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Page i Acknowledgement Page ii Table of Contents Page iii List of Tables Page iv Abstract Page v Chapter 1. Introduction Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Chapter 2. Derivation of the model equation Page 11 Page 12 Page 13 Page 14 Page 15 Page 16 Page 17 Page 18 Page 19 Page 20 Page 21 Page 22 Page 23 Page 24 Page 25 Page 26 Chapter 3. Derivation of differential equations Page 27 Page 28 Page 29 Page 30 Page 31 Page 32 Page 33 Page 34 Page 35 Page 36 Page 37 Page 38 Page 39 Chapter 4. Asymptotic solutions to the model equation Page 40 Page 41 Page 42 Page 43 Page 44 Page 45 Page 46 Page 47 Page 48 Page 49 Page 50 Page 51 Page 52 Page 53 Page 54 Page 55 Page 56 Chapter 5. The conclusion Page 57 Page 58 Page 59 Page 60 Page 61 Appendix A. Dirac matrices: Definition and identities Page 62 Page 63 Page 64 Appendix B. Derivation of Green's perturbation solutions Page 65 Page 66 Page 67 Page 68 Page 69 Page 70 Page 71 Page 72 Page 73 Page 74 Page 75 Page 76 Page 77 Page 78 Page 79 Page 80 Page 81 Appendix C. Asymptotic forms of Green's perturbation solutions Page 82 Page 83 Page 84 Page 85 Page 86 Page 87 Page 88 Page 89 Page 90 Appendix D. Asymptotic boundary conditions Page 91 Page 92 Page 93 Page 94 Page 95 Page 96 Page 97 Page 98 Page 99 Page 100 Appendix E. Hypergeometric series and associated legendre functions Page 101 Page 102 Page 103 Page 104 Page 105 Page 106 Page 107 Page 108 References Page 109 Page 110 Biographical sketch Page 111 Page 112 Page 113 
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AN APPROXIMATE VERTEX AMPLITUDE FROM THE SCHWINGERDYSON EQUATIONS OF QUANTUM ELECTRODYNAMICS By RUBEN A. MENDEZPLACIDO A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1988 OF FLORIDA L!Blf !; ACKNOWLEDGEMENTS I would like to express my sincere thanks to all who have helped me. I would specially like to thank Dr. Arthur Broyles for all the help, encouragement and support he always gave me, to Dr. H. S. Green for creating the mathematical foun dations upon which this work is based, and to my wife and family for the support they gave me during this long pursuit of knowledge. I would also like to thank Dr. Fernando Cesani for the leave of absence he so skillfully obtain for me from the University of Puerto Rico at Mayaguez. TABLE OF CONTENTS ACKNOWLEDGEMENTS ...... .................... LIST OF TABLES ........ ...................... ABSTRACT .......... ......................... CHAPTER I II III IV V APPENDIX A B C D E Page . . iv . .. iv * . V INTRODUCTION ................... DERIVATION OF THE MODEL EQUATION ......... 21 The SchwingerDyson Equations ... ............ 22 Approximations to the SchwingerDyson Equations . . 23 The Model Equation ....................... DERIVATION OF DIFFERENTIAL EQUATIONS ....... 31 Definition of variables in asymptotic regions ........ 32 Derivation of Differential Equations .............. 33 The Model Equation in the asymptotic region ....... ASYMPTOTIC SOLUTIONS TO THE MODEL EQUATION THE CONCLUSION ...... ................... DIRAC MATRICES: DEFINITION AND IDENTITIES . . DERIVATION OF GREEN'S PERTURBATION SOLUTIONS ASYMPTOTIC FORMS OF GREEN'S PERTURBATION SOLUTIONS .... .............. ASYMPTOTIC BOUNDARY CONDITIONS ....... HYPERGEOMETRIC SERIES AND ASSOCIATED LEGENDRE FUNCTIONS ..... ................ REFERENCES .......... ........................... BIOGRAPHICAL SKETCH ....... ...................... . 1 11 11 16 17 27 27 30 36 40 57 62 65 82 91 101 109 111 LIST OF TABLES Asymptotic Forms of Differential Equations ............. .39 Asymptotic Solution to the Model Equation .... .......... 54 Perturbation and Asymptotic Solutions in the Overlapping Region. 55 Perturbation Solutions to the Model Equations ........... ..75 The Perturbation Functions u, U1, u2, v3 and e as Functions of the Variable y for x = 1 and k2 = 0.1 .... ............ 77 The Perturbation Functions u, U1, U2, v3 and e as Functions of the Variable y for x = 10 and k = 0.1 ..... ........... 78 The Perturbation Functions u, u1, u2, v3 and e as Functions of the Variable y for x = 103 and k2 = 0.1 ............. .79 The Perturbation Functions u, U1, u2, v3 and e as Functions of the Variable y for x = 1010 and k2 = 0.1 ... .......... .80 The Perturbation Functions u, u1, u2, v3 and e as Functions of the Variable y for x = 1020 and k2 = 0.1 ... .......... .81 Table of Useful Integrals ...... .................. .88 Asymptotic Forms of the Perturbation Solutions .......... .90 Expansions for PI'(z) ......... ................... 105 Expansions for e2 Q'(z) ...... ................ ...106 Behavior of P/(z) and Q"(z) at the Singularities ... ....... 108 Table 31 Table 41 Table 42 Table B1 Table B2 Table B3 Table B4 Table B5 Table B6 Table C1 Table C2 Table E 1 Table E2 Table E3 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy AN APPROXIMATE VERTEX AMPLITUDE FROM THE SCHWINGERDYSON EQUATIONS OF QUANTUM ELECTRODYNAMICS By Ruben A. MendezPlacido August 1988 Chairman: Arthur A. Broyles Major Department: Physics An approximate set of invariant functions for the dressed vertex amplitude was found. An asymptotic solution to the unrenormalized SchwingerDyson equations of Quantum Electrodynamics was obtained which joined smoothly with the solutions found by a perturbation technique. The photon propagator is approximated by its form near the mass shell. The vertex equation was separated from higher order members of the hierarchy at the second order in the coupling constant with the aid of H. S. Green's generalization of Ward's Identity. No infinities were substracted to obtain the solutions. The function multiplying the matrix A is found to be dominant everywhere. CHAPTER I INTRODUCTION Before the arrival of quantum mechanics, there were two major problems that seem to point out the existence of flaws in the classical theory of electromagnetic fields. They were the blackbody radiation and the theory of electrons in atoms. The introduction of the finite quantum of action by Planck successfully resolved the first of these problems. The difficulties connected with describing the electron and other entities as "particles" are much more serious. Basically, the nature of the treatment of the electron as a classical particle with finite mass and extent leads to a dilemma. If we assume the electron to be a point without structure, then, in classical theory, the total energy of its electromagnetic field becomes infinite implying an infinitely massive particle. On the other hand, if we assume the electron to have a finite extent, its interaction with the electromagnetic field created by its own charge distribution will produce stresses tending to explode the charge. The quantum theory of the electron and the electromagnetic field was intro duced in the 1920s by Heisenberg and Pauli, Dirac and others. Unfortunately, they were not able to solve the equation that they derived in a manner that was free of infinite quantities. Although the energy of the electromagnetic field diverged as the logarithm of the electron radius in the quantum theory instead of as its reciprocal as it does in the classical theory, a second infinity appeared that is associated with the charge. This infinity appeared in integrals that diverged quadratically as the upper limit increased.' According to Dirac's hole theory,2 the creation of an electronpositron pair by a photon may be interpreted in the following way. The vacuum consists of an infinite sea of negativeenergy electrons. A photon may be absorbed by this vacuum raising a negative energy electron to a positive energy state and therefore creating a hole in the vacuum. The latter will appear as a positron, so that we obtain an electronpositron pair. Thus the vacuum may be considered as a sort of "polarizable" medium, because it potentially contains electronpositron pairs. A photon may now interact with this polarizable vacuum even if its energy is not sufficient to create a real electronpositron pair. In this case only a "virtual" pair is created and this annihilates soon afterwards. Although the apperance of virtual pairs improved the divergence of the selfenergy, it introduced new problems that do not have their counterpart in the classical theory. With the introduction of virtual pairs, it was found that the polarizability of the vacuum is infinite. The Dirac theory also predicts that, for hydrogenlike atoms, states with the same total quantum number n and angular momentum j are degenerate (same energy). It was noted, however, that the polarization of the vacuum discussed in the preceding paragraph would split this degeneracy. In particular the 22 S112 and the 22P1/2 levels should be separated by a small amount. In 1947, Lamb and Retherford3 made a direct measurement of this separation. They found that the 22S112 is above the 22P,/2 by 1058 megahertz. Actual theoretical calculations using a form of a perturbation expansion of the equations of quantum electrodynamics (QED) of the splitting gave rise to divergent integrals. Bethe4 circumvented the problem by simply limiting the range of integration over the divergent integrals. Bethe reasoned that at energies larger than the rest energy of the electron, relativistic effects become