46 ON mH . m: m meMHm e w>c=o \ m «5:: \ .1 _ m: 47 the ratio 0.587/(l2-l1) where A2 and A1 are the computed masses of the 2381 and 1381 states in this model. The curve labeled 3 is the cube root of the ratio of the observed leptonic decay rate of the 0(3095) + W++u- to the computed leptonic decay rate. Curves 1,3, and 4 all in principle give a value for k, the conversion factor between my dimensionless units and energy (Gev) units. Curves 3 and 4 however do not intersect, so only curves 1 and 3 are used to determine k. The intersection point of these two curves (1 and 3) determines k and the quark mass. The other curves in figure 5 can be used to fit any additional parameters that remain in the potential. The only other parameter currently available is the ratio of the scalar to vector coupling strength, RAT. By requiring that the 13P2 - 13Po mass difference come out to be the value observed for the x(3551) and x(3414) states, a value RAT(I) is determined. By requiring the computed I3P2 - 13?] mass difference to come out to the observed mass difference between x(3551) - x(3507), a different value RAT(2) is found. It is not possible to obtain a fit for the x(3551) - 0(3095) mass difference. It is also not possible to obtain a satisfactory value of RAT that will invert the calculated level structure, i.e. go from 1381, 13D1, 2381, 23D}, ... to 1381, 2381, 13D2, ... and still have a quark mass of about 1.8 Gev. When trial runs to obtain this result are made, the value of RAT is large and the k,m determinations produce very large values of the quark mass. This implies an extremely relativistic configuration, contrary to what is generally believed to be the case for the charmonium system. From the discussion in [29] it seems that their B/r potential is mainly responsible for the correct level structure 1381, 2381, 13D1, ... It was hoped to include this potential form in our model. The 48 problem preventing this usage to this point is an inability to perform numerically certain singular integrals. These singular integrals arise from the need to use a momentum representation for the B/r potential. When this is done, integrals with terms like 1n(p-p’) are encountered. Attempts were made to check out the integration routine for the coulomb potential in momentum space by computing numerically the-expectation values of l/r, l/r2, etc. whose values are known [17]. The results were inconsistent with those given in [17]. One therefore concluded that this method of dealing with singular integrals could not be trusted and the attempt was temporarily dropped. I had hoped to look into this matter again to see if this potential form can be included. I would like to look at this problem again to see if these difficulties can be overcome and an a/r form can be included in the potential structure. In figures 6, 7, and 8 are shown graphs of the wavefunctions for the first three singlet S states, singlet P1 states, and singlet D2 states. In [35] a procedure for the solution of the Schroedinger equation with a linear potential is given. The method developed in [35] can be used for values of L, the orbital angular momentum, different from zero. Reference [33] discusses these wavefunctions, and graphs of some of these wavefunctions are presented. Comparison of figures 7 and 8 with figures 2 and 3 of [33] shows that there is good agreement between our mPO, RAT-0 wavefunctions and their results. In the system of constraints used here to determine m, k, and RAT, the values of m determined were found to fall within the interval 3.0 < m < 7.0. Figures 9, 10, 11, and 12 show most of the same quantities that were plotted in figure 5 for this restricted mass range and for values of RAT-0.00, -O.16, -O.275, and -O.33. In figure 5, 49 15 r L -15 -20 l l I 1 U 10 20 30 40 50 Figure 6. Plot of the Wavefunctions of the first three states of the 180 system, for m . RAT - 0. 1150 ..... 2130 ----- 3130 2C1 . 50 I 15 I _10 -10 . -15 T -2EI ' l l l O 10 2C) BC] 40 5C] Figure 7. Plot of the Wavefunctions of the first three states of the 1P1 system, for m = RAT - 0. llPl 2151 ----- 3151 51 20 . i . . 15- J 10- - s- . -10 - .. -15 .- .. -20 I l I l U 10 20 30 40 50 Figure 8. Plot of the Wavefunctions of the first three states of the 1D2 system, for m - RAT - 0. 11D2 0000. 21132 ----- 31D2 52 qum:: mmmE mmoHCOMmcoEwo cw we onom Hmuconuon use .muwcs mmoHCmecoEwo a“ me -...-...- can -..-..- How o-eum Hmufiueo> use .>oo we -.-.- mam ..... ..... how o-mum HmuMupo> use moueum mom can can one you use o>asu -..-..- how we seem use -...-...- mucouommwo mama oou=QEou Aaam .Nmm- u a oucoaoMMwo mmee oo>uomno Anon .umm- u u when: u\ma-uv cause one -..-..- ueEwH uoon may a < no“: asp o>uzo ..... how we mean oz» x -.-.- m-o + +0 + Amman-eve mom can“: scoop oousnsou u m A o + +0 + ammom-svm you some: amuse oo>somoo mo Hosea page: u < upon: m\< mo woos mono one u x ..... moumum Hmma . mama any new mommme oousasou on» em oucopowwwo one ..... ampwcs mmoH:o«m:oEfio :wv Emuwoum one an mouzmsou monopowmwo mmme use an voow>fio n>oo ham.ov oocoaomeo mama a -.s oo>womoo ecu u x oo.o u e one .>ou mm -.-.i new . ..... ...... . pom o-mum anomueo> one moumum «mm one can on» how «so o>eso -..-..- you we osmm one -...-...- oocouomwwo mmee mouzaeou Aamm .Nmmv a oocoaomwwo mmme oo>womoo a-mm .Nmm- o onus: U\nn-u- swamp oak -..-..- pwEwH uoon o:u u < sues use o>a=o ..... now no oEom on» II .54 m-o + +o + amaomveva now seen: anooe eoesasoo m a-o + +o + ammom-s-e you new“: xeooo oo>aomoo mo awefia some: n < oposz m\< mo woos ooau osu n x ..... moumum ammH . «ama one pom mommme monomEoo one s“ oucoaommwo one ..... Amuflcs mmoflcowmcoEwo :flv anemone one an ooHSQEOU oucopommfio mans ozu an movw>wo n>oo 5mm.ov oucoaommHo mass 9 -‘s oo>aomoo one u a oH.o- n b<¢ .muouoeouom Homoeouom mo :owumewsuouoa .OH oesmMm 55 em mm ow mm om we Qv mm om OH apnoea - — F P - - @- om om ow om 56 Amuwcz moms mmoacoflmcoswo an we onmom HensoNflnon one .mufics mmoacowmcosflo an we -...-...- new -..-..i How oHoUm nonwaeo> one .>ou we -.-.- one . ..... ...... . How onmom noueuno> one mopmum «mm one can on» now won o>a=u -..-..- how mo oEmm ouconowwmo mmeE oousneou Anmm .Nmmv u a ounonowmflo mmms mo>eomno Anna .Nmm- u u anon: u\na-uv apnea qunH nosed one u < nun: usn o>esu ..... now we oeom onu n-o . .o + Ammonvesa now anew: saooe eoeoaeoo u m m-o + +o t AmmomVa-n now new“: amooo oo>eomno mo unEMH Moan: u < oaon: m\< mo pooh on:o one moumum nmmH . mama onu pom mommme oousasou on» :w oucoeomwno Amuncs mmoncoflmcoEwo any anemone one an nonsmEou ounoaowmwo mmme onu an poow>wo fi>oo nwm.o- oocouommMo moms 9 its oo>homno one me~.o- u H<¢ .mwouoEmemm Hewunouom mo :ofiaonwsnopoo .HH onswnm och; -..OIOOOI one -..-..- II Ad 57 em we on. mm om on. mm om mm so 0‘0 HH apnoea on em om on em om 58 Amuwns mmoE mmoH:0nm:oEno n“ we onoum newcoNHpon one .mumca mmoanonnoEwo an we -...-...- one -..-..- how oHoUm Hmonuao> one .>ou me -.-.- one . ..... ...... . how oamom Hooeuno> one moumum «mm one can onu you won o>p=u -..-..- now we oEem one -...-...- oucohommwo mama oopsnsoo mnmm .Nmmv n a ounouowao moms oo>nomno n-mm .Nmm- n u ouonz U\fia-uv enema one -..-..- uHEmH wozoa one u < now: won o>eso ..... How mm osom on» II Ad a-o + +o + Amman-syn now new“: zouoo mousnEou m n-o . .o + Amaomvasa «on anew: neooe eo>nonao no unann noon: n a ouon: m\< mo econ onzu on» u n ..... moumum Hmma . «nan onu pom mommme wousnsoo one an oonouommno one ..... mmuwns mmoanowmnoEwo :w- Eoumoun one an mousmEou ounohommHo mmee on» an wovfi>wo a>oo nmm.c- ounohommMo mmoe 9 its oo>nomno one u n mm.o- n H®v— OH HO mafia—3 GM mun wHwUm HmUthokr GEE. >ox u.oH u flovu manna meow oHuwuqu :w :o>wm r + cam + “mmN mo umEflH aozoq -..-..- >ox e.H~ n HHeH.ee eo>em we saw“: zmoou pocweuouow xaamacoEwuomxo r + cam + "man we quwH page: -.-.- E mmws xnmzd may mo :oHuucsm m mm sup“: xmuop > + «mm + Hmmm ..... E mama xumsd can mo :owuucsw a ma new“: Amoco r + ~mm + ”mmN ..... E mmma xumso any mo :owuocsw a mo cup“: xauop r + can + flmmN ee.e- u eox oH mo mugs: cw mw oamum amoeuho> one >ox n.oH u now“ manna «not oHufiuuma :« :o>ww r + can + "mMN mo HMEMH pozoq -..-..- >6; e.H~ u ”see on eo>aw mm sup“: xmuov vocweuouov xaamucoswuomxo r + can + ~mm~ mo quMH some: -.-.- E mde Macaw on» mo newuucsm a ma :uvmz xmuop r + «mm + "mmm ..... . E mmws xumzc one mo :owuucsw a no can“: xwuop r + Ham + ammm ..... a mass Manna any mo :owuucsm a no cap“: xmuop > + 0mm + ammm mnN.o- u h<¢ .moumum r + an + Hmmm now macaw: :owuwmcmpe .eH meow“; 67 es oeeeee om 68 .oo.o~ne op oo.ouE .mmme xumsc mmoH=0wmcoEmw ecu mw ofiwum anaconao; ozh .>®M Ca m0 mHMCD GM mw OHQUW HwUMHHQK/ 0:? >ox n.oa n meow manna mump oHUMHth cw :o>ww » + can + ammN mo quflH nozoq -..-..- >ox v.- n mHeH.:H co>wm we not“: known vocwsuouop AHkucoEwuoqxo r + can + mem mo pfiEwH song: -.-.- E mmms xumzc may we newuucsm a no cap“: xmuow r + ~am + ammm ..... . E mmme xumsc ecu mo :owuocsm a no can“: zmuop r + “mm + amm~ ..... E mmme xumad oz» mo sawuucsm a ma gap“: Amoco > + can + ammw mm.o- n b<¢ .moumum » + mm + _mm~ pom asap“: :owpwmcmah .mH oeswwm 69 me oteeee b ow 70 case 2 was done. For RAT a -O.16, the biggest difference between cases 1, 2, and 3 occurs in the ¢’(3685) + 3P1 + Y decay width, the low mass peak being about twice as big in cases 2 and 3. For cases 2 and 3 the difference in the calculated results for the decays leading to the 3P1 and 3P2 states is small. The decay width for the W’(3685) + 3P0 + Y process in case 1 is also larger than that computed by case 2, the peak being about 30 kev higher in case 1, and for m=5 the decay width is about 30 kev. This decay width, for case 3, is about 10 kev higher than that computed for case 2 in the low mass region (m85) where the fit is desired. For RAT - -O.33 the calculations in cases 2 and 3 produce identical results for the 3P2 process. The case 3 results for the other two decay widths are different from those obtained from case 2. The 3P1 width in case 2 is higher for m < S, but for m > 5 cases 2 and 3 give the same results. In the 3P0 case the oscillations present for m < 5 are more pronounced in case 2 than in case 3. For m > 5 case 3 yields a larger decay width than does case 2. At m - 5 the difference in the decay widths leading to the 3P0 state calculated by means of cases 2 and 3 is about 5 kev, and as m increases this difference drops off to a couple of kev at m - 20. For case 1 the computed decay widths are much larger, the 3P1 peak being about 2.5 times that of case 2. In case 1 the oscillatory behavior present for m < 5 in the 3P0 process is much larger in amplitude, the peak being at 125 kev instead of the 75 kev 'value of case 2. The ¢’(3685) + 3P2 + Y decay widths are also larger in case 1, being for m 8 5 about 80-90 kev compared to the case 2 value of about 50 kev. Figures 16, 17, and 18 show the results of the decay width calculations for the transitions 13P1 + 1381-+ Y and 13P2 +-1381 + Y for 71 .oo.o~nE op oo.ouE .mmmE xnmsc mmoacowmcoawp may mw onum Hansenwpo: one .>ox ooa mo mafia: :w ma macaw Hmowueo> one >ox com mo snow: >moo© m ow mwcommohpou o>uso mash -.-.- >ox ocH mo aura: xmuow m ou mucommohpou o>a=u maze ..... E mmmE thsc may mo :omuuczm a mu new“: xmuop » + Emma + Nana ..... E mmmE xemsc any mo :owuucsm n mm can“: amour r + Emma + Emma ofi.o- u Hox ooa mo mafia: cw ma onum Hmowpuo> och >ox com mo aura: xnuov m 0» mvcommopuou o>p=u maze -.-.- >0: so” mo some: xmuop a On mpcommoaaou o>u=o mash ..... E mme xamsc on» mo :owuucnm a ma gnaw: xmoov > + Emma + NamH ..... E mme sumac on» mo :owuucsm a mu sup“: knuop » + Emma + "an” mem.o- u Hox ooH mo muflcz cw ma QHEUm Houduuo> one >03 com mo sup“: amour m on mvcoamoppou o>u=u maze -.-.- >03 ocH mo pupa: xmoow m cu mwconmoauou o>uzu maze ..... E mme xumsc one mo acauucsw a mo new“: xnuop r + Emma + Nana ..... E mmmE xumsv on» mo :owuucsm m mm :uvw3 amooc r + Emma + Emma e S. In case 1 there is no characteristic low mass hump in the 3P1 decay. For large masses (m + 20), the 3P2 and 3P1 decay widths are about equal in each case. There are only slight differences in the results calculated in cases 2 and 3. The differences occur mainly in the region 3 < m < 5 and amount to no more than 10-20 kev out of the computed values of 300 kev for this mass range. For RAT - -0.33 and m < 5, case 1 gives larger values for the 3P2 +'1k3095) + Y decay width than does case 3. For m > 5, the decay widths are not appreciably different. For m < S, the 3P1 decay width in case 1 is negative because of the inversion of the 3P1 and 1381 levels. For case 1 the decay width of 3P1 + 1381 + Y is always less than 100 kev. The difference in the computed decay width in cases 2 and 3 is negligible. Figures 19, 20, and 21 show the results of the calculation for 13Po + 1331 + y, for RAT - -0.16, -0.275, and -o.33. The horizontal 78 .oo.o~uE ou oo.ouE .mmmE thsc mmoHEOMmEoEwc ecu ma mamum HwHEOpro: och .>ox cofi mo mafia: aw ma onum HmUMupo> mg? E mmmE xpmsc ecu mo :ofiuucsm a ma saw“: zmuow r + Emma + omma ..... >ox ooH u mafia mucouomom o~.o- u E + Emma + Emma you some: :ofiuwmcauh T?” madman 79 0H oaemea _ n 1o as so ooooo O O o o I OH 8O .oc.o~uE on oo.ouE .mmmE xpmoc mmoficowmcoswv on» ma onum amazoNflao: one .>ox ooH mo mafia: cw mw onum Haowupo> och E mmmE Mumsc ogu mo :oHHUESM m mm supflz amuop > + Emma + onH ..... >ox ocH n oEMH mucouowom mn~.o- u H<¢ .> + Emma + oEMH now can“: mewufimcmah .om ousmfim 81 ........... ......... 00000000 O. 00000000000000 ..... O... CI. 0.. so. 00 0. so no .0 0’ om manned - _ m: om om ow om om on ow om 82 .co.o~uE op oo.onE .mmoE xhosc mmo~E0mmEoEmv ozu m“ oHoUm HoucoNfiao; ozb .>ox ooa mo mafia: cw ma oaoom HoUwuuo> oak E mmoE xuosc ogu mo :owuucsm a mo and“: zouov > + Emma + coma ..... >ox OOH n ocwa oocouomom mm.o- n h<¢ .> + Emma + omma how can“: :owufimcouh .HN ousmwm 83 Hm oazmmm ..Mm: 8 em S cm 8 E. 84 line here corresponds to a decay width of 10.0 X 10.5 Gev. The experimentally determined width given in [41] is 9.7 i 3.8 X 10-5 Gev. In the 3P0 + 1381 + Y decay the computations were done at RAT a -0.16 and at -0.33 for all three cases. For RAT - -0.16, cases 1 and 3 produce quite different results. The main effect comes from the fact that the 3P0 level lies below the 1381 level for low masses. (See Table 14.) This results in the appearance of unphysical (negative) values for the decay widths for m < 5. The low mass hump seen in most of the decay plots is not present in the case 1 results. The graphs of the case 2 ' and 3 computations are very similar, the only difference being a slight bump in case 2 at m = 4.5 of about 30-50 kev.