very important and have to be included. These inclusions, at the end, will amount to the introduction of a cutoff in the integration limits. The final results were, therefore, dependent on a cutoff parameter. This dependency was logarithmic in the cutoff parameter and thus insensitive to the actual value of the parameter. This parameter could be described as the maximum energy of a photon that was emitted and absorbed by the atomic electron of hydrogen. He set this maximum energy to be the rest energy of the electron. With this technique, Bethe arrive at approximately the value measured by Lamb and Retherford (the socalled "Lamb shift"). The fundamental equations of QED can be written in different forms. One way of doing this is by writing the perturbation expansion in a series of powers of the electron charge. This expansion can be written in terms of propagators (Green's function) as developed by Feynman5 and interaction sites at which three "particles" can interact. Schwinger6 and Dyson7 derived an infinite hierarchy of integral equa tions that describes the interaction of an electron with the radiation field. In the future, we will refer to this hierarchy as the SchwingerDyson hierarchy as it is usually known. The equations in this hierarchy can be solved by straightforward iteration to obtain the perturbation series. This perturbative approach always leads to infinite integrals. Numerous attempts were made to eliminate the divergencies in a rigorous manner in the period following the invention of quantum field theory around 1925 until after World War II. This problem was solved about the time of the Lamb and Retherford experiments, by the development of the renormalization theory. The idea of renormalization can be interpreted as a rearrangement of the pertur bation series so that the new series converges and has finite terms. This rearrange ment amounts to expanding the electron and photon propagators and their join ing vertex around the mass shell (i.e. for values of the photon energymomentum near zero and electron's square energymomentum near its experimental square mass). The infinity associated with the charge is removed by rescaling the prop agators, wave functions and vertex parts',9 with the introduction of the socalled renormalization constants Z1, Z2 and Z3. They are associated with the vertex function, electron propagator and photon propagator respectively. Attempts to calculate these constants using perturbation theory have led to the conclusion that these constants must be infinite! The renormalization theory emphasized the covari ant aspects more strongly. Schwinger, Tomonaga'0 and Feynman," independently developed the first Lorentz covariant schemes designed to eliminate the divergencies in a more acceptable manner (the renormalization theory). Renormalization theory has enjoyed a remarkable success in the calculation of numerous effects such as the Lamb shift, the anomalous magnetic moment, the hyperfine structure of the hydrogen atom, and other relativistic phenomena. Quan tum electrodynamics has become a model for other field theories to follow. It is important then, to study the underlying mathematical structure of QED in or der to better understand field theories in general. The success of the nonAbelian gauge theories in unifying the electromagnetic interaction with the weak interaction further encourages the efforts to understand and resolve the ambiguities of QED. The current theories of electroweak and strong interactions are based on the same underlying mathematical structure. If the mathematical techniques (perturbation theories) used in QED were com plete and satisfactory theory, it would be a mathematically rigorous and logically consistent structure which allows at least in principle the calculation of all radiative processes. Because calculations have been based on the rearrangement of infinite series that may not remove all of the infinite quantities (Z1, Z2, and Z3 for exam ple), this cannot be said without reservations of the theory in the present state of development. There are two major points of view regarding the infinities found in QED. One states that there is something fundamentally wrong in the foundations of the theory. The other assumes that all the difficulties arise from the use of inadequate mathematical methods in solving the fundamental equations of the theory. The first point of view states that the solution to the problem must be found in a modified theory or in a completely new one (if possible). The arguments against this radical approach are found in the actual success of renormalization theory in