‘ For RAT - -0.33 the calculations produce quite different results for m < 5. The results of case 3 look very much like the curves for 3P2 +'w-+ Y in the previous plots. Case 1 for m < 4.5 is negative and large. The decay width for m = 5 is about 100 kev and for m.> 6 is small (<30 kev). The results of case 2 are shown in figure 21. The behaviors in cases 1 and 2 for m < 4.5 are almost mirror images of each other. For case 3 the computed decay widths at m - 4.0 and m - 5.0 are 166 kev and 120 kev respectively. In figures 22, 23, and 24 are shown the computed decay widths for the transitions 2331 +1130 + y, 1351 +1130 + y, and 2331 + 2130 + Y. The horizontal lines in these figures are the experimental limits quoted in [41] for the decay ¢’(3685) + nc’(3592) + Y, namely processes are still not determined very precisely, but their values, even within their given errors, still fall within the .43 to 2.795 range. The graphs in figure 22 were computed with a value of RAT - -0.16; the plots in figures 23 and 24 were done with values of RAT = -O.275 and -0.33 85 .OO.ONHE OH OO.OHE .mmmE v—hNDU meHCOMmfiwczmmu OSH mm OHNUm HQHCONMhOr— 02% E mmoE xuosc on» mo :oHuocsm m on swag: zouow r + E mmoE xposc on» mo :ofiuucsw a mo sup“: xouov > + E mmoE xuosc on» mo Eofluucsm o we saw“: xouop r + oa.o- .m>ou Em :uvfiz zooop wousmEooV camoH ocu ma oHoum Hoofluao> oak me-EH x mos.mv as-eH x oem.ev H<¢ .msuvwz :owuwmcosh oHomwn owuocwoz omqm + gmmN MO sole + _mmH mo emse + swam mo .NN osswwm eawofi camofl eamofi camoH camoH 86 OH Nu oszmpu _ DEE: QQHI om: 87 .oo.o~uE ow oo.ouE .mmoE xsmsc mmoHEmecoEww onu mw onum Hooconno: ozb E mmoE Macaw osa mo sawuucsm m on sup“: xouow r + E mmoE xaosc onu mo :owuocsw m on now“: snoop > + E mmME gems: ozu mo EOMHUEEM m mo Eur“: zooop > + mm~.o- .a>oo cw can“: xouom wousmEoov odon on“ ma oHoum Houflueo> ocb ne-ee x was.mc as-ofi x oom.ev hou EM :uwwz kooop wousanuv onOH one ma onum Hoofluao> one ne-EH x mes.me as-ofi x oom.ev E mmoE th53 on» mo :owuocsm o no can“: kouop > + omEN + Emmm mo E mmoE xsmzc ozu mo Eowuucsm a no cap“: xmuov > + Ema“ + Emma mo E mmoE xemzc on» mo :owuucsm o no can“: havoc » + omHH + “mam mo mm.o- a 8. From figures 22, 23, and 24, it appears that the magnetic dipole widths do have a dependence upon RAT. To get a better feeling as to the actual variation of these magnetic dipole transition widths, we give the actual values of these transition widths for the quark masses 3.0 to 7.0 in TABLE 15 for RAT - -0.16, -O.275, and -O.33. For an explanation of the notation used in this table see TABLE 16. Additional information on these radiative decay processes can be found in Tables 17 to 21. Tables 17 to 20 give the values of the radial integrals for RAT - 0.00, -O.16, -O.275, and ~O.33. The key to the organization of these radial integral tables is shown in TABLE 16. 'IABLE 21, for RAT - 0.00 and m - 5.0 shows the way the radial integral 18 built up. It is seen from TABLE 21 that the main contribution comes TABLE 15 VARIATION OF MAGNETIC DIPOLE NIDTHS WITH RAT(GEV) PROCESS RSSZ RATEC RATEA RSSZ RATEC RATEA RSSZ RATEC RATEA RSSZ RATEC RATEA RSSZ RATEC RATEA .857OE-05 .1043E-O5 .3871E-06 .2081E-05 .5491E-06 .221OE-06 .2043E-06 .2594E-06 .13365-06 .1369E-06 .1824E-O6 .3508E-07 .5557E-07 .1182E-06 .5682E-07 RAT--O.16 RAT--O.275 H-3 .8247E-05 .8399E-06 .3184E-06 H-4 .51OOE-05 .3989E-06 .1731E-06 HIS .2879E-06 .2249E-06 .1034E-06 H-6 -93995-07 .I344E-06 .64275—07 H-7 .3682E-O7 .8576E-O7 .4218E-O7 THE ERRORS IN RSSZ AND RATEC ARE ABOUT 25 AND 40 PER CENT RESPECTIVELY RAT--O.33 .1041E-O4 .7371E-O6 .2850E-06 .1060E-O4 .3371E-06 .1322E-O6 .2507E-06 .1854E-06 .8672E-07 ~7772E'07 .1094E-06 .5307E-O7 .2761E-O7 .6906E-O7 .3434E-O7 OBSERVED .645E-06 .756E-06 .43 TO 2.8E-06 .6455-06 .756E-06 .43 TO 2.8E-06 .645E-O6 .756E-06 .43 TO 2.8E-06 .645E-O6 .756E-06 .43 TO 2.8E-06 .645E-06 .756E-06 .43 T0 2.8E-06 quark mass quark mass quark mass quark mass (m) (m) (m) (m+1) RSPZ RSPO RSPT RSPZ 93 TABLE 16 RADIAL INTEGRALS KEY RPZS RPOS RPTS RPZS 812C. RATEA RSSZ RATEC RATEA where the correSponding processes are identified RSPZ RSPO RSPT RPZS RPOS RPTS RSSZ RATEC RATBA o’(3685) w’(3685) w’(3685) x (3414) X (3507) X (3551) ¢’(3685) .w (3095) w’(3685) X(3414) + x(3507) + x(3551) + 6(3095) + 4(3095) + 6(3095) + Y Y nc(2934) + y nc(2984) + Y né(3592) + Y TABLE 17 94 THE RADIAL INTEGRALS AS A FUNCTION OF THE QUARK MASS FOR RAT=0.00 .1000OOE+01 .1000OOE+01 .1000OOE+01 .200000E+01 .ZOOOOOE+01 .ZOOOOOE+01. .3000OOE+OI .3000OOE+01 .3000OOE+01 .4000OOE+01 .4000OOE+01 .4000OOE+OI .500000E+01 ' .5oooooe+01 .5oooooz+01 .6oooooa+01 .6oooooe+01 .6ooooos+01 .7ooooos+01 .7000605+01 .7oooooa+01 .Boooooe+01 .Boooooe+01 .aoooooe+01 .9ooooos+01 .9oooooe+01 .9ooooos+01 .217319£+oo .150658E-02 '.225797E+oo .4671135—01 .113364£+oo .397167£+oo .2249195+00 .2736605+oo .404432£+oo .2686OZE+00 .2841685+oo .3372715+oo .23764IE+00 .247163£+oo .280406E+00 .2281515+oo .232434E+00 .247249£+oo .209056E+00 .211284£+oo .2195485+oo .191764£+oo .193ou5e+oo .198048E+00 .1768485+00 .1776415+oo .180858£+oo .164087E+OO .164605£+oo .166772£+oo .747014E+00 .796518E+00 .575166E+00 .824063E+00 .852695E+00 .821269E+00 .604400E+00 .603421E+00 .604089E+00 .458781E+00 .457804E+00 .462235E+00 .406179E+OO .378131E+00 .402359E+00 .318388E+00 .317823E+00 .315047E+OO .279730E+00 .279550E+00 .278809E+00 .250550E+00 .250463E+00 .250136E+00 .227644E+00 .227596E+00 .227422E+00 .209112E+00 .209082E+00 .2089805+00 .1024705+oo .935898E-01 .148867£+oo .242934£+oo .287832£+oo ~37357SE+00 .266567E+00 . .145168£+oo .352293E+00 .228200E+00 .6217255-01 .277449£+oo .1923oos+oo .341702E-01 .224539£+oo .164943£+oo .118132E-01 .182945E+00 .143602£+oo .105477E-01 .155771£+oo .126702£+oo .725529E-02_ .135372£+oo .113175E+00 .5170155-02 .119594£+oo .102154E+00 .3820915-02 .107056E+00 95 TABLE 17 (CONT') .loooooe+oz -.153126£+oo .19376ZE+00 -.930260E-01 .Ioooooe+oz -.153h79£+oo .1937425+oo .291216E-02 .Ioooooe+02 -.1549945+oo .193677£+oo .9686645-01 .110000E+02 -.143639£+oo -.180808E+00 .8535575-01 .IIOOOOE+02 -.143888£+oo -.180794£+oo .227723E-02 .110000E+02 .1449805+oo .180751£+oo .8842855-01 .Izoooos+02 -.135359£+oo -.169709£+oo .788272E-OI .120000£+02 -.135540£+oo -.169699£+oo .181939E-02 .120000£+02 .136348£+oo .169669£+oo .813302E-01 .13oooos+02 -.128075£+oo .160077£+oo .7320805-01 .13oooos+02 -.128209£+oo .16007os+oo .1480245-02 .130000£+02 -.128820£+oo .160048E+00 .752780E-01 .140000£+02 -.121617£+oo -.15163os+oo .683234E-01 .I40000E+02 -.121720£+oo -.151624£+oo .1223165-02 .Iuoooos+oz -.122190£+oo -.151608£+oo .2005805-01 .Isooooe+oz -.115853£+oo -.144152£+oo .640403E-Ol .Isooooe+oz -.115932£+oo -.144148£+oo .102437E-02 .150000E+02 -.116301£+oo -.144135£+oo .6551065-01 .160000£+oz -.110674£+oo -.137480£+oo .6025532-01 .1600005+oz -.110736£+oo -.137476E+00 .8679595-03 .160000E+02 -.11103oa+oo -.137467E+00 .6151435-01 .17oooos+02 -.105995£+oo -.131484£+oo .5688735-01 .17oooos+02 -.106045£+oo -.131482£+oo .743OI3E-03 .170000E+02 -.106281E+00 -.131474£+oo .5797505-01 .IBOOOOE+02 -.101745£+oo .126064£+oo .538718E-OI .180000£+02 -.101785£+oo .1260625+oo .641852E-03 .180000E+02 -.101978£+oo .126057E+00 .5481895-01 .Igoooos+oz -.978668£-01 .121137E+00 .511565E-01 .190000E+02 -.978997E-Ol .121136£+oo .558965E-03 .Igoooos+oz .9805835-01 -.121131E+00 .519872E-01 .2ooooos+oz -.943119E-OI -.116636£+oo .486993E-01 .200000£+02 -.943391E-01 -.116635£+oo .490320E-03 .2ooooos+02 .9447095-01 .116631£+oo .4943255-01 TABLE 18 96 THE RADIAL INTEGRALS AS A FUNCTION OF THE QUARK MASS FOR RAT--O.16 .100000E+01 .100000E+01 .1000006+01 .2oooooa+01 .2ooooos+01 .2ooooos+01 .3oooooe+01 .3oooooe+01 .3ooooos+01 .400000E+01 .4ooooos+01 .4000005+01 .5ooooos+01 .5ooooos+01 .sooooos+01 >.600000E+OI .6ooooos+01 .6ooooos+01 .7ooooos+01 .700000E+OI .7ooooos+01 .Boooooe+01 .Booooos+01 .Booooos+01 .9ooooos+01 .9oooooe+01 .9oooooe+01 .314656£+oo‘ .984360E-03 .713235£+60 .698416E-01 .198053E+00 .424731£+oo ’ .289642£+oo .3230795+oo .414009E+00 .305935£+oo .315706£+oo .351846£+00 .286690E+00 .289127s+oo .3037ooe+oo .253805£+oo .254696E+00 .263312E+00 .228462£+oo .22957oz+oo .235375£+oo .2ogsso£+oo .210113E+00 .213595£+oo .193298£+oo .I936IIE+OO .195843E+00 .I79427E+00 .179613£+oo .181113£+oo .90573OE+00 .IOOOOOE+01 .186857E+01 .841996E+00 .847571E+00 .809197E+OO .618097E+OO .617968E+OO .609615E+OO .482617E+OO .482904E+00 .482382E+006 .403025E+OO .403384E+00 .403937E+OO .340072E+OO .337297E+OO .312365E+OO .302819E+OO .302556E+OO .301617E+OO .272436E+OO .272331E+OO .271961E+OO ‘ .248212E+OO .248159E+00 .247971E+OO .228436E+OO .228405E+OO .228298E+00 .1217oue+oo .II9327E+00 .187547£+oo .269781£+oo .253649£+oo .39346ZE+00 .258072£+oo .117468E+00 .344420E+00 .218984£+oo .539162E-01 .269863E+oo .186035E+00 .271822E-OI .217333E+00 .160092£+oo .3774946-01 .180083E+00 .139905£+oo .998030E-02 .153816£+oo .123947E+00 .739OI6E-02 .133967E+00 .111071£+oo .541835E-02 .118551E+00 .100511£+oo .408369E-02 .106259£+oo .Ioooooe+oz .Ioooooe+oz .Ioooooc+oz .110000E+02 .110000E+02 .110000E+02 .Izoooos+oz .120000E+02 .120000E+02 .13oooos+oz .13oooos+oz .13oooos+oz .I40000E+02 .140000E+02 .140000E+02 .150000E+02 .150000E+02 .I5000OE+02 .16ooooe+oz .16ooooe+02 .16ooooe+oz .I7ooooe+oz .17oooos+oz .Iyoooos+oz .180000E+02 .180000£+02 .180000E+02 .Igooooe+oz .Igoooos+oz .190000E+02 .200000£+oz .2oooooe+oz .2oooooe+oz TABLE 18(CONT'D) .167523E+00 .167638E+00 .1686865+00 .157220E+00 .157295E+00 .158050E+00 .148226E+OO .148276E+00 .148834E+00 .140308E+00 .140343E+00 .140765E+00 .I33284E+00 .133309E+00 .I33634E+00 .127009E+00 .127027E+00 .127281E+00 .121368E+00 .121381E+00 .121583E+00 .116265E+00 .116275E+00 .116438E+00 .111626E+00 .111634E+00 .111766E+00 .107388E+00 .107394E+00 .107503E+00 .103499E+00 .103503E+00 .103593E+00 97 -.211956E+00 -.211937E+OO -.21187OE+OO 197937E+00 .197975E+00 .197931E+00 .1859BOE+00 .185971E+00 .185941E+00 .175534E+00 .175528E+OO .175506E+00 .166354E+00 .166349E+OO .166333E+OO .158214E+OO .153211E+OO .158199E+OO .150942E+OO .15094OE+OO .150931E+OO .144401E+OO .144399E+OO .144392E+OO .138482E+OO .138480E+OO .138475E+OO .133097E+OO .133095E+OO .133091E+OO .128174E+OO .128173E+OO .128169E+OO .917180E-01 .3I6OI6E-02 .962422E-OI .84296OE-O1 .250176E-02 .879295E-O1 -779557E‘01 .201923E-02 .80924OE-01 .724818E-01 .165695E-02 .749425E-01 .677114E-01 .137918E-02 .697772E-01 .635194E-O1 .116227E-02 .652728E-01 .598081E-O1 .990150E-O3 .613IO8E-01 .565005E-01 .851619E-O3 ~577993E‘01 .5353505-01 .738715E-O3 .546661E-O1 .508616E-OI .645661E’O3 .518533E-O1 .484393E-01 .568189E-O3 .493144E-01 TABLE 19 98 THE RADIAL INTEGRALS AS A FUNCTION OF THE QUARK MASS FOR RAT=-O.275 .754984E+00 .791531E+OO .Iooooos+01 .Ioooooa+01 .IOOOOOE+OI .2000065+01 .2ooooos+01 .2oooooa+01 .3oooooe+01 .3ooooos+01 .3ooooos+01 .4000005+01 .4000005+01 .Aoooooe+01 .sooooos+01 .5oooooe+01 .5oooooz+01 .eoooooe+01 .6oooooe+01 .Booooos+01 .7oooooe+01 .7ooooos+01 .7oooooe+01 .8ooooos+01 .aooooos+01 .aooooos+01 ..900000E+01 .9000005+01 .900000E+01 .189974E+00 .240058E-02 .226665E+00 .145047£+oo .4288135-01 .3758505+00 .162740£+oo .225274E+00 .39527ZE+00 .238835E+00 .2588IIE+00 .326959£+oo .1840]8£+oo .205073E+oo .262320E+00 .207322E+00 .212864E+00 .230626E+00 .190202E+00 .193149E+oo .202977E+00 .174421E+00 .176150E+00 .182067E+00 .160761E+00 .16184SE+00 .165633E+00 149063E+00 149779E+OO 152320E+OO .571706E+OO .782606£+oo .845562£+oo .823551E+00 .582081£+oo .592731E+00 .597849E+00 .43184ZE+00 .433901E+00 .441314E+00 .346402£+oo .345171£+oo .335028£+oo .292947E+00 .292821£+oo .291347E+OO .255893E+00 .255827E+00 .255343£+oo .228414£+oo .228372E+oo .228147E+00 .207064E+OO .207036E+OO .206912E+00 .189907E+00 .189888E+OO .189812E+OO .1042235+00 .9094525-01 .147512£+oo .200503E+00 .311263E+00 .3582505+oo .271109£+oo .175095£+oo .35845oe+oo .238024£+oo .7013945-01 .286382£+oo .l97583£+oo .6210335-01 .222228£+oo .1698226+oo .1641105-01 .185544E+00 .I47056E+00 .105668E-01 .157548£+oo .129252£+oo ~699797E-02 .1366405+oo .115107£+oo .48708IE-02 .120531E+00 .103653£+oo .3535845-02 .10777os+oo .IOOOOOE+02 .IOOOOOE+02 .1000OOE+02 .1IOOOOE+02 .110000E+02 .11000OE+02 .120000E+02 .120000E+02 .120000E+02 .13000OE+02 .13000OE+02 .13000OE+02 .14000OE+02 .14000OE+02 .14000OE+02 .150000E+02 .1SOOOOE+02 .150000E+02 .16000OE+02 .160000E+02 .160000E+02 .17000OE+02 .17000OE+02 .17000OE+OZ .1BOOOOE+02 .180000E+02 .180000E+02 .190000E+02 .190000E+02 .19000OE+OZ .ZOOOOOE+02 .ZOOOOOE+02 .ZOOOOOE+02 TABLE 19(CONT'D) 139016E+00 .139508£+oo .141279£+oo .130326£+oo .130677£+oo .1319505+oo .1227495+oo .123006£+oo .123945£+oo .116089£+oo .116281£+oo .1169915+oo .110191£+oo .110338£+oo .1108845+oo .10493os+oo .105044E+00 .10547IE+00 .Ioozo7e+oo .1002985+oo .100637£+oo .9594355-01 .960160E-01 .962892E-01 .9207365-01 .921324E-01 .9235485-01 .8854415-01 .8859236-01 .8877535-01 .8531082-01 .8535085-01 .8550295-01 99 .175765£+oo .175751£+oo .175701£+oo .163872E+00 .163862£+oo .163828E+oo .153710E+00 .153702£+oo .153678E+00 .1449095+oo .I44903E+00 .1448865+00 .137203E+00 .137199£+oo .137185£+oo . .130391£+oo .130388£+oo .130377£+oo .124320£+oo .124317E+00 .124309£+oo‘ .118869£+oo .118867£+oo .1188605+oo .1139452+oo .113944£+oo .1139385+oo .109472£+oo .1094715+oo .109466£+oo .105388£+oo .105387E+00 .105383£+oo .9421415-01 .2656375-02 .9742375-01 .8631442-01 .2052642-02 .8887305-01 .796i325-01 .1623665-02 .8169125-01 .7386135-01 .130983E-02 -7557585-01 .6887315-01 .1074482-02 .7030712-01 .6450775-01 .894180E-03 .657213E-Ol .606566E-01 .7534825-03 .6169435-01 .5723485-01 .64189754o3 .5813035-01 , .541749E-01 .5521325—03 .5495405-01 .5142285-01 .4790035-03 .5210555-01 .4893465-01 .4187525-03 .495369E-01 TABLE 20 100 THE RADIAL INTEGRALS AS A FUNCTION OF THE QUARK MASS FOR RAT-~O.33 .IOOOOOE+01 .IOOOOOE+OI .IOOOOOE+OI .ZOOOOOE+OI .ZOOOOOE+O1 .ZOOOOOE+01 .3000OOE+OI lsoooooa+01» .3000OOE+01 .4000OOE+01 .4000OOE+01 .4000OOE+01 .SOOOOOE+OI .5000OOE+01 .500000E+OI .GOOOOOE+OI .600000E+01 .600000E+01 .7000OOE+OI .7000OOE+O1 .7000OOE+01 .8000OOE+01 .BOOOOOE+OI .BOOOOOE+01 .900000E+01 .900000E+OI .900000E+OI -.177196E+OO .293672E'02 .227065E+00 .196307E+OO .490867E-02 .363868E+OO .129844E+00 .196436E+OO .389602E+OO .346331E+00 .276173E+00 .320458E+OO .213941E+OO .232601E+00 .265352E+00 .194519E+OO .200548E+00 .219430E+00 .178440E+OO .181666E+00 .192076E+OO .163560E+OO .165456E+00 .171687E+00 .150665E+OO .151856E+OO .155825E+OO .139624E+00 .140412E+OO .143066E+OO .757340£+00 .7888856+00 .569838E+00 .743145E+00 .834391£+oo .817294E+00 .567324E+00 .579536£+oo .593293£+00 .215341£+01 .415476E+oo .425050£+oo -327997E+00 .328508E+OO .319799E+oo .276078E+00 .276179E+oo .275286E+oo .240454£+oo .240468E+00 .2401655+oo .214235E+OO .2142285+oo .214083£+oo .193969E+OO .193958£+oo .193875E+oo .177742E+OO .I777325+oo .177679E+OO .104912£+oo .896735E-Ol .146809£+oo .1662085+oo .320354£+oo .341977E+00 .2718185+oo .195668E+00 .361503E+00 .247057E+00 .864422E-01 .294315E+00 .I90705E+00 .9889355-01 .225568£+oo .I7273IE+00 .1700455-01 .18707u£+oo .149125£+oo .1044935-01 .158589£+oo .130768£+oo .6789305-02 -137379E+00 .116249£+oo .4661445-02 .121075£+oo .104534E+00 .3347415-02 .108183E+00 .IOOOOOE+02 .100000E+02 .Ioooooe+oz .110000E+02 .IIOOOOE+02 .110000£+02 .Izooooe+oz .Izooooe+oz .Izooooz+oz .I3ooooe+02 .1300005+02 .13oooos+oz .Iuooooe+oz .I4000OE+02 .Iuoooos+oz .Isoooos+oz .150000E+02 .150000£+02 .16ooooe+02 .16oooos+02 .1600005+02 .Iyooooa+oz .17oooos+oz .Iyooooa+oz .180000E+02 .180000£+02 .1800005+02 .1900005+02 .Igoooos+oz .1900005+02 .2000005+02 .2oooooe+oz .2ooooos+02 101 TABLE 20(CONT'D) .I30149E+00 .130691E+00 .132536E+00 .121961E+00 .122348E+OO .123671E+OO .114828E+00 .I15111E+OO .116087E+00 .108564E+00 .108776E+00 .109511E+00 .103019E+00 .103181E+00 .103747E+00 .980772E-01 .982036E-01 .986458E‘01 .936438E-01 .937438E'01 .940948E-01 .896431E-01 .897233E-01 .900056E-01 .860135E-OI .860786E-OI .863033E-O1 .827046E°01 .827580E-OI .829470E-01 .796745E-01 .797188E-01 .798757E-01 .164400E+00 .164392E+00 .164356E+00 .153202E+00 .153196E+00 .I53170E+00 .I43647E+00 .14364ZE+00 .143623E+00 .135383E+00 .135379E+00 .135364E+00 .128153E+00 .128149E+00 .128138E+00 .121766E+00 .121763E+00 .121754E+00 .116077E+00 .116074E+00 .116068E+00 .110972E+00 .110970E+00 .110965E+00 .106363E+00 .106361E+00 .106357E+00 .102177E+00 .102176E+00 .102172E+00 .983567E-01 .983556E-01 .983524E-01 .9490955-01 .2492535-02 .9774565-01 .868738E-01 .1911735-02‘ .891294E-01 .8007065-01 .1502632-02 .818992E-01 .742407E-01 .1205615-02 .7574725-01 .691917E-01 .984323E-03 .704502E-01 .6477835-01 .8157826-03 .658423E-01 .6088875-01 .6849325-03 .6179776-01 .5743555-01 .5816312-03 .5821945-01 .5434995-01 .498870E-03 .5503145-01 .515764E-01 .4316955-03 .521734E-01 .490703E-OI .3765325'03 .495966E-Ol 102 TABLE 21 ACCUMULATION OF THE RADIALINTEGRALS 2 °-.1A6373£-08 2 -.I46184E-08 2 .14099IE-08 3 -.259785£—05 3 -.259450E-05 3 .250269E-05 A -.142608£-03 A -.I42413E-03 A .l37428E-03 5 -.203201E-02 5 -.202888£-02 5 .1960135-02 6 -.124057E-01 6 -.123856E-01 6 .1199922-01 7 -.363156E-OI 7 -.3627OIE-01 7 .3531715-01 8 -.431407E-01 8 -.431024E-OI 8 .420757E-OI 9 .I97459E-01 9 .2012356-01 9 .-.217585E-01 10 .126663E+00 10 .12835Ae+oo 10 -.13A085£+oo 0. O. O. -.123103E-08 -.120817E-08 .111657E-08 -.219372E-05 -.215313E-05 .199048E-05 ~.123063E-03 -.120818E-03 .111824E-03 -.186883E-02 -.183602E-02 .170479E-02 -.132231E-01 -.130097E-01 .121578E-O1 -.531043E-01 “.523713E-01 .494519E-01 -.133232E+00 -.131822E+00 .126193E+00 -.226364E+00 -.224777E+00 .218342E+00 -.288749E+00 -.287584E+00 .282634E+00 O. 0. .695466E-05 --4798595'05 -.444891E-05 .449301E-03 -.311564E-03 -.289903E-03 .429800E-02 -.305538E-02 -.289486E-02 .182293E-01 -.138511E-01 -.138052E-01 .428070E-01 -.371321E-01 -.413986E-01 .593118E-01 -.624716E-01 -.865457E-01 .594584E-01 -.663341E-01 -.l34284E+00 .764343E-01 -.456313E'01 -.166148E+00 .119040E+00 -.238495E-01 -.179224E+00 11 11 11 12 12 12 13 13 13 14 1A 14 15 15 15 16 16 16 I7 17 I7 18 18 18 19 19 19 103 TABLE 21(CONT'D) .199517E+00 .202785E+00 -.213966E+00 .223716E+00 .227787E+00 -.2A1834£+oo .22779IE+00 .232051E+00 -.2A6784£+oo .228137£+oo .232418£+oo -.2A7229£+oo .228151E+00 .23243A£+oo -.2A7248£+oo .228151£+oo .232434E+00 -.247249E+00 .22815IE+00 .23243A£+oo -.247249E+00 .228151E+oo .232434E+00 -.2A7249£+oo .228151£+oo' .232434E+00 -.2A7249£+oo -.312611E+00 -.311863E+00 .308422E+00 -.317744E+00 -.317149E+00 .314268E+00 -.318349E+00 -.317781E+00 .314997E+00 -.318387E+00 -.317821E+00 .315046E+00 -.318388E+00 -.317823E+00 .315047E+00 -.318388E+00 -.317823E+00 .315047E+00 -.318388E+00 -.317823E+00 .315047E+00 -.318388E+00 -.317823E+00 .315047E+00 -.318388E+00 -.317823E+00 .315047E+00 .15177BE+00 -.I42276E-01 -.182442£+OQ .162939E+00 -.120792£-01 -.182905E+00 .164786E+00 -.118287E-01 -.182943E+00 .164936E+oo -.118137E-01 -.182945E+00 .164942E+00 -.118132£-01 -.182945E+00 .I64943E+00 -.118132E-01 -.182945£+00 .1649435+00 -.118132E-OI -.182945E+00 .164943E+00 -.118132E-01 -.182945£+00 .164943E+00 -.118132£-01 -.1829455+00 20 20 20 21 21 21 22 22 22 23 23 23 24 24 24 25 25 25 26 26 26 27 27 27 28 28 28 29 29 29 104 TABLE 21(CONT'D) .228151E+00 .232434E+00 -.247249E+00 .228151E+00 .232434E+00 -.247249E+00 .228151E+00 .232434E+00 -.247249E+00 .228151E+OO .232434E+00 -.247249E+00 .228151E+00 .232434E+00 -.247249E+00 .228151E+00 .232434E+00 ~.247249E+00 .228151E+00 .232434E+00 -.247249E+00 .228151E+00 .232434E+00 -.247249E+00 .228151E+00 .232434E+00 -.247249E+00 .228151E+OO .232434E+00 -.247249E+00 -.3I8388E+00 -.317823E+00 .315047E+00 —.318388£+00 -.317823E+00 .315047E+00 -.318388E+oo -.317823E+00 .315047E+00 -.318388E+00 -.317823E+00 .315047E+00 -.318388E+00 -.317823E+00 .315047E+00 -.318388£+oo -.317823E+00 .315047E+00 -.318388E+00 -.317823E+00 .315047E+00 -.318388£+00 -.317823E+00 .315047E+00 -.318388E+00 -.317823E+00 .315047E+00 -.318388E+oo -.317823E+00 .315047E+00 .164943E+00 -.118132E-01 -.182945E+oo .164943E+00 -.118132E-ol -.182945E+00 .164943E+00 -.118132E-o1 -.182945E+00 .164943E+00 -.118132E-o1 -.1829A5E+oo .164943E+00 -.118132E-o1 -.1829A5E+oo .16A9A3E+oo -.118132E-o1 -.182945E+00 .I64943E+00 -.118132E-o1 -.182945E+oo .I64943E+00 -.118132E-o1 -.182945E+00 .164943E+00 -.118132E-o1 -.182945E+00 .164943E+00 -.118132E-o1 -.182945E+00 30 30' 30 31 31 31 32 32 105 TABLE 21(CONT'D) .228151E+oo -.318388E+00 .164943E+oo .23zA3AE+oo -.317823E+00 -.118132E-o1 -.2A72A9E+oo .315OA7E+oo -.1829A5E+oo .228151E+oo -.318388E+oo .164943E+oo .232434E+00 -.317823E+oo -.118132E-o1 -.247249E+00 .315047E+00 -.182945E+00 .228151E+oo -.318388E+oo .164943E+00 .232A3AE+oo -.317823E+oo -.118132E-o1 -.247249E+00 .315047E+00 -.132945E+00 106 from the eighth to the fifteenth terms. This same region is where the major contributions for the other cases also occur (e.g. RAT . -0.16, case 1 or case 2). Tables 22 and 23 give the results of the 2Y and 2 gluon decay widths for some charmonium states. The results were computed by expressions similar to those used in the leptonic decay widths reported in Appendix 11.1. The expressions used in these calculations can be found in [46]. The computations of the radiative decay widths reported in [30] make use of a substitution that is common in discussions of atomic spectra, namely the replacement of by u . u(xi- Af). In [30] 1.1 - Af is replaced by mi-mf. With this substitution the expression for the radiative decay width is . 2 2 2 _ 2 2 3 2 r(m1+ nf+ v) <2/3) eq am, inf/Am (of In1 1) WM! P (A) where AA is 42-11, i.e. the difference in the computed masses and R is the radial integral tabulated in tables 17 to 20, and P is the polarization sum factor. These calculations were done using the subroutine Cheat listed in Appendix III, along with the rest of the computer program which is also included in Appendix III. The results of these calculations are tabulated in tables 25, 27, and 29, while tables 26, 28, and 30 give the ratio of the calculations shown in figures 13 to 21 to those calculated by (A). TABLE 24 gives the key to the meaning of the various columns in tables 25 to 30. The quantity listed in the first column of tables 25 to 30 is not the quark mass but the quark mass plus 1.0. It is clear from these tables that the two ways of doing the calculation give quite different results. 107 In order to determine the quark mass in Gev, one needs to know a certain conversion factor. This factor as stated above changes with the quark mass. For the dimensionless quark mass from 3 to 7, one can see how this factor varies with the quark mass from Figures 9 to 12. The average value for this factor is between 0.40 and 0.50, depending upon RAT and m. For RAT - -O.16 we have for m - 4.0, 5.0, 6.0, and 20.0, the values 0.32510, 0.38849, 0.42379, and 0.78668 respectively. A simple calculation shows that the "predicted" quark mass from the model falls in the range of 1.5 to 2.0 Gev, which is what almost all charmonium model calculations give. MASS 3.0 NO‘U‘ COO NO‘U‘J-‘UJ 00000 3.0 4.0 5.0 6.0 7.0 TABLE 22 1 (150) 3.714 5.840 8.785 12.615 17.381 4.314 6.928' 10.610 15.482 21.622 4.758 7.756 12.028 17.749 25.025 2(150) 5.504 7. .226 15. 20. RATI-O. .698 .713 .940 19. 26. RATI-O. .625 .121 16. .675 .904 11 13 11 22 30 914 481 717 275 455 306 33 100 108 11 21 15 29. .891 94. 54 36 121 TWO-GAMMA DECAY WIDTHS FOR SOME CHARMONIUM STATES(KEV) RAT--O.16 I (3P0) 5. .229 .596 38. 65. 476 723 119 .076 -033 803 403 ~375 18. 255 .975 69. .212 384 1(392) 3.885 5-539 11-393 19-374 31.600 6.006 9.660 16.659 28.476 46.923 8.037 12.304 21.422 36.787 61.099 109 TABLE 23 TWO-GLUON DECAY WIDTHS FOR SOME CHARMONIUM STATES(MEV) MASS 1(150) 2(150) 1(3P0) 1(3P2) RAT--0.16 3.0 2.543 3.767 7.497 2.659 4.0 3.998 5.418 15.374 4.476 5.0 6.013 7.684 29.566 7.799 6.0 8.636 10.598 53.014 13.262 7.0 11.898 14.181 89.152 21.631 RAT--0.275 3.0 2.953 4.585 9.687 4.111 4.0 4.743 6.649 20.581 6.613 5.0 7.263 9.543 40.802 11.404 6.0 10.598 13.318 75.149 19.493 7.0 14.801 18.008 129.243 32.120 ‘ RAT--0.33 3.0 3.257 5.219 11.466 5.501 4.0 5.309 7.613 24.992 8.559 5.0 8.233 11.021 50.621 14.664 6.0 12.150 15.522 94.991 25.182 7.0 17.130 21.155 165.947 41.824 OBSERVED VALUE OF THE DECAY RATE 12.4. < 8.0 16.3 1.8 THE ERRORS ARE ABOUT 25 PERCENT IN THE EXPERIMENTIAL VALUES 110 TABLE 24 KEY TO TABLES 25 TO 30 Tables 25, 27, and 29 give the following electric dipole transition widths as calculated by the subroutine "cheat." The meaning of the columns is: column 1 The value of the quark mass in dimensionless units + 1.0 column 2 2351 + 13Po + y (RXSPZ in cheat) column 3 2351 +~13P1 + y (RXSPO in cheat) column 4 2351-+ 13?; + y (RXSPT in cheat) column 5 13Po-+ 1351 + y (RXPZS in cheat) co1umn 6 13P1 + 13S1 + Y (RXPOS in cheat) column 7 13P2-+ 1351 + y (RXPTS in cheat) In tables 26, 28, and 30, the columns 2 through 7 give the ratios of the decay widths calculated in the main body of the program to those given by column 1 The column 2 The column 3 The column 4 The column 5 The column 6 The column 7 The using value ratio ratio ratio ratio ratio ratio where RSPZ,...,RPTS are processes as determined the subroutine "cheat." of the quark mass in dimensionless units + 1.0 RSPZ/RXSPO RSPO/RXSPO RSPT/RXSPT RPZS/RXPZS RPOS/RXPOS RPTS/RXPTS the transition widths for the corresponding decay in Orthog and as shown in figures 13 to 21. 111 no-mo004o.. co-uonnevo. vo-uvnno0u. vo-uouucso. eo-mnmvvuu. vo-m.nso0v. so-moosacu. vo-mmwoovc. vo-musono~. co-moouoov. vo-uvmonon. vo-m40summ. eo-mn0vm.o. vo-moooo«o. vo-usossen. so-m.nnsOA. so-mnuomon. vo-moooo0o. eo-umom.nv. eo-mm.o.eo. no-uanomon. mo-u.soe... so-uosmumm. no-mosomn.. so-momoeno. no-mo.ooo.. vo-mm0mons. no-mno.o_u. so-u..v0no. no-usu.oon. no-uoenmo.. no-mocsmcv. mo-uso.an.. no-m.cosos. no-m«vmvn.. no-uoo.ese. no-mc.n«ou. «o-oooeneo. no-uncvooo. oo+mmon.«n. no-uvvooon. no+uvuu.o.. op.01 u h. moinmvmoa. «Ouwnvnoop. v01uNOho¢w. mcuwwwuvwh. motwmhomoh. .OumvmnOvp. v01W¢woouN. moiwflmhmua. motwmuwnoh. «Onmoohnmm. vOuwmmOOvv. moiwhmhmum. mOIuOvuhmn. 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One such simplification is to replace the interaction term by some effective potential form to describe the interaction between the two particles. A review of effective potential methods and their application to the hydrogen atom is described in a review article by Crotch and Yennie [28]. The effective potential chosen in this work is a linear combination of two harmonic oscillator potentials. Using this potential between two spin 1/2 particles, the spectrum of this two particle system is determined. By using the coupling constants in the potential, and by shifting the overall zero point of the potential, a fit is attempted for the c4: charmonium system. In this fit the 3? states are lower than the observed states. The 1180 and 21So states come close to the experimental values. Using the wavefunctions determined by solving, numerically, the radial Bethe-Salpeter equations, radiative decay widths are determined for all those transitions observed experimentally. The computed decay widths are within a factor of three to five of the experimentally determined decay widths. The main defects of this model are the following. (i) It predicts the wrong ordering of the 331 and 3D1 states. (ii) The 3P0, 3P1 and 3P2 splittings are incorrect, as well as the 351 and 3PJ splittings. (iii) The systematic behavior of the computed and observed ¢’(3685) + 3P + Y decay widths are inconsistent. Experiments indicate J that these decay widths are approximately the same for all three 3PJ states, but the model here gives results that increase with J. The only 131 electric dipole decay widths in agreement with experiment are the 4’(3685) + 380 +'Y and the 320 + 0(3095) + y, the best fit being for RAT - -0.16 and m E 5.50. (iv) The magnetic dipole decay widths, when compared to the experimental values, are too low by a factor of three to five. (v) The predicted widths of the 1180, 2180, l3po,-and 13P2 based upon results given in [46] are too big. 0n the more positive side, one of the most interesting results is the ratio of the radiative decay widths computed by and (£1311) given in tables 26, 28, and 30. These ratios show the method used in some potential model charmonium calculations, e.g. [30], to be suspect. The arguments used in support of this substitution of for (£1611) are qualitative and not based upon actual calculation. As far as I know this work is the first one to attempt to do both calculations for the same potential function and compare the results. The substitution when used in this calculation changed the radiative decay widths appreciably. An example of this variation can be seen by looking at TABLE 26 for m95.0. In this case this substitution produced lower rates for the decays into the triplet P states from the ¢’(3685) level. This substitution also produced lower widths for the 3?; +-1K3095) + Y decays. When this substitution was made in the 3P0 + 0(3095) + Y and 3P1 + 4(3095) + Y decay widths calculations, the widths were increased by a factor of 2 to 4. This suggests that this substitution yields very unreliable results and any agreement of these calculations with experimental data is strictly coincidental. In terms of comparison of the calculations of others with mine on the charmonium radiative decay width, the appropriate comparisons are not with my correct calculations displayed in figures 13 to 24 but 132 rather with the Cheat values listed in tables 25, 27, and 29. The reason for this is that almost all calculations done to date make the replacement by as is done in the Cheat calculations. As can be seen from tables 26, 28, and 30, there can be, and usually are, appreciable differences resulting in these two methods of determining the radiative decay widths. Hence the validity of these published calculations is very questionable. A comparison of three different radiative decay width calculations with experimental results is shown in figure 29. Another item peculiar to this work is to show how the decay widths vary as a function of the quark masses. For the electric dipole cases the decay widths are rather flat, as a function of the quark mass, for large values of the quark mass. This would indicate that when information becomes available for the higher mass analogues of the charmonium system, i.e. b-b and t-E, the radiative decay widths could be compatible with experiments over a larger range in the quark masses. In the low quark mass region where the c-E system is fit, a change in quark mass of l to 2 units produces a large change in some of the decay widths (factors of 3 or more in some cases). This variation of the widths with quark mass can be used to place upper and lower limits to the quark masses. If the quark mass is too low, the radiative decay widths become much too large in many cases. If the quark mass is too large, the radiative decay widths will be smaller than the experimental decay widths. The most sensitive cases are the 3PJ decays into the 4(3095) state. To be able to make use of the calculations in this way, however, the experimental uncertainties in the 3PJ decay widths will have to be improved considerably. 