calculating such effects as the Lamb shifts, the anomalous magnetic moment of the electron, and the finestructure constant. In order to get a flavor of the kind of agreement that can be accomplished with this theory the following results are presented:12 1. Lamb shift in hydrogen bEep = 1 057 845 (9)knz, (1.1) bEth = 1 057 849 (11) kHz (1.2) 2. Electron's anomalous magnetic moment aep = 1 159 652 200 ( 40) X 1012 (1.3) ath = 1 159 652 460 (127) x 1012 (1.4) 3. Fine structure constant aeXp = 137.035 993 (5), (1.5) a' = 137.035 989 (3), (1.6) where the subscripts "exp" and "th" stand for experimental and theoretical values. The quantity enclosed in parentheses represents the uncertainty in the final digit of numerical value. The value of a' is based on the very accurate measurements of 2e/h (0.03ppm) by the ac Josephson effect. As can be seen from the results given previously, theoretical calculations using renormalization theory match the experimental results on the order of O.1ppm. No other theory, to our knowledge, can claim such a success. If we accept this remarkable agreement between theory and experiment as evi dence that the fundamental equations are correct, we are led to favor the conclusion that the infinities that arise in the theory come from an inadequate way of solving the fundamental equations. The study of other model field theories has led to the suspicion that the perturbation expansions after renormalization become series that are only asymptotically convergent. This has led to a search for a better technique of solving the equations. The technique should, in principle, allow the calculation of the bare (noninteracting) mass, charge and constants ZI, Z2 and Z3. In this search, GellMann and Low13 sought to demonstrate that the renor malization constants are infinite. They stated that, although they could not rule out the possibility of infinite coupling constants, its was possible to isolate a nec essary condition for which the vacuum polarization is finite. In a long series of papers, Johnson, Baker and Willey,1417 extended their work and showed that if an eigenvalue condition is satisfied, then all renormalization constants in QED can be finite. The eigenvalue condition defines a function that is the coefficient of the logarithmically divergent integral appearing in the photon propagator calculated in massless QED due to only those graphs with one closed fermion loop. This eigen value condition is expressed in terms of the bare coupling constant. Adler'" then showed that the zero, if it exists, must be an essential singularity of this function with all its derivatives zero at the singularity. The existence of this function has not been proven. Nevertheless, it is possible to speculate that the existence of an infinite order zero will never be seen in any finite order of perturbation expan sion. These results generate great interest in a number of people'9 for finding, in a nonperturbative manner, the solutions to the SchwingerDyson equations. From them, Z1, Z2, and Z3 could be identified and it could be determined whether or not infinities are inherent in the theory. Following this approach, H. S. Green, J. F. Cartier, and A. A. Broyles started the long term project of solving the SchwingerDyson hierarchy of equations using a nonperturbative method. The equations in the hierarchy are written in terms of momentum space variables, that is, in terms of Fourier transforms. Figure 1.1 shows a general block diagram that can be use in describing the state of development of the project. Each block represents an important step in the solution of the hierarchy. Starting with the SchwingerDyson hierarchy of equations (block 1), H.S. Green, J.F. Cartier, and A.A. Broyles2(herein referred to as Ref. I) were able to determine the unrenormalized electron propagator using an approximate form of the photon propagator. In order to obtain this solution, two approximations were made (block 2). Their first step was to truncate the infinite hierarchy with the aid of Ward's identity.21 Ward's identity relates the vertex function to the next lower equation in the hierarchy (i.e. to the electron propagator) in the limit of zero mo mentum transfer to the electromagnetic field. The second approximation replaced the photon propagator with its massshell form. With these approximations, they were able to calculate the electron propaga tor over the entire range of the variables upon which it depends (block 3). They found that in order to have a solution, the bare mass of the electron must be set equal to zero and a particular gauge must be chosen, the socalled Landau