133 co_us_:o~ou _Po:cou n e .upzmog peacoEwcoqxo u > .copuo_:u_ou «coco u u .coapspaupsu acacia u o essencesee c. scoe_= scone oseoseoss .sw oc=s_a Nam mam omm H ome cm “mp 11 6:0»0. m.m~ m.~ H m.m_ mm.m eo.¢m “an mam ooh Pom cup 1 N auo ll o-- omm pep mm H mm owe cop 1111 escape umm m— mm. H emu mw omH m~.— 134 The inclusion of a coulomb type potential form might eliminate some or all of these problems. It is uncertain how the inclusion of an u/r potential would affect the radiative decay widths. The a/r potential would affect the wavefunctions most near the origin where the energy levels would be affected, but in this region the decay matrix element contribution is small. Therefore the inclusion of an o/r potential might remedy some of these defects. APPENDICES Appendix I.l Kinematic Constraints on the Singlet and Triplet Wavefunctions The number of known exact solutions of the Bethe-Salpeter equation is very very limited. In view of this, almost all calculations involving the Bethe-Salpeter equation start out with some-truncated or approximate version of the complete equation. Two common assumptions made are: (i) In a first approximation, retardation effects are ignored initially, and if they are required, then they are computed as perturbation effects. (ii) One demands that the equation be correct up to order (v/c)2 and be linear in the potential function describing the interaction of the two particles. This combination of restrictions allows one to write the two-particle Bethe-Salpeter equation in the form: (8(0) + p+vp+>o = Koo (1) where p+ - 11(26- +H)/8w2, p_ - -H(20.1 - H)/8002 (2) H = pr+ - 2wp_, p0 = (4m2 - H2)/4w2 where p+, p_, and p0 are projection operators onto the eigenvalues 2m, -2m, and O of H(p). The application of po and p_ to equation (1) will yield the following equations: P011103) + p+vp+I¢ = K0904 = 0 135 136 So PDH(p)¢ + pop+vp+¢ = 0 + H(p)po¢ = 0 + H2(p)¢/4w2 = ¢ p_[H(p) + p+vp+l¢ = Kop_¢ . o if K0 > o D_H(p)¢ - H(p)p_¢ = 0 + p_¢ = 0 + (82(01/84210 - (zen/84210 = 0 So (H2(p)/4mz)¢ = (H/Zw)o - o + H4 = Zoo (3) The consequences of equation (3) upon the wavefunctions can be found by expanding 4 in terms of various tensor forms, namely 4(0) - 08(0) + YuVu(p)C + 8-i(p)0/2 + E-i’(p)C/2 + YuAu(P)YSC + Y5CP(P) (4) The matrix C in this decomposition represents the operation of charge conjugation. In terms of the two sided matrix notation, equation (3) with ¢(p) as given by equation (4) is: H(p)0 - (3-3 + 8m)[CS + Yuvuc + Eofc +3-T’ + YuAuYSC + YSCP] + [CS + yuvuc + E-TC + E-T’c + YuAuYSC + ySCP](-3T-§ + BTm) 137 2411(1))- The expression (6-; + Bm)¢(p) + ¢(p)(-dT-S + BTm) is reduced to a simpler form as follows: (i) M0ve the matrix C in the second term past -d¢ and ET. This converts ~6T into 8 and BT into -8. With this the second term has the form 0(p)(;-3 - Bm)C where W(p) - ¢(p)C-1. (ii) we now combine the same terms in 0(p), viz., [(d-E+Bm);-7(;¥6-Bm)lc and make use of the following identities: (E-K)(8-B) + (Ema-K) - 2(4-8) (8-K)(3-§) + (GEMS-I) - -275(X-B) (Hui-i) + (i-BMM) = 41151408) 1118(4'6) - m(6°6)8 = 417'; In 8(3-6)+(6'6)8 - o The reduction is now rather easily done and the terms with the same Y matrix structure are then collected together. The results are: 138 mm = [(3.3 + 8m)SC + (an? - 8m)SC1 + [ME-i160 + iio<3ximc -in(3-i)c + i?-(Ex7)y5c - 18(S-i)c - im(3-i)c] +[-ys(3-i)c + iE-(Exf)c - im(?-T)ysc - (3-T)750 -iEo(3xi)c + im(?-T)Tsc] ...-(5.ch + id-(SXT’N + uni-inc + (35100 -i3-(5xi’)c + im(?-T’)C] +[-iysB(3-K)c +iy-(3xl)c + im(3-K)c + 1158(3-X)c -i?-(Kx3)c - im(3-K)c] +1-i(ToE)Anysc + mAbYSC + i(§-$)Any5c + mAnYSC] +[-(6-5)PC + mBYSCP ' (3‘5)PC + 18'YSPC] - 2(§-$)sc + [21?o(3x6)ysc - 21m(3-i)C] + P26411501 + [us-inc + uni-i001 + [mi-(Exbo +[2mAnY5C] + [-2(6°6)PC + ZmBYsC] . 2[5-i’c + i(mT’ + (SxK))-7c - S-Epc + 3-(38 - imi)c 139 + i(3xi)o?ysc + mP8Y5C + (mAn - E-T)Y501 - 2w[SC + 7-7 0 + ynvnc + To; c + E-T’c + T-X Y5C + AnBYsc + YsPC1. The last equality is just Ho 8 20¢ (equation 3). coefficients we see that p-T’ - wS imT’ + 1(3xX) - oi -im§ + SS - wT’ i<8xi> - AK -3 P - m T mAg - S'T 3 NP mP 3 ”An (51) (511) (5111) (51v) (61) (611) (6111) Comparing the The four equations in set (5) come from comparing the coefficients of the following Y matrices: C, 1C, 6C, and ? Y5C respectively. The set of equations in (6) likewise comes from comparing the coefficients of the following Y matrices: 6C, YSC, and BY5C respectively. In this derivation we assumed that the YuVu term has only three components, i.e. YuVu a 1-9 + 8V4 with Vn = 0. 140 From the set of equations (5) one can see that the scalar S, the axial vector A, and the "pseudotensor three vector" T’ can all be expressed as a function of the vector term V, viz., s - -(i/n>3-1 (71> X - (i/o); x 17 (711) ia-umom4+56$» nun Equation (71) comes from (51) by substituting in (51) for E-T’ the relation found by dotting the vector 5 into (511). Equation (7111) comes from (511) by substituting into (511) for A from (51v). The set of equations (6) can also be expressed in terms of a single potential function. In this case we choose P, and we are led to the following relations: 1 . {-5/4) P (81) A..- (m/o) P (811) Equation (81) is just the same as (61). Equation (811) comes from (611) using (61) to eliminate the T dependence in favor of the P dependence. One can therefore write the wavefunction for the Bethe-Salpeter equation strictly in terms of the four functions P(p) and $(p): 19 = <1/(2/5111-3-3/s + mBYs/w + Yslpc _ (91) + + 34 - (l/(2/2111-13-i/m + i-i -¢—3 F’H. K,PV-U,PW,PX=U,PY,PT-U,PU=U V PP.PQ Z-A*B*C ID,A-1.0+I*G(J,4)*G(J.PQ)/V+M*G(J,4)/V 9 DE U.NU B C, ,P 142 143 10,6-1.0+I*G(J,4)*G(J.PQ)/V-M*G(J,4)/v 10,3.1455(J)*G(J.4)*G(J.PQ)/V+M*G(J,4)*GS(J)/V+65(J) ID,TRICK,J IDOG(JDPQ)*G(Jth)--G(Jih)*c(JDPQ) ID.G(J.PQ)*G(J.PQ1-PQDPQ ID,G(J.PP)*G(J,4)--G(J.4)*G(J.PP) 10,G(J.PQ)*G(J,PP)-PQDPP+65(J)*G(J,4)*G(J.PV) ID,G(J,PP)*G(J,PQ)-PQDPP-GS(J)*G(J,4)*G(J,PV) ID,FUNCT.PV(HU+)-EPF(MU.PQ,PP,4) ID,PPDPP-W**2-M**2 ID.PQDPq-V*V-H*H ID,PP(4)-0.00 *END Z-A*B*C ID,A-1.0+I*G(J,4)*G(J,PP)/W+G(J,4)*M/W lD,C-1.0+I*G(J.4)*G(J,PP)/W-G(J,4)*H/W 10 .B- 1 *05 (J) *0 (J , 4) *0 (J , PQ) /V+M*G (J , 4) 1405 (J) /V+GS (J) ID.TR|CK.J 10,5(J,PQ)*G(J,h)"G(J,4)*G(J,PQ) ID.G(J.PQ)*G(J.PQ)-PQDPQ ID,G(J,PP)*G(J,4)--G(J,4)*G(J,PP) ID,G(J,PQ)*G(J,PP)-PQDPP+GS(J)*G(J,4)*G(J,PV) ID.G(J.PP)*G(J,PQ)-PQDPP-GS(J)*G(J.4)*G(J,PV) ID,FUNCT,PV(HU+)-EPF(MU.PQ,PP,4) lD.PPDPP-W**2-M**2 ID,PQDPq-V*V-M*M ID.PQ(4)-0 ID,PP(4)-0.00 *END IN THE NEXT TWO PROGRAMS, AN OVERALL MINUS SIGN HAS BEEN OMITTED. THE OUTPUT OF THESE PARTS GENERATES THE C TERM FOR THE BETHE-SALPETER EQUATION IN THE WORK OF CUNG ET.AL. CITED IN THE BIBLIOGRAPHY. 2-A+C+F ID.A-G(J,1)*B*G(J.1) [Dac-G(J92)*B*G(JDZ) 144 ID.F-G(J.3)*B*G(J.3) ID,B-|*GS(J)*G(J,4)*G(J.PQ)/V+H*G(J.4)*65(J)/V+G5(J) ID.TRICK.J ID.G(J.PQ)*G(J.4)--G(J.4)*G(J.PQ) lD,PQ(4)-0.00 *END Z-AXBXC ID,A-1.0+1*G(J.4)*G(J,PP)/W+G(J,4)*M/W ID,B--3.0*GS(J)-3.0*H*05(J)*G(J,4)/V-I*05(J)*G(J.4)*G(J.PQ)/V 10.0-1.0+I*G(J.4)*G(J,PP)/W-G(J,4)*M/W ID.TRICK.J ID.G(J.PQ)*G(J.4)--G(J.4)*G(J.PQ) lD,G(J,PP)*G(J,4)--G(J,4)*G(J,PP) 10.6(J.PQ)*G(J.PP)-PPDPQ+I*G§(J)*G(J,4)*G(J.PX) 10,0(J,PP)*G(J,PQ)-PPDPQ-IXGS(J)*G(J,4)*G(J.Px) ID,FUNCT,PX(MU+)-EPF(HU,PQ,PP,4) 1D,DDT PR,PX(MU+)-EPF(MU.PQ,PP,4) ID.G(J.PP)*G(J.PP)-W**2-M**2 lD,PPDPP-W*W-M*M ID,PQ(4)-0.oo 10.9?(4)-o.oo *END z-AXBXC ID.A-l*G(J,4)*G(J,PK) ID.C-I*G(J,4)*G(J.PK) ID,B-I*65(J)*G(J.4)*G(J.PQ)/V+M*G(J,4)*05(J)/V+GS(J) ID.TRICK.J lD.G(J,PQ)*G(J,4)--G(J,4)*G(J.PQ) lD,G(J,PK)*G(J.4)--G(J,4)*G(J.PK) In.c(J.PQ)*G(J.PQ)-PQDPQ ID,PQDPO-V**2-M**2 ID.PKDPK-1.0 lD.RK(4)-0.00 ID.PQ(4)-o.oo *END z-A*B*c ID.A-1.0+l*G(J.4)*G(J,PP)/W+G(J,4)*M/W lD,C-1.0+I*G(J,4)*G(J,PP)/W-G(J,4)*M/W 10.8-GS(J)+M*GS(J)*G(J.4)/V-I*GS(J)*G(J.4)*G(J.PQ)/V 145 +2.0*|*GS(J)*G(J,4)*G(J,PK)*PQDPK/V ID.TR1CK.J ID,G(J,PP)*G(J.4)--G(J,4)*G(J.PP) 10.6(J.PK)*G(J,4)--G(J,4)*G(J,PK) 10.6(J.PQ)*G(J.4)--G(J.4)*G(J.PQ) ID.G(J.PQ)*G(J.PQ)-PQDPQ 10,0(J,PQ)*G(J,PP)-PQDPP+I*GS(J)*G(J,4)*G(J,PV) 10,0(J.PP)*G(J,PQ)-PQDPP-I*G5(J)*G(J,4)*G(J,PV) ID,FUNCT.PV(MU+)-EPF(MU.PQ,PP,4) 1D.DDT PR,PV(MU+)-EPF(MU,PQ,PP,4) 10,0(J.PP)*G(J,PK)-PPDPK+85(J)*G(J,4)*G(J.PU) ID,G(J,PK)*G(J.PP)-PPDPK-GS(J)*G(J,4)*G(J.PU) lD,FUNCT,PU(HU+)-EPF(MU.PP,PK,4) ID,PQDPq-V**2-M**2 ID,PPDPP-W**2-M**2 ID.PKDPK-1.0 ID.PP(4)-o.oo ID,PQ(4)-0.00 ID,PK(4)-0.00 *END Appendix I.3 (1) Derivation g: the 1JJ Equation for_§ Coupled Vector-Scalar Interaction The basic two-body 43 Bethe-Salpeter equation with both a vector and a scalar harmonic oscillator potential is + ’ I 2 2 ’ + +’ - +’ 2(w-u)f(p)+(v/(ww))1(2ww -m ) + A(m +4141 - P'P )If(p ) - 0 (1) Here v = -(k/2)V§, 5(3)(3 - 6’) and A is one half of the ratio of the scalar to vector coupling constant k. The derivation will consist of rewriting VS, as Vi + (1/p2)V§ and performing the operations of the radial and angular parts of the Laplacian in this equation separately and then combining the results at the end. To simplify things, rewrite (1) as 2f<3) + )[<2+A>-nm'- max-A) - A Mun?) - o This equation’s radial part can be written as: {-2/k)12(w-u)]f + (2+A)V§f - (l-A)(m2/w)V§(f/w) - (AP/owiof/o (2) At this point we shall make use of the following identities: v§(f/o) - (1/o)v§f - (2p/m3)df/dp - (3m2/w5)f (3) v§(pf/o) - (p/owgf + (2n2/o3)df/dp + [(2m”/p) - (mzp )lf/m5 (4) vficpi/Mm») =- vgf - mV§(f/w) 146 147 = vgf - (n/o)v§f 4 (zap/o3)df/dp + (3m3/w5)f (5) Inserting (3), (4), and (5) into (2) we have: (-2/k)[2(w-u)1f + (2+A)v§f - (1-A)1

[21f + v§f1<2+A> - <1-A)(m2/42) - - A E-S’1f<$’> = [(2+A)/p2]vgf - <1-A>v§f - (Apz/(n2p2))pivgsif (7) At this point we make use of the following identity: 148 IVE. 011 - -201 + 2vi (a) With identity (8) the expression (7) becomes: (1/(42p211[<2+A>wZ—m2<1-A>1v3f — A)0110,Vi-20,+ 2v11f - (l/(m2p2))[(2+A)m2 - m2(1-A) — Ap21v§f + (2A/w2)f + o (9) In (9) we have made use of the fact that piv 0. 18 Now combining (9) with (6) we get the desired final expression for the 1JJ equation: (-2/k)[2(w-u)]f + [Vif + (1/p2)v§f]((202 - am2)/m2) + df/dp(2m2p/w“)a + [(3m7/ws)c + (Apz/w“)]f - 0 (10) The quantity a appearing in expression (6) and equation (9) is just 1-2Ao ~ 149 (ii) Derivation of 3JJ Equation for a Coupled Vector-Scalar Interaction The basic two-body 43 Bethe-Salpeter equation with both a vector and a scalar harmonic oscillator potential is: + 2f(3) + {[1 + Ai(mz + 44' - 3-8'11 - (83’-8’S)1-f(5’> - o (1) The quantity A is one half of the scalar to vector coupling strength. Collecting terms simplifies equation (1) somewhat. When this is done, the result is: 2f<3> + (v/(ww’)){[ww’(L+A) + S-i’cl-A) + Am211 + <1-A><53’-3’3>1-¥<3'> = 0. (2) We now concentrate on the term involving the v in equation (2). The quantity v for a harmonic oscillator potential takes the form (-k/2)5(3)(§-3’)V;,. For ease of handling we shall split the Laplacian up into its radial and angular parts and perform the reductions separately. We begin with the radial component reduction. (-2/k)[2[(1/w>vgf - (2p/0314f/dp - (3m2/ws)f] - -(2/k)12(m-u)]f + V§f[(l+A) + (l-A)(pzlwz) + A(m2/wz)] + df/dp[(1-A)<2n2p/o“) — A<2n2p/o“)1 + [(l-A)((2m“/w6) - (mzpz/w5)) - A(3m“/w6)1f <4) The coefficients for Vif, df/dp, and f can be further simplified as follows: " vgf: 1+(p2/m2)-(2m2-m2)/m2; A(1-p2/n2 + mZ/m2)=A(w2-p2+m2)/w2 - 2Am2/o2 df/dp: (l-A)<2m20/w“> - A(2m2p/m“) - <1-2A)<2m2p/o“> f: <1-A)<<2m“/46>-(m202/ws>> - <3m“A/wfi> = <1-A)<3m“/06 —n2/n“) - (3n4A/06) a (3n4/o6)(1-2A) - (1-A)(n2/n“) 151 Setting a = (1-2A), expression (4) can be rewritten as: -(2/k)[2(w-u)]f + V§f((2w2-am2)/m2) + df/dp[2m2pa/w“] + [shin/06 -(1-A)(n2/n“)1f (5) To complete the derivation of the 3J equation we need to reduce J the part that comes from the angular part of the Laplacian. The expression to be reduced is, from (3), (mm/0211131 + (l-A)(pHth/phfifipjf1 + (Am2/(wzp211fif1 + (l-A)(1/m2)[61\7§pjfj - pjvgpifj] (6) The reduction of this expression is aided by the use of the following identities: (i) [V3, 81] = -2pi + 2vi (11) [via fij] . 511 ' fiifij Inserting these into (6) we get: ,(1+A)(1/p2)v§f1 + (1-A)(1/o2)pj[pjv§fi-2pjfi +2iji] +(Am2/(0202DVEE1 + (l-A)(p2/(w2p2))[(V£81+281-2V1)fijfj - 2 - (mintsj ZVj)fiifj] - (l/pZ)VEf1[(1+A)+(pZ/AZ>(1-A>+fi 152 _ 2 - + (1 A)[(1/o )( ZVifijfj + 2v 01f )1 J J - (1/02)<(202-am2)/42)V§f1 - (1-A)(2/o2)fi - 2 - + (1 A)(l/w )(zvjpifj 2vipjfj) Combining this with (5) we have the final expression for the 3JJ system. -(2/k)[2(w~u)]fi+ Vif1((2m2-cm2)/wz) + dfildp(2m2pc/m“) + [an-“on.-6 - (1-A.)n2/o‘*]fi -(2(1-A)/o2)f1 + (1/p2)((2nZ-sn2)/02)v§f1 + (1-A)(1/m2)(2Vj131fJ-2V18j fj)-0 and where vjpifj- W3 3 -(L- §)f - (1/2)[j(j+1) - 1(1+1) -s(s+1)]f1 (iiia) The Radial Equation for the 3(J11)J Bethe-Salpeter Equation with Vector Harmonic Potential The starting equation is: 2(w-u)?))v§fJ (A) <-v/ - <3m3/w5) + (3m3/m5)(p2i(w) + (3m“/w5) - (m2/w“)] . fiifijfj[5ij(3m“/m5 - mz/m“) - (3m3/m5) + (e-m)3m3/m6 + (3m“/w5) - (m2/m“)l - fiifijijiJOmes - mZ/e“) - (3m3/w5) + (3m3/m5) - (saw/m6) + (3m“/w6) - (m2/w“)] ' fiifijijijOmu/w6 - mz/m“) - mZ/m“] Now for the angular terms, we proceed in a similar manner. (1) ) - <1/p2)v§f <2) ) = (fiilw)(V§/p2)(fi1/w)f - (pZ/m2><1/p2>p1v§eif . (1/m2)§i[piv§ -251 +2v11£ 157 a (1/m2)v§f - (2/m2)f +0 (3) -)3$ =- -)mifijmpzwgfj = '(1/(w(w)))fiifijV§fj . -(1/(m(uH1n)))[V§fiifij + 45151 -2Vifij 415913”.j 2 .. .. - -(1/(w(url1n)))[V(flifijfj + 451$ij $711331?j 251ij J1 <4) -))5’3' =- -><1/p2>VEfi1fiifi = '(1/(w(w+m)))Vifiifijfj <5) > p“/pi§k<1/p2)v§ fikfijfj Pz/(w2(w+m)2)fi1fikvfi pkfijf j Pz/(“2(w+m)2)[vifi1fik Mfiifik -2Vifik ‘2fiivklfikfijfj p2/(w2(m)2)[Vifiifijfj+4§1§jfj-ZVifijfj-Zfiivkfikfijfj] pz/(m2(m)2) [V3 115 f rap 19 if j-zv its ji 3 158 - pz/(w2(m+m)2)[Vifiifijf1+4fiifijfj-2Vifijfj'0 ~4fiifijfj] - Pz/(w2(w+m)2)[Vifiifiij-Zvifijfj] <6) 2)35’ - 2p1v3pjfj - <2/m2)[V§151 -2§1 -2Vi]fijfj . 2 2 _ (2/u )[V‘fiipjfj +2131§j£j 2vipjfj] (7) -§'3 = ’(Pz/N2)fij(1/p2)Vifl1fj a - 2 2 - (l/w )[V(fiJ +25j Zijifj n— 2 2 :- (l/m )[V‘fiifijfj+2fiifijfj 2vjp1£jl Now combine terms, looking at the coefficients of Vifi, Vifi vjfiifj’ and fiifijfj' in' sz [5 < 3 ii <1/w2>1 - V26 f ((2w2-m2>/(p2e2)) (l/pz) + 61 ( 11 j J vigipjtj[-(1/(m(u+m)))-(1/(w/> +<1/w2)1 - vfipipjij-(z/(w>) +<1/(w -> +<1/m2)1 - Vfifiifijfjl-(1/(w(whm)) -Cm/(w2(w+m))) +(1/w2)] 159 vfipipjfj(1/(m2(w+m)))[-m-mi-m] ==0. Vifijfj[+(2/(w(w+m))) -<2p2/2) -<4/w2>1 - Vipjfj[+(2/(w(w+m))) -2(m-m)/[pivj+a j fiifijlfj +fi1Vij(2/(w(w+m))) " Zfi V f [(1/w2)+ (1/(m(w+m)))l +(2/m2)5 _ 2 1 J i ijfj ‘2’” )fiifijfj - ZfiiijJ[(2aH1n)/(w2(whn))l +<2/m2)csijfj -<2/m2)131t5jfJ Now we do the 5151 terms. Note that we have obtained two additional terms of this type in the V reduction. 351‘: fiifijij-(Z/wzm -4/(w(w+m)) +(4/w2) -(2/w2) +(2/w2)61j-(2/w2)] 1L1 - fiifijfjl-(A/(w(whn))l To proceed further we look at the term: 2 - 2((2urFm)/(w (w‘hn)))(fii\7jfj Vifijfj) 160 This term looks like 5 x F, i.e. an angular momentum or an t-§ type structure. To proceed further just consider the fiin 3 j -V1pjfj part. pivjfj - vipjfj . [V151 - 51j+sipj]fj - vipjfj - Vjp‘ifj - Vifijfj - aijfj + fiifijfj = I Up to this point we have been somewhat sloppy in our notation for typing purposes. The symbol V used up to this point is as defined 1 above to be given by V: - V§‘+ (l/p2)(Vi-Vi). Strictly speaking V here i should be explicitly written as (V()i, but was written as V1 for convenience. To proceed further we need to distinguish between the angular quantity (V()1 and the various components of the gradient operator V . With this in mind we proceed on with the derivation of the i ‘f-§ term. (vp1 - vi - §1<$-$>. j- H I [VJ-fij(p-V)]fiifj -[V1-fii(p'V)]fijfj -cij + pisjfj + + Vjfiifj 41stj - sjcp-3>pifj +fi1(p°V)fiij -613£j+915jfj 1(xjfiifj-xipjfj)-1[fij(p x)§i -§i( p 'x)fij]fj-51jf j +fi1fijf j -(i- -§)1j- il

/[-<1t-§>ij -6ij+fiifij]fj =[-2<2m>/(w2(umn>) (1/2)[j(j+1)-1(1+1)1+2(2w+m)/(wz(urhn)]51jfj -2<2m)/(m2)6 +2<2m)/(w2tsits ijfj ij --(2w+m)/(w2(w+1n))[3(J+1)-1(1+1)]611fj Hum)/(m2///(w2p2))[1(1+1) -j<3+1> +1<1+1>1<51jfj = {-<<2w2-m2>/ + <<2m2-m2>//> - <<2m>/(u2>> I! <1/w2)[[-wp2>/<p2)1 <1/w2p2>[)/1 (l/w2p2)[w2 -w(w-m)] = (1/w2p2)[mw] = m/(wpz) So we have for Vi<(2m7~mz)/(mzpz>>fi - <(2m>/(w2(m+mmlj/(m2p2>>j<3+1)csijfj + (m/wpzNijfj[j(j+1)-l(l+l)I With these last substitutions we have for the 3(Jil)J vector 163 potential equation -(2/k)2(m-M)fi(p) + ((2m2-m2)/w2)v§f -<(2m2—mZ)/(m2pz))j(j+1)fi +(2m2p/w“>af1/ap + £1 - (mZ/w“>1 _ 2 n. (m /w )tiifijfj + (hum/(Maura)”1311511?j +)[j(j+1) -1(1+1)1fi - o. (iiib) The Radial Equation for the 3(J11)J Bethe-Salpeter Equation with 2 Scalar Harmonic Potential The starting point for the derivation is the equation 2(m-u>f<3> +A[m2+mw’-3-$’>I -(53’-E’$) +(1/<)<253' - E$2-E'S’>1-f<3)-o <1) The reduction, as with the other cases discussed, is carried out by first considering the radial part, then the angular part, and then adding up the two contributions. The radial part reduction is aided by the use of the following identities. 164 (i) v§(f/m) = (1/u)v§f - (2p/m3)df/dp — (3m2/w5)f <11) V§(pf/w) - (p/wwfi + (2m2/m3>df/dp + f(pnzm‘vwsp) _ m21,/ms] (iii) v§ = V? - mvgcf/uo - Vii - (m/m)v§f + (2mp/m3)df/dp + (3m3/m5)f Writing v as ~(k/2)5(3)(3 - 3’)v§, with v§,f a Vif + (l/p'2)Vif and considering for the moment only the V: terms, we have from (1): O -(2/k)[2£ (2) The remaining terms sum up to zero. To see this note that (A/(w(urhn)))[2(fiifikfikfijp2>V§(p2f/(w(w+1n))) - fiipjvgozf‘nmm») - pzfiifijV§(p2f/(w(ur+m)))l is the contribution from the last three terms, and using pkfik - 1, we have the resulting zero contribution here. The terms in (2) can be reduced with the aid of (i) and (ii), giving 165 -(2/k)2(w-u)f +A[(1/w)V§i - <2p/w3)df/dp - <3m2/w5)£1 +Av§f -A(p/m)[(p/w)v§f + (2m2/m3)df/dp + f(p)(2m“/(w5p) -m2p/m5)] . -(2/k)2(w~u)f +Av§f[m2/u2 +1 -p2/w2] -2A(2m2p/m“)df/dp -2Ai(p)[3m“/m6 - mZ/(2w“)] (3) We now proceed with the angular part of the Laplacian. Av§f1 +AV2f -A(Pz/(w2p p2))fijV2 5 f -Apz/(mzpz) (pivfi pjfj- pjv( fiifj ) -+Ap“/(m2(m+m)2p2)[zpipkfi pkfijfj- piijzf- vi pifijfj] (4) The reduction of (4) is made easier by the use of the identities: 2 ._ (i) [V‘,pi] 291 +2vi (11) [V1953] a 611 - $151 2 - - (111) [v(’fiifij] 45151 +2V15j +2fiivj The first three terms in (4) yield: 166 A(m2/(w2p2))v§fi +-A(1/p2)v§fi -A(p2/(m2p2))fij[5jvg -2pj +2v ]fi 1 = Av§f1[m2/(w2p2) +(1/p2) - (pz/(m2p2))] + 2Af1/m2. The fourth and fifth terms in (4) yield: -Ap2/{tv§ a, +213i wins;j - [vi 51 +253 -2v1151£j} =-A 2 m2 2 -2V f +2v f P /( p )[ ifij j jfii j] The last terms yield the following: I = Ap“/2p2>[zfiifik2p2>lzpivi 51:1 -4§1fijfj-V§ fiifijfj -4fi1fijfj ..2 +2V1fijfj +251ijj v( pipjfj] - Ap“/(m2(m+m)2p2)[2(v§ gipjfj -2v1§jfj) -4515j£j —2v§pifij£j -4§ipjfj +2v1§jfj +2p1ijj] . Ap2/(w2(w+m)2)[-4V1fijfj -4§1§jfj +2V1pjfj +2injfj] 167 - A 2/(2w2(w+m 2 -2v f +2 v f -4* f p ) )[ ifii 3 fit 3 J pifij J] The net result of these angular reductions is vgfi[2Am2/(m2p2)1 + (2A/u2)f1 -(Ap2/(w2p2))[2Vjfiifj -2V15jfjl 2 2 2 - _ +(Ap /(w (w+m) )[2p1VJfj ZVipjfj Apipjfjl. The term Zinjfj can be rewritten as 2(Vjpj -Gij + pipj)fj, which gives v§£1<2Am2/f1 - (Apz/(mzpzntzvjtsifj -2V1fijfjl +A2m2ur+m22V f-Zf -2v f-Z f 5 < p /< ( > )I 151 j 1 193 1 2121 j] < ) Combining the radial and angular parts we obtain: (-2/k)2(w—u)fi(p) - 2(Am2/u2)v§f1 -2(2Am2p/m“)d£/dp -2A£1(p)[3m“/m5 - mZ/zm“] + (2Am2/(m2p2))v§f1 +2A(f1/m2) .. 2 _ 2 2 2 - (A/m )[zvjfijfj zvifijfj] +(Ap /(m (m+m) )[ZVinfJ 2fi -2vj§jfj -2§1pjfj] (6) The equation (6), or rather some of its angular terms, can still be simplified. 2 (1) The V‘fi term. As noted in connection with the vector potential equation ' _L 168 v22 =- -1(1+1)fj - -[j(j+1) -j(j+1) +1(1+1)]f < j J ' -j(j+1)f + [j(j+l) -1(1+1)]f j .1 (ii) (Vjpjfj - Vipjfj) terms - 2 _ 2 2 2 _ 2[A/w Ap /(w (urhn) )IIVJfiifj Vifijfj] .- 2 2- 2 2 _ 2(A/w )[((urhn) p )/(uH1n) ](ijiifj Vifijfj) =-2(A/m2)[((uflm)[(m)-(w-m)])/((ur+m)(ur+m))](Vjfiifj-vifijfj) .- 2 - 2(A/m )(2m/(m+m))[Vjfi1fj vipjfj] --2(2mA//(w (mung pifijlfj Using (f-§)ij- (j2-12-32)(1/2)6ij we have 159 (2/k)2(m~u)f1(p) - 2(Am2/m2)v§£1-2(2AmZp/m“)dfi/dp -'21‘.fi(p)[3m"/m6 - mZ/Zw'fl -(2Am2/(w2p2))1(j+1) +(2A/m2)f1 +(2mA/(m2(m))>li1fi +(2Am2/(w2p2))[1(1+1)-1(1+1)]fi +(2A(m-m)/(m2(u-Hn)))fiipjfj. «moo-ml(«120mmm?1 + 2(2mA/(w2(urm>))f1(p) Combining the [j(j+l) -l(l+l)] terms we have for its coefficient 2mA/(wp2). Finally we have (2/k)2(m-u)£i - (2Am2/m2)v§£1-2(2Am2p/m“)dfi/dp -2A£1(p)[3m“/m6 - m2/2w“] - (2Am2/cw2p2>)i +[i(i+1)-1(1+1)1(2mA/cwp2» + (ZACw-m)/(w2(w+1n)))fiifijf J (111C) Combining the 3(J31)J Scalar and Vector Potential Forms We now combine the terms arising from the potential in the two previous equations, i.e. the Bethe-Salpeter equation with a pure vector potential and the same equation with a pure scalar potential. we proceed directly by writing out the terms indicated. 170 amazon-mi - [<2w2-m2)/w2 + 2Am2/wzlvrf1(P) + [(2w2-m2)/m2 - 2Am2/13(3+1>£1

+ [2m2p/w" - 2(2Am2/w2)]df1(p)/dp + i1(p)[3m‘*/w6 - mZ/m“ -2A(3m‘*/w5 - 113/m“)! + Ira/(mp2) + 2mA/cwp2)1[iI - JET—Di» /(2j+1) “x — <1/<2<2i+1>>> oz] - (1/2>I With this the final form is the following: 171 (2/k>2(w~u)fi

=((sz-am2)/w2)V§f1(p> + ((2w2-0m2)/w2)j(j+1)fi(p) + (2m2pa/m“)df1(p)/dp + [3m"a/w6 - mZB/m“]f1(p) + (mY/(wp2))[J(i+l)-1(l+1)lf1(p) +[2m5/(w2wm) +2A/(m1[<1/2)I -2f1 + [vii-1 - <1/p2)i<3+1>1<<2u2—am2)/w2> + (Zmzpa/w“)df1/dp + [3m“a/w5 - (1--A)(m2/m'*)]f1 - o (2) (-2/k)2f1 + [Vifi - <1/p2)i1((2w2-am2>/w2) + (2m2pa/u“)df1/dp + [3m‘*a/w5 - mZB/w“ + (mY/(wp2))(j(j+l) - 1(1+1))l + [2m6/(m2(w+m)) + 2A/(w(uH-m))][(1/2)Gij - (W/(znlnox - (l/(2(23+1)))ozlijfj + (m2/w“>[/a-1/(<2yhi2>> yields the relation A + B(2y+fiz) = 1 so that y(A+ZB) - 0, mzA + E1213 - 1 From the first of these relations we see that A = -ZB and from the second -2sz + 523 . 1, so that B - -<1/(2m2-fiz)) = -(1/am2). and A ' +(2/am2) We can now write 175 I<1/(<2y+n2>))dy = f[(2/am2)<1/(2y+n2) - (l/am2)(1/(y+m2))]dy - (2/au2)[<1/2>1n<2y+nz>1 -<1/am2)1n - (llamz)[1n((2y+nZ)/(y+m2))l Combining things we see that a(p) a I<2w2-auzl/m21“lexptln + ln((2P2+fiz)/(P2+m2))] = [(Zuz-am2)/w2]'lexp[1n(p2(2w2-am2)/w2)] .p2 (b) The Sturm-Liouville Form and the Lagrangian The Sturm-Liouville forms of the 1J J: 3JJ. and 3(Jil)J equations are: -(2p2/k)2(w-u)f + d/dptp2((sz-am2>/u2)di/dp1 + [3m“ap2/w6 + Ap“/w“1f - <<2w2-am2>/w2>i2ti + d/dp[pz((ZwZ-am2)/w2)dfi/dp] -[(2w2-am2)/w2]j(j+l)f1 + [3n“op2/u6 - nzpze/u“ 176 +(tnY/m)(_1(j+1)'-l(1+1))]fi + [(Zmpzé/(w2(ur+m))) + (ZApz/(w(w+m)))][(1/2)5ij- (MT—Ti 3+1 /<2i+1>)[<>(2m~E)f 3(J11)J: In this case one can also see by counting powers of p that the leading behavior is given by the same factor as in the 1JJ and the 3.]J cases. In these three cases it is appropriate to replace m by p for large p and for finite m << p. In each case then the equation appropriate to the limit p + "113 2p2 dzf/dpz + 4p df/dp - (sz/k)(2p-E)f . 0 or dzf/dp2 + (2/p)df7dp - (l/k)(2p-E)f - 0 (1) If we set f(p) - u(p)/p (2) then df/dp - -(1/p2)u(p) + (l/p)dU(p)/dp (3) de/dp2 - <2/p3)udu/dp - <1/p21du/dp + (l/pmzu/dp2 - <1/p1d2u/dp2 - <2/p>du/dp + <2/p3>ud2u/dp2 - <2/p>du/dp + <2/p3)u(p) + <2/p>1<1/p>du/dp - <1/p2)u1 - (l/k)<2p-E>u

/p = 0 or dzuxdpz - ((2p/k) - (E/k>)u

= o ' (5) (ii) Case of p + O, for finite m > 0. Look at the coefficients of the first and second derivative terms first, which from part (i) are: p2(2wz-am2)/w2 (second derivative) and 2p((2w2-am2)/w2) + (p2/w2)4p - (p2(2m2-am2)/w“)213 (first derivative) As p + 0, these coefficients become P2((2m2-am2)/m2) and 2p<<2m2—an2)/m2> + <4p3/m2> - 2p3<2m2-am2)/m“ respectively. Keeping only the leading term in p for these two cases, we have that For For For For 180 p2(2-a) = coefficient of second derivative term 2p(2-a) - coefficient of first derivative term the 1JJ equation the coefficient of f(p) is 3m“ap2/w6 + Ap“/m“ -((2w2-am2)/w2)j(j+l) - 4p2m/k 1>2p2E/k small p this goes over into [-(2-a)j(j+l)]. the 3.]J case we have [3m"ap2/u5 - (l-A)m2p2/cn‘* - ((2u2-m2)/w2)1(j+1) - (2132/1020) + 2pZE/k1 + -<2-a>1/w2)ilf1 - [2mp26/{NEWS/unn)(ax)ij+<1/(2<23+1))) 0, namely «fit/cm2 + (2/p)df/dp - (1(1+1)/p2» = 0 (6) (iii) Case of m + a Proceeding as in the previous cases we start with the coefficients of the first and second derivatives of f. coefficient of dzf/dp2 - p2((2m2-am2)/w2) + (2-a)p2 coefficient of df/dp - 2p((2m2-am2)/m2) + 4p3/w2 - 2p3(2u2-an2)/u“ + 2p(2-o) + 4p3/n2 - 2p3(2-o)/n2 - 2p(2-o) + 2p3a/m2. The coefficient of the undifferentiated term is 182 [3ml’cp2/m6 + Apu/w” - ((Zmz-am2)/m2)j(j+1) - (2p2/k)2w + (2p2/k)E] + [aapz/m2 + Ap“/m“ - (2-a)j(j+1) - <2p2/k)<2m + (pzlm) + -E)] where we have made the approximation w = m and in (2p2/k)2m we have made use of the expansion w a /m2+p2 a m'+ (p2/2m) + ... Here we have set E a 2m + A. Collecting terms we now have (2-1f

= 0 <7) Comparing this last equation with that for the radial equation of the three dimensional harmonic oscillator such as is found in [38], we see that they are of the same form if the force constant is taken to be k(2-a) - k(l+2A). In a like manner one proceeds to show that the 3JJ and 3(J11)J equations give rise to this same limiting form. (iv) Case of m + O 183 Proceeding as in the previous three cases we look at the coefficients of the first and second derivatives of f. coefficient of de/dpz: p2((2w2-am2)/m2) + 2p2 coefficient of df/dp: 2p((2u2-cm2)/u2) - (4p3/u2) - p2((2w2-am2)/w“)2p + 4p - 4p - 4p - 4p coefficient of undifferentiated term: [3111“0I132/m6 + Ap“/w‘* - ((ZwZ-amz)/w2)J(J+1) - (2p2/k)2w + <2p2/k1E1 + o + A - zi For the 1JJ equation in the limit m + 0 we have the form: 2p2d2f/dp2 + 4pdf/dp + [A - 2j(j+1) - 2p2(2p-E)/k]f - 0 or de/dp2 + <2/p1df/dp + 1A/(2p21 - J(J+1)/p2 - (zp-E)/k1f - o (a) In this case the other two equations do not reduce to the same limiting form. For the 3JJ equation, the coefficient of the undifferentiated term is -21(1+1> - (sz/k)<2p-E). For the 3(J31)J case this same coefficient is: 184 {-2i<1+1)1f1 + AMI/2Nij - (/j/(21+1))(ox)ij ‘<1/<2(25+1)>)(°z>131f3 - (2p2/k)(2p-E>fi In this limit these equations become 3JJ= de/dp2 +.(2/p)df/dp + 1-1/k1f = o (9) 3(J11)J: dzfildpz + (2/p)df1/dp + [-(j(3+1)/p2)5ij +(A/sz){(1/2)51j - <43?3¥T7'/<2i+1))(ax>1j - <1/<2<23+1>>)/k)61.jlfJ - o (10) 11 Note if A - 0 then all three equations in the m - 0 limit have the same form again. (v) Lagrangians of the Radial Equations The general form of these, or any, Sturm-Liouville equation is: d/dx[p(x)df/dx] + [q(x) + Xr(x)]f - 0. It is well known that such equations arise as the Euler-Lagrange equations associated with the Isoperimetric Problem in the calculus of variations. The statement of this problem is to find a function f(x) such that the following integral I - (1/21fB [p(x)(df/dx)2 - q(x)f2]dx a [3 £(x,y,y’)dx (11) A A is an extremum, subject to the further constraint that 185 (1/2)fB r(x)f2 dx = 1. (12) A The limits A and B may be finite or infinite, p(x) must be differentiable for x in the region A < x < B, and q(x) and r(x) should be continuous in the same region. The associated Lagrangians for the equations are respectively £1 - <1/2)[p2<<2w2-am2)/w2)2 - <1/2113n‘icp2/o6 + Ash/u“ - <<2w2-m2>/m2)i(1+1) - (zpznomlf2 £2 - (1/2)[p2<<2w2-m2>/w2>1(df/dp>2 — (l/zmn‘ispZ/m6 -<1-A)m2p2/m“ - ((2w2-am5)/w2)1 - (2p2/k12w1f2 £3 - (1/2)[P2((2w2-0m2)/w2)](dfi/dp)2 - (1/2)[3n"o1p2/u6 - anZB/u“ + (mY/w)(j(j+1)-1(1+l)) - (2p2/k)2w - ((ZwZ-am2)/w2)j(j+1)]f2 - <1/2)1<2mp26/)> + ApZ/wcwnnu/mij _ (v"iTj’i?1‘5‘/<2i+1))(ax)ij - <1/<2<23+1)))(°a’ii]f12 - (1/2){(m292/w‘*)[(F——j(i+l)/(21+1))(Ox)ij 2 + <1/(2(2;1+1)))(oz)ij]1fj 186 where a = 1-2A, 8 = (3-2A)/2, Y - 1+2A, 6 = 1-A, and the common constraint condition is (zn/k)IB pzlf1(p)12 dp = 1. A Appendix 1.6 Solutions of Limiting Forms of the Radial Bethe-Salpeter Equations and the Choice of the Variational Trial Wave Functions We first obtain solutions of the limiting forms of the equations derived in appendix I.5(i) and (ii). We first discuss the case of p + 0. In this case the equations have the limiting form [equation (5) of appendix I.5(i)]: dzu/dpz - (2/k)pu + (E/k)u = o (1) If we set y - a(p-E/Z), equation (1) becomes dZu/dy2 - yu - o (2) for the choice a - 34(27E5. The solution of equation (2) is the Airy function. If p + 0, then the limiting forms of these equations again all take on the same form, namely equation (6) of appendix I.5(ii): dzf/dp2 + (2/p>df/dp - [2(2+1>/p21f - o (3) The general solution of equation (3) is f(p) = apz + bp-(z+1) We impose the requirement that f(p) + 0 as p2 which requires that b E 0. The idea now is to make use of these two limiting forms of the Bethe-Salpeter equations to obtain "good" approximate solutions to these equations. One method is to write the approximate solutions in the form: 187 188 f(p) = [p‘13

[Ai<3/?§7E7'p>/p1 <4) This particular form does not preserve the above limiting forms however. This particular problem can be easily taken care of by modifying equation (4) to read: 1 f(p) - [p‘13

[Ai(3/?§7E7 p>1<1/> ' (5) with a > O. The form of the Airy function is somewhat awkward to deal with for the following reasons. The constant k which appears in the argument is one of the parameters that is to be determined by comparing the eigenvalue spectrum and decay rates calculated by these equations to those observed experimentally. A second reason is that Ai(x) for x =_3S 300 is very small (8 10- ). The expected value of k from previous calculations is about .3. The numerical integration routine to be used is a Gauss-Laguerre algorithm whose abscissa range will pass this limit. The more points used in the numerical integration, the better the energy eigenvalues will be, especially if those points are near x - 0. This factor of 3/(57E) can be eliminated from the argument of the Airy function by making the following change of scale. p - 3VZE725 x m = 341E725 K (6) Inserting (6) into the term d/dp[p2((2mZ-am2)/w2)df/dp] we have d/dx[x2((ZwZ-aK2)/w2)df/dx] 189 2 where w = x2 + K2 now. In the coefficient of the f(p) terms we have IJJ: [amiapz/ue + Ap“/w'* - ((2w2-0m2)/w2))j(j+1) - (zpzlkXZw-Enflp) + [Bruaxz/ws + saw/u“ - ((sz-aK2)/w2))j(j+l) - 2x2m - 34(27k) Ex2]f(x) The éJJ and the 3(Jil)J equations behave in a similar manner, i.e. with the substitutions p + x and m + K, all of the terms in the coefficient remain of the same form with the exception of the (2p2/k)(2m~E) term. This term, as in the lJJ case, becomes 2x2m - 3J127k) sz In terms of the variables x and K the variational problem now reads: Find those functions f(x) which render the following integrals an extremum f0 £1>dx (7) subject to the constraint 3% E I0 x2f2(x)dx a 1 (3) The functions £1(x,f(x),f’(x)) are 190 i1(X.f(X).f’(x)) - (1/2)(df/dX)2[x2(2w2-am2)/w2] - (1/2)[3t“cx2/u6 + Ax“/u“ - ((ZmZ-aK2)/m2)j(j+l) - 2x2u1f2 (9) £z.f’(x)> - (1/2)(df/dx)2[x2<2w2-am2)/w21 - (1/2)[3K“ax2/m6 - (l-A)(K2x2/w“) - ((sz-aK2)/w2)J(J+1) - 2x2u1f2 (10) £a/w21(dfi/dx)2 (1/2)[31<"ax2/u16 - szzslm“ + (Ky/m)[j(j+l)-£(£+l)] - ((2w2-0K2)/m2)3(3+1) - 2x2m1f12 <1/2112tx25/cw2 + Ax2/>>ij 1 In terms of the variables x and K the trial form of the function f(p) given in equation (5) reads: 191 f(x) =- x‘B(x)Ai(x)(l/(a+x“l)) ' (12) The function B(x) ideally should go to unity as x + O and as x + 6. 0f the functions tried, none was found that obeyed these restrictions and yielded satisfactory results. In view of this, the function B(x) was chosen to be a polynomial B(X) . 80 + 31)! + 32x2 + 000 + 39x90 Details concerning this choice of B(x) and its calculational results are discussed above in Chapter II of the thesis. Appendix 11.1 The transition rate for 3¢1(3095)+e++e- +— We start from the S-matrix element for the process 3Wl(3095)+e +e , = = 1flfi 193 The terms omitted from this last expression will give a zero expectation value because they do not contain the requisite number of e+ and e- creation operators, in the first or second order terms in this expansion, to produce the e+e- final state. The terms in . this last expression can be factorized as follows: = 8de4xd4y<0|Au(x)Av(y)|0> <0|j3(y)|3¢1(3095)> + a second term with x and y interchanged. Since the only term in this product that contains two arguments, x and y, is the photon term An(x)AV(y), the time ordering need only be considered for this term. The quantity is called the photon propagator. We now discuss each of these three terms separately. In what follows we shall use the Fourier expansion of the functions ¢(x) that arise in the current matrix element. The form that we shall employ for this purpose is 3/2 2 l 3 m . w) - (2 s31“ p 3(9) (bs(p)us(p) exp <-1px) + + as (p)vs(p)exp(+1px) (i) The term a = I The Fourier expansion of I is 3 2 2 ._l . 3 3 , m + - + I ‘21: 1eId pd p S 83:33 B(p)E(p') vs.(p') exp (-i(p+p')x)|0>. 194 plus other terms which involve one creation operator of either type and an annihilation operator or no creation operators of either type . + (i.e. bS ...- Let - b;|0> 3 3 . + 5 (p-q)5 (p'-q'). rs r's' Now with the integrations over p and p' we arrive at the result 13 2 ( - g) 1e {gge u (q)Yv (q') exp (-i(q+q')x> u r' (ii) The photon propagator term 4 <0|TAu(x)Av(y)|o> a (.27]; Id4k Siptéflx-yn k -ie The expression on the right hand side of the photon propagator term (ii) is obtained in the following way. we expand Ah(X) in a Fourier expansion using 3/2 3 + “p(X) . (5%» f 1355- (e:) 3 3 Au(x)1\,(y) = (2-11; 2a II 3;" 35:: (eX “(k)a)‘(k)exp(ikx) X + ++ _- (1'): +I 'l 0+! + I _" +6“ (k)al (k)exP( 1kx))(€v(k )aa(k )exP(1k Y) + €V(k )aa(t )exP( 1k Y)) 3 3 3 . 1 2 I Q_§§_E_ [e:(§)a + a‘+ + . . (k)e (k')a (k')exp(1kx - lk'y) X00, /m' x V a (k)a 81+ a“ (k);: (k )aa (k )exp(-ikx + ik y)] + A 195 TAu(x)Av(y) = B(xo-yomuumvm + 6(yo-XO)AV(Y)AH(X) 3 3 3 l d kd k' A + + a + + . . = (— 2: I———-—-— (e (k)a (Me (ma (kwexpukx-xk'mm -y) 2" M1 mm. H x v a o o + NE: W: “K' E' 'k +'k"( - Eu )ax( )ev( )aa( )exp( 1 x 1 y)e yo xo)) The vacuum expectation value of the time ordered product is zero, except for the two terms explicitly written out above. In this case we have <0er mm ( )|o> = (i 3 2 II 9.331311% Ni?) “(I‘mz‘ (TE-Eu I.l Vy . 2‘" “'1 aw. €11 EV ad ° 1 + a + . 3 + +0 1k - k' - + k k' k-k exp< x 1 y)6(xo yo) eu( )ev( ’Ga:x‘ ) exp(-ikx + ik'y) B(Yo-xo)) (1 3 z I (2“) I k2_1€ auv (iii) The Hadronic Current Term: H - <0|jh(y)|3 (3095)> . u “’1 a = <0|jg a <0|exp(-ipwy)jg(0)exp(ipwy)I3¢1(3095)> a <0|j$(0)|3¢1(3095)> exp(ip¢y) Now combine (i), (ii), and (iii) to get 2 + - 1 3.— me .h 3 s a = II‘EF’ eur(q)yuvr.(q') EE3<°|3v‘°’| w1(3095)> 4 . . _ , l 4 expgikgx-yzz expupwyww exp( “em )1!) (21,) Id k k2 _ 1e 636)! 197 Note that the 8 has been omitted. We have a sum of two terms, which are equal, the second term coming from interchanging x and y. Next we perform the integrations over y, k, and x, in that order, giving us 17 m2 s . (— 11,) e6 <0|jh(0)|3w (3095)>fi (qw v (q'w -9 v 1 r ]J r' E' I d‘kd4xexp(- -'i(q+q )x)-exp(ikx)Id yexp(ip¢y)exp(- iky) 1 7 n: =- 15? e6m <0|j$<01|3¢1ooss1>u (qwuv ,.>u (qwuv ,.(q') —. exp(-1(q+q‘-p )x Id‘x Q p2 W 2 3 .h 3 - , ["2 , (51,1) eGW<0|3u| ¢1(3095)>urG‘( ) v ( ') -9 W p 1 r q Y“ r. q 33' —26(q+<1'-p¢) 9x» We now want to find the quantity II2. We will sum this over the possible final state polarizations and integrate over 198 the allowed momentum states so as to determine the total decay rate of ¢+e++e-. Let T I |I2 suitably summed and integrated over. m2 2 2 + - h — , __§_ '1‘ ‘15:. {(211) e (ur Eu 2 P1: 91 2 2 (4) 6 (q+q'-p) 2 2 me h h w . (21:) e E,<0|jv(0) |¢> 2 9 W 2: ,5} (9”qu (q'fir. (‘1')Yv“,191 J r.r Look first at the term summed over r and r'. and note that this double sum is equivalent to the taking of the trace of the matrix product -1 . -1 , Yu ('2'; (firm-m) )Yv[ fiwq +m)] . Note that —- l zur> m: and we have neglected m: in comparison to 32. With this T becomes 2 2 m: 5 (Py'q'q” 2 h 2 232 2 T . -(2n) e EE'( e—) ]< o|j (0)|¢>| {-5— (l-cos 9)) DW me We now need to give an interpretation to the factor (64%, -q-q')) 2. This is done in the standard way of rewriting [64(pw-q-q'))2 as 64(pw-q-q') 6 (pw-q-q ) <— #1 44Id x expx) = 2 4 (vxr) ( 1) where er is a normalization space-time. 201 Now we would like the quantity T/t. T 2 ”:92 n 2 23: 2 1 '1' - -(21r) —2— 1| —-2- (l-cos e) —; a m p e e W 4 4 5 (Pw-q-q ' ) V (F3: 0 To obtain the total decay rate we must sum over the possible sets of (unpolarized) final states available. If we choose the normalizing volume to be (2n)3. the density of final states is given by d3qd3q' and we have 2 2e2 6m . $- . 431) —-; (l-cosze)||26‘(p¢-q-q')daqd3q' 96 a = 1%“; |<0|jh(0) |¢>|2(1-cosze)6‘(pw-q-q'm3qd3q' 9 W We now integrate over d3q' using the delta function, and we get dw I _;§%_ |<0|jh(0)|¢>|2(l-cosze)54(aw—B-E')d3q. "P 6 We next integrate over d3q I qqud(cose)d¢. The d¢ integral just gives us 2n. For the other two integrals we have ‘11 n 3 2 cos 1 l 4 I (l-oos e)d(cose) = [case - “73110 - (my-(17) . .3 O 2 2 2 x dx 1 2 ”(31723” dq - I6 (aw-22m dz = stw-x) :- 2- - gfaww-xu dx 8 x—z- ' 8 ‘E-‘i a a ' 202 2 _ p Finally w I ;2%_ |<0|jh(0)lw>lz(2fl)(-4/3)(—%1 Pp . 52.3! ||2 91» The final step is to determine |<0|jh(0)|¢>|2. The matrix element <0|jh(0)|¢> is related to the wavefunction of the decaying particle as follows: <0|jh(0) |¢> - 3<0|jh(0) N8) + 3< o|3h(0)|‘pD> 2 . fiie3/: { Ida? f8(P) " IdBP p f8 (p) 3B(;+m) } (2n) flu flu fine 2 r (p) i 4 p + 3!! I? ‘39 sum» The factor ofu/3 that appears in these terms is to take into account the currently accepted hypothesis that there exist three different “colored" quarks that can give rise to the same qq system. In terms of ¢(0) we have m - 3% I<0Ijhl2 3m V 21 2 ' ;;§'{/3 i qu§¢1011 v 21 t 2 2 - — 3e 2|¢(0)| ) 3m2 q 203 where 2 2 2 2 g__ 2 3 4e ‘ 4 eq - (eQ) = 4“ Q (4n) 9(4“) (4“) '§u(4fl) and where Q = who Thusw .. 32 3 inunnhpm) [2 3m2 9 V 2 - lsaz (4n>lw<0)|2 9m V 2 9m V Appendix II.2 Derivation of the Transition Rate for the Radiative Decays in the Bethe-Salpeter Equation We start this derivation from the S-matrix element of the appropriate decay process, namely <7,B|S|§>, where E is the initial state and 1,3 represents the final state of the transition in 'question. The S-matrix in the interaction representation is given by the time ordered product of expl-iIHI(t)dt] where HI(t) is the interaction hamiltonian. The interaction hamiltonian is given by -Iju(x)Au(x)d3x I -Ij:(x)Au(x)d3x to the lowest order in the 0 electromagnetic current,Au(x). In the second term,j:(x0 stands for the hadronic component of the electromagnetic current. With these considerations one can express the S—matrix element for the desired transition in the form I =- 1<1,B|Id‘*x HI(x)|‘fi> = 1Id“x The expression is the hadronic contribution to the electromagnetic current matrix element between the states B and B. 204 205 The S-matrix element can be diagrammatically represented as /Y / I %_____L " :3 To translate this into a more useful form for calculational purposes, we assume that to lowest order this can be represented in the form I Y I I X2, :’ X0— —-1- ‘41 N ..0 >4 X :4L-—~—1LH This diagram is evaluated just like the analogous one for electron-positron scattering, with the emission of a photon, in quantum electrodynamics. we start by looking at the top (or q-) line in the diagram. This contribution yields —- + _ 1"? (X3)Yu311 Saba-3(2)] A (X2)[1 SC(X2-X1)]YVI¢ (XI) and a similar contribution from the bottom line _+ u s + II .. 1P (x6) Ya [1 Sc(Xu-Xs)] YB '1’ (X5) The interaction part between lines I and II is given by [iguUDc(x3 - xs)][iguaDC(xl- x“)]. 206 (1) Conversion of‘E to we The first step in this derivation is to convert the expression for the positively charged line into one involving the charge conjugate wavefunction, since the Bethe-Salpeter equation will be written in a form that makes use of w and we rather than one that makes use of w and '$. The charge conjugate wavefunction we is defined as - ¢c(x) I C $T(x). This says that $ I [wc]T[C.l]T. we will also need to know the explicit form of ¢(x) expressed in terms of wc(x). This follows directly with a little algebra. (The second equality in the first line below is just the definition of WKx).) p(x) - [wc1Tt_c‘11T = [puffs p(x)* - BTC-l¢c(x) p(x) . eTlc'11*wc* - (sTc'1>*wc* - d$m(x> p(x) - ¥$;(x)C‘1 . 4$;(x)c 207 (ii) Conversion of II We now make use of these relations between wc(x) and w(x) in the positively charged part of the diagram representing the radiative decay of our system. I: - Von.) Y.“ [1 SCCXu'XSH 185 ¢+(xs) - wcTcm.) [c’llT Qt: sclvBSI-*6cTwc I -CT$;T(x5) Y85T[i Sc(xs-xt)]TYa“T C-1¢c(xu) ' 4$;(xs)C YBSTli SC(X5‘xu)]TYa“T C-l¢c(Xu) 5T . .$c(xs)c YB c'lcli sc(x5-x.,)]Tc'lc YQ“TC'ch(xu) - Team-135m Sc(xs-xn.)](-Ya“)T ween.) 208 (iii) Combining terms and the Bethe-Salpeter equation With (1) and (ii) above the full matrix element can be written as: I II Interaction I $-IX3)Yu[i Sc(x3-x2)] £+(x2)[i Sc(x2-x1)] vaw'cx1){-Wc(xs>vvu sees-m.)Havana.)1}~ iguch(xa'xs)1gaBDc(X1'Xu) At this point we display the matrix indices in order to keep things straight. we rewrite this matrix element in the form: Int .. ..[TF-(xanlflYulABu Sé(x3-x2)]BC[d+(x2)ICD[1 Sam-mung I101mlw'cxl)lec’sznunvlmu sees-mongrel” 1¢c(xt)lp iguch(X3‘Xs)1gasDc(X1-Xu) We now collect terms that appear as "initial" or "final" state terms. The "initial" state terms are those with arguments x1 and xh and adjacent matrix terms: (1-4) [YQIEF w’(x1>F[vBIOch1Anu1AB Wflxsnmnvlm igwnc <1-4) (3'5) 209 (a) The (1-4) Initial State System [YQIEFI1P(X1)1F[YBIOP1WC(XI+)]P11808DC(X1‘XI+)] [YalgFlMX1)]F[¢C(XL.)]‘PIYBTlPolngBDJM-Xt)1 [v 1 [wcx )1 [w (x )1 c‘lctv T1 c’lcti p (x -x )1 aEFch“? 8P0 Basel“ [:1 Warm [1(on C-1(CYTC-l) c [is 90‘1"ka aEF F c PPQ B QRRO ch [YalEFIMM)1FWC1PC;é[-YBIQRCROIisasDC(X1'Xu)1 I -[Ya]EF[ Mn .XuNFPC- PQIYBIQR CRoligaBDc(X1‘Xu)l 1 -1gaBDc(x1-Xu){[YGIEFIMXhXOIF-p C. pQ ”316R CR0} [(114m) Wm.) c‘1<-zm)c1E0 (b) The (3-5) Final State System [WIX3)IAIYHIA31$;(X5)IMIleMNliguch(X3-Xs)1 [YuTlBAl-Wxaficks)IMIYVIMN [iguch(X3-xs)] c’lchuT] BAc'lcmxaficcxs) lmlvvlmlingc(xa-Xs)l c‘ll—y IBActhaflccxsnmn u ]MN[iguch(x3-x5)] V 210 -1 ._ I C YuC¢(x3,x5)Yv[-iguch(X3-Xs)1 (c) An Aside on the Bethe-Salpeter Equation We shall need to make use of the adjoint form of the Bethe- Salpeter equation and this section is devoted to obtaining such a form. We start from the Bethe-Salpeter equation in the form: (almwd'IC-amm -- -gZDc(x1-x2)vu¢c"lvuc where g2 is a constant. We first note that 71(x1,x2) " YuTW*(xl,X2)Yu . and C* I C-l. We start by taking the complex conjugate of the above Bethe-Salpeter equation, giving (31* +m)¢*(C-l)*(-¢2* +m)C* =- ~gZDcx(x1-xz)yuew(c’1”Yuma. We first consider the left hand side of this equation and express it in terms of $1x1,x2). L ' (11* + m)w*(C-1)*(-32* + m)C* 3 (31* + m)¢*C(-22*+m)C—1 ' (31* + m)YuT$ YuC(-32* 1"!!!)0-l " (31* + m>YuT$('CYuT)(-32* + HOC-1 211 ' (31* + m)YuTEI‘C) YuTC‘32* + m)C-'l Now note that T T T T Yt Yu*3u = Ya (Yi 81-7634) = rt ('Yiai‘Yhau) T T T = Yu(-Yiai-Y636) = (Yiai-Yuau)vu s {-11 ai'Yu 3t>1t where Yu I YhT and 11 I ”YiT- So we have that, moving YnT past '32* L = (21* _+ m>mTI<-C)(+12T+ m)YI+TC-l - (11* + mmT‘w'<-C)(+32T+ aux-€111.) In a similar manner T T T Yu*3u*Yt = (11 ai-Ytat)vt I (‘Yiai-Yuau)Yu T T ' Yu(Y131-Yu3u) . Yu(-Y1*31-Yu3u) = Yu(-Y1 31'Yu 3») r T a 3 Ya (Yu v) This gives, now moving the YuT past 31* L - xtTmTau‘ + ma? t-c1 [+321+ mH-C'llvu - vuTc'lcnuTaul + m1 c'lc‘ui CUzT + m1 C'ln. ‘ YuTC—l(31+m)d$(32+m)Yu We next turn our attention to the right hand side of the Bethe-Salpeter equation. 212 T -1 T R I ‘829c*(X1IX2)Yu W* C Y“ C ‘ Igch*(x1-x2)vuTytT'$ Yr C YuT C-l . Isch*(X1-X2){(YuY1)r+r+T}c‘1 - -32Dc*{-(yiv.)‘+(vtvu)”$I-c1{-(v.v1)‘+¢ a [‘1' "1 Ann] awakens)lutvvlmugwnccxa-xs)1 - _1[C-1(35+m)d$(X3,X5)(‘a3+m)]NB Although it is not explicitly written down, there is an implied integration over the repeated variables in the interaction term I II Int. The above manipulations have been done on the integrand of this expression. (iv) Final reduction of the current matrix element Combining the various pieces from (iii) we have -Int - I II Int I [c'l(15m)c$(x3,x5)(-a3+m)]NB[isc(x3-x2)]BC [1+(X2)]CD[iSc(X2IX1)]DE[(31+m)¢(xl.Xu)C-1(I3t+m)C1EO [iSc(x5-XQ)INO p 214 The reduction is aided by the use of repeated integrations over the variables x1,...,x5. Before we begin this process, we note two properties of Sc(x-y) that will be useful in our reduction procedure. The quantity Sc(x2-x1) has the Fourier representation Sc(xz-x1) = (I/ZN)“Id“p (mrifi>exp(ip(xz-x1))/(p?+m2-16) Also note that the application of the operator 3 + m to Sc(x2-x1) yields (51+m)8c(xz-x1) = (1/21)“Id“p(m-ifi)(ifi+m)exp(1p(xz-x1))/(p2+m2-1e) - (l/21)“fd“p(p2+m2)exp(ip(x2-x1))/(P2+m2-ie) - 6“(x2-x1) Furthermore note that under charge conjugation we have C-ISc[-za +m)]NB[iSc(xa-xz)IBCIi+(xz)]CD [15(x2-X1)]DE[W(X1sxu)i5(x5'xu)lEN Now integrate over x1 and x5. This gives -Int - [C-1(35 +m)CEKX3.xu)(-13 +m)]NBI1SCCX3-X2)]Bcld+(xz)]CD Wc$1ia+ expc-ikx>¢1 ' [C-l(1k’/2 - Wax + m)Cexp(1k’X)?(X)][iIl+(X + (1/2)x)exp(-1kX)¢(X)1 = [C-1(ik’/2 - z/ax + m)Cexp(1k'xy$tx)1[1é exp(ik(X+(l/2)x) exp(-ikX)¢(x)] - [Elm/2nf - a/ax + m)CEKx) ié exp(ikx/2)] exp(i(pf+k-pi) We now integrate over X. Note that fd“X d“x (integrand) - fduxz d“xu (integrand), i.e. the Jacobian of the transformation (X,x) + (x2,xu) is 217 one. The result of our integration over X is just the addition of a delta function in place of the exponential term (modulo 21’s). -Int - c’1(1¢f/2 - a/ax + m)d$(x) ié exp(ikx/2)(2w)“6(pf+k -p1)¢(x) At this point we take the trace of this expression. This was done using the 1977 version of Schoonschip we have on the CDC here. ’The evaluation was done by using the momentum representation of the above expression in the form -Int - ¢f((1/2)8ET + 1 $°3 - Bpo + m) ¢1?-é This form is obtained by making use of the identity trace(ABC) - trace(CBA) for Y matrices. In more detail we have -Int - c'1[1§f/2 - z/ax-+ mld$(x)1é exp(1kx/2)(2w)“6(pf+k -p1)¢(x) Note trace[-Int] 8 trace{C-1[ipf/2 - 3/3x + mld$ ié ¢(x)1}(2fi)“5(Pf+k‘Pi) - (constants) trace{[ipf/2 - fl/Bx + m]C$ ié Q(x) C-ll} mm”1 s ¢1 trace[-Int] 8 (constants) trace{[ipf/2 - l/Bx + mIC3'ié ¢i] = (constants) trace{¢1 ié'$ C[ipf/2 - 3/3x + m]} . (constants) tracef$Clipf/2 - 3/3x + mloi ié} 218 = (constants) trace{¢f[(i/2)BET + ;-E - p08 + m]¢i ié] where 3C 8 ¢f and the operators in the square brackets act to the left. The constants in this last equation are (21!)l+ 6(pf+k-p1) 2(2e/3), and the delta function is the four-dimensional energy momentum delta function. Appendix II.3 The Calculation of the Angular Momentum States of the Vector Particles Arising from qq States In the calculation of the transition amplitude, we shall need to make use of the explicit polarization states of the particles. In this appendix we derive the required polar- ization states and the appropriate normalizing constants. The triplet particle states that we will be considering are 3P0, 381, 3P1, 3P2, and 3D1. The calculations make use of the relation ' N 'b 'b ’b M-m A + m TJLM i C(LlJ, M-m, m)YL (r): with l i for 3 = +1 0 N 'b E m = for m = O l -i for 3 = -1 O (1) 331. Let J - 1,_L = o. ++1 +0 +-1 T = 1//4n g T = 1//4n g T a ll/an g 101 100 10-1 so that the appropriate polarization vector for these particles 219 220 is 1//Z? Em. (2i) 3D}. Let J=1, L82. M=l case. m m . _N N M-m “ +m TJLM = i C(LlJ, M m, m)YL (r)€ 1121 = /1/10[Yg($)3+1] - /3/10[Y;($)E°] + /3/5[Y:(§)E‘1] . -1//§ 8 V1710[/57(4n) 8(3cosze -1)' -i//2 ] o o -/3710[-/157(8n) sine cose e+1¢ o ] 1 1/JE +/3/5[/15/(2nj ksinze e21¢ -1//§ ] o m.l- 1 - -<1//§>(ll/I6>(1/2)/5/(4w) cap: -1> 1 o 0. +/3/1o /15/(8fl) pz(px+ipy) o 1 1 -1 +/3/§ /f§7<2¢><1/4>(px+ipy)2<1//§> 221 ‘ — (3/(8/17))(p,2(+2i|tvxpy - pg; -p§ + 1/3) = (~3i/(8G))(p§ + Zipxpy - pf, + 13% -1/3) 113/(45))“:x + ipy)pz J .4 ~"(a/(8%?)Hp; + Zipo + p; -1 + 1/3) .Y = (-31/(8/?))(2ipxpy - p; + 1 - p; - 1/3) L.(3/(4v’17))(px + my)!»z __, r(3/(8/1r'))'(2p§ + 21px;)y -2/3) .1 = (-3i/(8/F))(1 - 2p; +21pxpy -1/3) L(3/(4/1?))(px +1'py)pz J, px(px +1py) -l/3 (3/(4/?)) -iE1/3 + 1pyy)pz (Zii) 301. J=1, L=2. M=0 case. 1120 - x3716 v;1(6)2*‘ - «273 vg<5>E° + «3716 v;(6)z'1 -1//2 #3710EJIS713n15in9 cose e'1¢ -i//2 J 0 222 0 - /2/5[(1/2) /5/(4n) (3cosze -1) 0 J 1 1//§ ‘+ /3/10E-/15/18w1 sine cose e+i¢ -i//2 J 0 aux-my)“ o 1 = (-1//§)/§7T6 15/ Ba ipz(px-ipy) -/57§(1/2)/§7T£?7' o 0 - 3p§ -1 J ' L. J r* pz(px+ipy)W -/3710 #15718n1 (ll/2) -ipz(px+ipy) L. 0 J l73/(4/2—)) ( ' + 4/2— +- 1 w pz px-ipy) (3/( n))pz(px Ivy) (31/(4/2F))pz(px-ipy) - (31/(4/5?))pz(px+ipy) L(-'3/(21/-2_1r-)_)(P§'1/3) .1 2pxpz 1 I ' rpxpz i = _3/(4/f;) 2pypz = -3/(2/2?) { Psz 2( 2 _1/3) 2 -1/3 L ”z -1 U" ‘1 For a 301 state one can write the poiarization vector as (2-6)B - 2/3 times some constant k. For a definition of ++m ' e , see section 6 of appendix II.3. The point of the above caicu1ation is to determine the appropriate constant k. 223 A E'- = (-1//7)(px + ipy) ‘1 (~1//'§)(px+ipy)px + (1/3)(1//7) + (e'-6>6 - E'xs (-1//'2')1€° + vjwfiré‘lj -1//2 (1//§)[/§7T§?7 sine (cos¢ -isin¢) -i//2 o 224 o - J3/14n1 cose 0 1 _ 1//§ - /§7(§?7 sine(cos¢ + isino) i//§ J o Px - ipy o = [bum/W 81! (II/2') pr-ipy) -(1//§)/“l—3/ 41:7 0 0 pz px+ipy -(1//§)/3/lsn5(1//§) -i(px+ipy) J o 2px px = -1/(4/F) Zpy = -1/(2/?) py 2pz pz 50 for 3P0 our polarization vector is Em = B//E? . (4) 3P1 case. Here L=J=1. (a) m = +1 case TJLM ‘ T11+1 = (ll/7)EY:(B)E° - Yg(5)31] o (ll/2)E/3/(8n) sine(cos¢ + isin¢) o 1 225 ll/2 - V3/(4w) cose -i//2 ] 0 0 - pz -(1//§)/3718n) 0 - ipz px + ipy - 0 p2 = (1/4)/37n ipz -px-ipy In this case the angular part of the wave function is just a constant times E x 3 . To see this look at the above case. ‘ s 3 .2 -1//2 -i//2 0 = i(-ipz//?) + 3(pZI/7)-+ £(-py//Z + ipxl/Y) px py pz (-1//§)Ei(pz) + 3(ipz) + §(-px-ipy)] -pz = (il/2) -ipz px+ipy -pz BUt T111 = ‘(1/4)V3/W 'ipz px+ipy Therefore comparing terms, the required constant is 226 obtained from (il/2)k = -(1/4)/i7; -+ k = /§(-i)(-1/4)/§7; = (l/(2/§))V3/n. (5) 3P2 case. Here J=2,L=1. JLM T212 The The polarization matrix is+?-p for this case. Q“-> = T21” = § C(112;M - fi.h)vr;§(p)sm = 1 ‘ ' Y1(p)€ -1//2 = -JETTEFT sine (cos¢ + isin¢1 'l//7 o px+ipy = (1//2)/§7ra?7 1(px+ipy) ' o 1 = (1/4)/§7n (px+ipy) i o determination of the constant of proportionality k in the polarization matrix kTZm.E is as follows: " . 1 T' “r F’ . r 1 1 O px px+1py 1212 = (k/z) 1 -1 o p, = (k/z) 1(px+ipy) O 0 0 0 L .1 132-1 L_ ,. 227 1" ‘1 px+ipy = (1/4)/3/n i(px+ipy) L. 0 -1 k/Z = (1/4)/3/v + k = (1/21/3/1 (6) Determination of“?m +zm is obtained from the following considerations. ++m _ . & M-% h .uv - § C(112.M.m)au av F’ '1 1/2 i/2 0 For m=2 this just gives us i/2 ~1/2 0 0 0 L. __ 1 since E'= - i (II/2) and the Clebsch-Gordon coefficient 0 is one. Appendix 11.4 Radiative Decay Rate Formulae (1) Results of the Schoonschip Evaluation of the Current Matrix Element In the program used to evaluate the Dirac algebra associated A with the current matrix elements, certain numerical factors were omitted from the main body of the program for convenience. TABLE 9 below lists those factors that depend upon the polarization state involved. In addition an overall 41/8 was omitted. The 1/8 comes from the ‘normalization of the two body wavefunctions and the factor of 4! comes from the angular integration that was done with Schoonschip. The results of the Schoonschip evaluation of the Dirac algebra and the angular integration over the internal momentum are given in TABLE 10. (ii) The P0 Integration As seen from the previous section, the Schoonschip program for the current matrix elements yields results that have the following form: (constants)(p/E)[po+(1/2)ET-E]f1(p)ff(p)(polarization factors) For example in the 2381 + 13Po case we have explicitly -(8/3)iff(p)f1(p)(p/E)[po+(1/2)ET-E]E';(4I/32n)(4e/3) The functions f1(p) and ff(p) are actually functions of p as well as p0. These functions can be decomposed into the product structure f1(3.Po) - f1(13|)[l/(po+(1/2)ET-E+ie) - 1/(po+E-(l/2)ET-ie)1 A similar structure holds for ff(;,po), the only modification being that E is replaced by E In the current matrix element we have the T f' 228 229 products f1(p)ff(p), so structures of the following form are encountered C(po) . [1/(po+(l/2)ET-E+i£) - 1/(po-(1/2)ET +E—ie)] [1/(po+(l/2)Ef-E+ie) - l/(po-(1/2)Ef +E-ie)] [Po +(1/2)ET-El (1) The decay width is obtained by integrating over the four momentum (p,po). The angular integrations were relatively easy to do and were done concurrently with the current matrix element calculation in Schoonschip. What remains are the radial IEI and the po integrals. The only terms involving po are explicitly displayed in (1). The pa integral is over the real line, the ISI from 0 to m. For simplicity we break the integral up into four parts as follows. IdpoG/<2u) IdpoET-E+1e))/<2n) Idpo/<>/<2x) IdP0(Po+(1/2)ET-E)/((po-(1/2)ET+E-16)(po+(1/2)Ef-E+i€))/(2W) + fdpo(po+(l/2)ET-E)/((po-(1/2)ET+E-16)(po-(1/2)Ef+E-i€))/(2*) The first two terms give just -i. The second two terms yield the following, using the method of residues. 230 Idpo(po+(1/2)ET-E)/((po-(1/2)ET+E-1€)(po+(l/2)Ef-E+1€))/(2n) - +1 Residue[(Po+(1/2)ET-E)/(po+(1/2)Ef-E)] - +1 [(ET/z - E +ET/2 -E)/(ET/2 - E +Ef/2 -E)] - +1 [(ET-ZE)/((ET+Ef)/2 -2E)] IdpoET+2)Ef+z>)/<2w) - +1 [2(ET/2 - unwrap/2) +((Ef/2 ‘E)+(Er’2 #E)>/((Ef-ET)/2)l +ET)/2 -2E]] - +1 2/(ET-Ef)[ET-2E -[(Ef - +1 2/(ET-Ef)[ET-za'-Ef/2 - ET/Z +23] - +1 2/(ET—Ef)[(l/2)(ET-Ef)1 = 1 Adding together the results of the integrations we have +1[-1+1-(ET-2E)/((ET+Ef)/2 -2E) - -1 (ET-ZE)/((ET+Ef)/2 -2E) Therefore Idpoc(po)/(2n) - -i(ET-2E)/((ET+Ef)/2 -2E) (2) (iii) The Transition Rate Formulae 231 The S-matrix element for the radiative decay B’ + B + Y is given by -(2fl)"i[current matrix element] 6“(pf-pi+k) The usual expression for the decay width is given in terms of a matrix element denoted by Mfi' (See Appendix D of [45].) If one expresses the wavefunctions f1(p) and ff(p) in the current matrix element in terms of the radial wavefunctions of the states B and B’, then our current matrix element is precisely this Hematrix element. According to [45], the decay width in terms of this M~matrix is given by my + B + v) = <1/2>(1/<4u>)2 IBI/Mfi. Ida Z 1an spins This expression however assumes a specific normalization for the states we have labeled B and B'. Sakurai’s expression assumes these states are plane waves and are normalized accordingly. In the case of bound states a different normalization is conventionally used, namely a * f0 1(1))" (p) dp - 1- If we use this normalization condition then the above equation for F reads r(n' +B+v> - (8mB/mB.)(e/41I)2151 fan 2 1an <3) spins Note here that we have factored out of our matrix M a 2e. The sum over spins here means a sum over the final states and an average over the initial states. The quantity IEI is the magnitude of the momentum of one of the decay products in the final state. The angular integral and the spin average are easily done and in our cases we obtain the results for the spin averages listed in TABLE 11. In the case of the magnetic dipole transitions, an extra factor of k2 arises from the 232 polarization term which has not been incorpowidthd into TABLE ll. The cases in which a 381 state is replaced by a 3D1 state lead to the same Spin average results, e.g. 381 + 3P0 and 3D1 + 3P0 both have a spin average of 2/3. We now carry out the calculation of the decay width for the special case of 381 + 3P0, and just list the results for the remaining cases. we start with the M matrix element for the transition 381 + 3Po. M - (1/12)<81/3>Idp Pzif1(p)ff(p) (p/E) {-1(ET-2a)/< E Idp p2{£1

ff

(p/E) [(ET-zz)/<(ET+Ef)/2 -2E)]} <6) Inserting (5) into (3) we have for the decay width r(B' +B+ v) - (8mB/mB.)(e/(4W))21;1(4")(2/9)2R2(P)(2/3) - (64/243) “(mu/my) ISI R2 (p) (7) where a - e2/(lm) Equation (7) is for the 351 + 3P0 case. We can however write the decay width equation in the form 233 r(B’ + n+1) = 8)l$l IceIZRZ

P = 8alEl Icel 2112(1))? (8) In equation (8), if we specify P and Ce’ which are constants coming from TABLE 11 and TABLE 12 respectively, then we can write all of the electric dipole transition widths by specifying these two constants, with the obvious understanding that the appropriate radial integrals are chosen. For the magnetic dipole transitions the functional form changes to m + n+1) = 8e2/(Aw>2I3I3 IchZR2 - 8(mB/mB’N1313 Icml 2112(1))? <9) In this case the constants Cm are given in TABLE 13. The polarization constants P are those quantities given in TABLE 11. .PS 62,80 - APPENDIX ll.5 SCHOONSCHIP REDUCTION PROGRAM TRIPLET 5 ONE TO TRIPLET P zERO EE,ES,M,HEE,COEF,P,PO J,Jl,MU,NU,LA,MA,MM,HN,0,N,J2 PP,TF,T|,PS,V,W,E,PT,PQ-U PF,P|,EP,EPS,EPP THE FOLLOWING EXPRESSION NEEDS TO BE DIVIDED BY THE QUANTITY (I/(h*PI)*COEF, WHERE COEF Is THE COMMON DENONINATOR OF THE ORIGINAL CURRENT MATRIX ELEMENT I.E. (M**2*EE**h*(EE+H)**2 TRIPLET S ONE TO THE TRIPLET P ZERO CASE. 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In the past few years several elementary books on elementary particles have been published that discuss the material in chapter 1 in much more detail, e.g. [l] Polkinghore, J.C., "The Particle Play," W.H. Freeman, San Francisco, Calif. (1979). [2] Trefil, J., "From Atoms to Quarks: An Introduction to the Strange World of Particle Physics," Scribners, New York (1982). [3] Segre, E., "From X-rays to Quarks" W.H. Freeman, San Francisco, Calif. (1980). The book by Segre [3] deals extensively with the early development of 'nuclear physics, quantum mechanics and particle physics. The most recent developments are not treated as extensively as the earlier pre-world war II material. The.other two books deal with the recent developments more extensively and are easy to read. The book [2] by Trefil contains somewhat more detail than that of Polkinghore [l] and is somewhat more recent. More theoretical surveys of many of the topics discussed in these books and in chapter 1 are to be found in the following books by Roman, Sakurai, Cell-Mann and Ne‘eman, and Lichtenberg. [4] Roman, Paul, "Theory of Elementary Particles," revised edition, Interscience, New York (1961). [S] Sakurai, J.J., "Invariance Principles in Elementary Particle Physics," Princeton University Press, Princeton, New Jersey (1964). '1. .L I‘L , hm. 284 [6] Cell-Mann, M. and Y. Ne'eman, "The Eight-Fold way," W.A. Benjamin, New York (1964). [7] Lichtenberg, D.B., "Unitary Symmetry and Elementary Particles," second edition, Academic Press, New York (1978). The book by Roman was written before the development of unitary symmetry but contains a very detailed discussion of isospin, strangeness, and the discrete symmetries of elementary particles. The book by Cell-Mann and Ne'eman is a collection of basic papers in unitary symmetry published in or before 1964. The book by Sakurai, although written about the same time as [6] does not discuss unitary symmetry but covers much the same groups as Roman's book. The book by Lichtenberg is a recent, and readable, textbook on the applications of unitary symmetry to particle physics. Some of the basic works dealing with special relativity and quantum mechanics can be found translated and reprinted in the following books. [8] Einstein, A., et a1., "Theory of Relativity," Dover reprint of 1923 edition published by Methuen and company, London. [9] Van der waerden, B.L., "Sources of Quantum.Mechanics," Dover 1968 reprint of a book published by North Holland Publishing Company, Amstersdam in 1967. [10] Schroedinger, E., "Collected Papers in Wave Mechanics," translated from the second German edition, Blackie and Son limited, London (1928). Dirac's article on his relativistic wave equation appears in [11] Dirac, P.A.M., "The Quantum Theory of the Electron," Proceedings of the Royal Society of London, Series A, volume 118, p. 351 (1928). and is reprinted as paper 1 in [12] Hofstadter, R., editor, "Electron Scattering and Nuclear and Nucleon Structure," W.A. Benjamin, New York (1963). 285 Two recent discussions on the history of the Dirac equation are [13] Kragh, Helge, "The Genesis of Dirac's Relativistic Theory of Electrons," Arkive for the History of Exact Sciences, volume 24, p. 31 (1981). [14] Moyer, Donald Franklin (1) "Origins of Dirac's Electron 1925-1928," American Journal of Physics, volume 49, p. 944 (1981). (ii) "Evolution of Dirac's Electron 1928-1932" ibid, volume 49, p. 1055 (1981). (iii) "Vindication of Dirac's Electron 1932-1934" ibid, volume 49, p. 1120 (1981). The following article [15] Nakaniski, N., "The Bethe-Salpeter Equation," supplements to Progress in Theoretical Physics #43 (1969). is a review of work dealing with the Bethe~Sa1peter equation up to 1969. Reference [15] contains an extensive bibliography of papers dealing with the Bethe—Salpeter equation, and one of the original purposes of this review article was to compile and publish this bibliography in one place. The first attempts to construct a relativistic two body wave equation go back to G. Breit's work in 1929 on the hyperfine structure of helium. [16] Breit, G., "The Effect of Retardation on the Interaction of Two Electrons," Physical Review, volume 34, p. 553 (1929). Discussions of this early work of Breit and other authors can be found in the Encyclopedia of Physics article by H.A. Bethe and E.E. Salpeter, which was published separately in book form. [17] Bethe, H.A. and E.E. Salpeter, "Quantum Mechanics of One and Two Electron Systems," Academic Press, New York (1957). Also discussed in [17] is the more recent work of these two authors and others. The more recent work begins with Bethe and Salpeter's basic work on the relativistic two body wave equation. 286 [18] Salpeter, E.E. and H.A. Bethe, "A Relativistic Equation for Bound State Problems," Physical Review, volume 84, p. 1232 (1951), A great deal of the literature on the Bethe-Salpeter equation deals with radiative corrections to positronium. Some of the first articles on this system dealing with radiative corrections are [19] Karplus, Robert and Abraham Klein, "Electrodynamic Displacement of Atomic Energy Levels III. The Hyperfine Structure of Positronium," Physical Review, volume 87, p. 848 (1952). [20] Fulton, Thomas and Robert Karplus, "Bound State Corrections in Two Body Systems," Physical Review, volume 93, p. 1109 (1954). [21] Fulton, Thomas and Paul C. Martin, "Two Body Systems in Quantum Electrodynamics, Energy Levels of Positronium," , Physical Review, Volume 95, p. 811 (1954). A survey of work on the positronium system can be found in [22] Stroscio, Michael A., "Positronium: A Review of the Theory," Physics Reports, volume 22C, p. 215 (1975). This review also contains a good introduction to the Bethe-Salpeter equation. The major advance in the study of relativistic multi—particle systems began with [17] and was based upon development of a covariant renormaliza? tion scheme by F.J. Dyson, R.P. Feynman, J. Schwinger and others in the late 1940's. Many of the basic articles detailing this development can be found in the collection [23] Schwinger, J., "Quantum Electrodynamics," Dover, New York (1958) Also reprinted in [23] is one of the early articles dealing with radiative corrections to positronium referred to previously, namely [18]. In determining higher order radiative corrections to the positronium system, a more systematic development of the Bethe-Salpeter equation and related perturbation expansions was needed. Work along this line can be found in 287 [24] Feldman, Gordon, and Thomas Fulton and John Townsend, "Simplified Bethe-Salpeter Equation for Positronium," Annals of Physics [N.Y.], volume 82, p. 501 (1974). [25] Cung, V.K., Thomas Fulton, Wayne W. Repko, and Donald Schnitzler, "Complete Reduction of the Fermion - Antifermion Bethe-Salpeter Equation with Static Kernel I," Annals of Physics (N.Y.), volume 96, p. 264 (1976). [26] Cung, V.K., Thomas Fulton, Wayne W. Repko, Alan Schaum, and Alberto Devoto, "Complete Reduction of the Fermion - Antifermion Bethe-Salpeter Equation with Static Kernel II," Annals of Physics (N.Y.), Volume 98, p. 516 (1976). [27] Repko, Wayne W., "Exact Relativistic Hamiltonian Arising from the two Fermion Bethe-Salpeter Equation with a Static Kernel," Unpublished report prepared for Aspen W0rkshop on Quantum Electrodynamics (1976). This thesis makes use of results in [26] and [27] as the starting point for this numerical investigation of the Bethe-Salpeter equation with a static kernel and harmonic oscillator potential. The review article of H. Crotch and D.R. Yennie, referred to in the text, is [28] Grotch, H. and D.R. Yennie, "Effective Potential model for Calculating Nuclear Corrections to the Energy Levels of Hydrogen," Reviews of Modern Physics, volume 41, p. 350 (1969). The above citations mainly deal with background material and serve- to place this work in the context of current elementary particle physics development. Most of the theoretical calculations dealing with the charmonium system make use of the nonrelativistic Schroedinger wave equation with a potential of the form V(r) 8 ar + b/r. The most extensive treatment of this standard model is [29] Eichten, E. and K. Gottfried, T. Kinoshita, K.D. Lane, and T.M. Yan, "Charmonium: The Model," Physical Review, third series, volume D17, p. 3090 (1978). [30] Eichten, E. and K. Gottfried, T. Kinoshita, K.D. Lane, and T.M. Yan, "Charmonium: Comparison with Experiment," third series, volume D21, p. 203 (1980). In these articles the authors make a great many predictions for the charmonium system involving mass spectra and various types of decay 288 rates for all of the allowed states. One item that hampered this earlier work was the incorrect identification of the IS; states. The radiative decay rates predicted by these authors are also incorrect, being off by a factor of two or three from the experimentally determined decay rates. In the following work, Hoestler and Repko tried to see how a proper treatment of relativistic effects would affect a charmoniumrlike system. [31] Hoestler, Levere C. and Wayne W. Repko, "Bethe-Salpeter Equation with Instantaneous Harmonic Oscillator Exchange," Annals of Physics (N.Y.), volume 130, p. 329 (1980). In [31] they assumed a harmonic potential and calculated the spectra and P((3095) +'e+ + e—). Using these two quantities they determined, in exactly the same way as that described in chapter 2, the quark mass and the harmonic coupling constant. The convergence for an initially purely linear potential (which occurs when n 8 o) to a purely harmonic potential level structure (limiting case for m‘+ on) was shown to be very rapid. They made no attempt to determine wavefunctions or other quantities that could, in principle, be calculated for this system.- In our method of solution we choose to make use of the fact that a limiting form of the equations possessed Airy functions as their solutions. To make use of this fact we needed accurate values of these functions. To accomplish this we made use of the work of Yudell L. Luke [32]. [32] Luke, Yudell L., "Mathematical Functions and their approximations," Academic Press, New York (1975). The method used in this book to determine the Airy function is to expand IBessel functions of orders f 1/3 and f 2/3 in a series of Chebyshev polynomials. The appropriate coefficients are found in [32] on pages 351 and 352 (for the t 1/3 cases) and on pages 354 and 355 (for the f 2/3 289 cases). The definitions and properties of the Chebyshev polynomials used here are given on pages 453 to 464 of [32]. The recurrence relations used in the Airy function computations are given on pages 453 and 459 of [32]. As a check on the wavefunctions at m = o and L + 0 one can compare our results in figures 7 and 8 with those shown in [33] Maurone, Philip A. and Alain J. Phares, "The Linear Potential Wavefunctions," Journal of Mathematical Physics, Volume 21, p. 830 (1980). The method of solutions employed in [33] was developed only recently and can be used to solve recurrence relations that result from series solutions of differential equations involving three or more terms. This method of solution is described in the following articles, [34] Antippa, Adel F. and Alain J. Phares, "General Formalism Solving Linear Recursion Relations," Journal of Mathematical Physics, Volume 18, p. 173 (1977). [35] Antippa, Adel F. and Alain J. Phares, "The Linear Potential: A Solution in Terms of Combinatorics Functions," Journal of Mathematical Physics, volume 19, p. 308 (1978). [36] Phares, Alain J., "The Energy Eigenvalue Equation for the Linear Potential," Journal of Mathematical Physics, volume 19, p. 2239 (1978). [37] Antippa, Adel F. and.Toan Nguyen Ky, "The Linear Potential Eigenenergy Equation. 1: the coefficients Kn (31')," Canadian Journal of Physics, volume 57, p. 417 (1979). and some unpublished Villanova University reports referred to in these articles. The other limiting case of our equations is the three—dimensional Schroedinger equation with a harmonic oscillator potential. This equation is discussed in several places. Two convenient sources for this equation are [38] Fong, Peter P., "Elementary Quantum Mechanics," Addison Wesley, Reading, Mass. (1962). 290 [39] Morse, P.M. and H. Feshbach, "Methods of Theoretical Physics," volume 2, p. 1662, McGraw-Hill, New York (1953). Some of the numerical data used, e.g. masses, spins, etc. of the various charmonium states can be found in the periodically issued Particle Data tables compiled by the Particle Data Group at Berkeley and Cern. The last published compilation is [40] Particle Data Group, "Review of Particle Properties," Reviews of Modern Physics, volume 52 (April 1980). The more recent work dealing with the 130 states as well as the radiative decays of various charmonium states can be found in [41] Gaiser, John E., "Charmonium Spectroscopy from Inclusive Photons in J/w and w' Decays," preprint SLAC-PUB. 2887, invited talk presented at the XVIIth Rencontre de Moriond: Workshop on New Flavours, Les Arcs, France, January 24—30, 1982. [42] Konigsmann, Kay C., "J/w Radiative Transitions to Pseudoscalars," preprint SLAC-PUB. 2910, invited talk presented at the XVIIth Rencontre de Moriond: Workshop on New Spectroscopy, Les Arcs, France, March 20-26, 1982. Reference [41] includes a compilation of theoretical models and their predictions of the charmonium system properties along with the more recent Crystal Ball experimental results. [41] is the only source that I know that contains information on the radiative decay rates of the various triplet P states. This work explicitly assumes the existence of quarks as constituents of elementary particles. Even though the quark model is discussed in many books and articles, e.g. [l], [2], [3], [7], [29], and [30] above, the only experiment giving any positive indication for the existence of free quarks is that by LaRue, et. a1., [43] La Rue, 0.3., and W.M. Fairbank and A.F. Hebard, "Evidence for the Existence of Fractional Charge on Matter," Physical Review Letters, volume 38, p. 1011 (1977). 29] Additional information on the quark model can be found in R.P. Feynman's book [44] Feynman. Richard P., "Photon-Hadron Interactions,” H.A. Benjamin, Reading, Mass. (1972). In particular the ratio (e+ + e' +|hadrons)/(e+ + e- +.u+ + u') is calculated in lecture 35 of this book. A derivation of the radiative decay width given in appendix II.4 can be found in the book by Sakurai. [45] Sakurai, J.J., "Advanced Quantum Mechanics," Addison-Wesley, Reading, Mass. (1967). The two-gluon and two-gamma widths far the 1150, 2130, l3Po, and l3P2 states were computed from equations given in [46] Bakbieri, R., R. Gatto, and R. Kogerler, “Calculation of the Annihilation Rate of P-wave quark anti-quark Bound States," Physics Letters, Volume 608, p. 183 (l976).