gauge. These results are in agreement with the findings of Baker and Johnson.17 As an additional result of the determination of the electron propagator, the value of the renormalization constants Z, and Z2 were calculated. They conclude that within the boundaries of the approximations made, Z, = Z2 = 1. No infinities appeared in obtaining the solutions. Start 1 2 SchwingerDyson Initial Approximations hierarchy 1. Truncation of hierarchy of equations 2. Photon Propagator 9 Go to next level in SchwingerDyson hierarchy A I I Calculate new I electron propagator I I I Model Equation] I Solve vertex equation o Euaton 1. Perturb. Soln. 7 1 I Calculate new photon IEquation I propagator using new 1 1. Asympt. Soln. vertex function 8 Decision Box Iterate? Next level in hierarchy? End? 1 r Figure 1.1 Block Diagram of iteration method for solving the SchwingerDyson hierarchy of equations. In order to determine the photon wavefunction renormalization constant, as sociated with the photon propagator Z3, it is necessary to devise a more accurate form for the photon propagator. The SchwingerDyson equation for it involves, however, the electron propagator and vertex amplitude. The electron propagator found in Ref. I is available, but a vertex amplitude must be found. It is possible to demonstrate, by using symmetry and invariance arguments, that this amplitude can be expressed in terms of eight scalar functions of the electron and photon mo menta. Some progress has already been made on this subject. Using the same type of approximations which led to the determination of the electron propagator, in Ref. I, and a generalized form of Ward's identity derived by H. S. Green,22 J. Cartier et al.23'24 (herein referred to as Ref. II), determined all eight scalar functions needed to define the vertex function in the perturbation region. Since a direct analytical solution to the vertex amplitude is, at the present moment, not feasable (block 4), a somewhat different approach was used. Making some reasonable assumptions about the behavior of the vertex function near the mass shell, an approximate equation to the vertex equation was derived. We call this equation the model equation (block 5). The solutions to this equation will indicate the functional form of the vertex amplitude and, in principle, could be used as starting solutions in a numerical it eration procedure. Also, a parametrized version of these solutions can be use in a minimization procedure. The model equation was solved analytically by an iteration procedure and tested to see if the the approximations made were valid. The solutions found proved to represent reasonably well the vertex function near the mass shell and to be con sistent with the Ward Identity. Once again, no infinities were incurred in obtaining these solutions. The next logical step to follow is to find the form of the vertex function in the asymptotic (i.e. large momentum) region and which join smoothly with the perturbation solutions found in Ref II (block 6). This is the problem that we address in this work. The knowledge of the form of the vertex function in the asymptotic region will open the possibility of determining the photon propagator, the renormalization constant Z3 and thus the bare charge of the electron (block 7). Finally, to show that the procedure is selfconsistent, at least one iteration is necessary. With the new vertex function and photon propagator a recalculation of the electron propagator is possible. The present stage of the project, including the present work, is delimited in the block diagram by the broken line. The present work is arranged in the following way. Chapter II gives a general description of the method used in truncating the SchwingerDyson hierarchy of equations. In this chapter, we also show how to convert the truncated hierarchy into a set of coupled differential equations and the derivation of the model equation. In Chapter III, the model equation is transformed into a set of nine scalar differential equations. Also, a simplified version of these differential equations is found to be valid in the asymptotic region. In Chapter IV, the solutions to these equations are found. In the last chapter we summarize and discuss the nature of the solutions. It is felt that this work can make a positive contribution to a better under standing of the interaction between electrons and photons, and that it will shed some light on the question: Is QED a finite and mathematically sound theory? In addition, the techniques developed may prove to be useful in the solution of other problems in QED. CHAPTER II DERIVATION OF THE MODEL EQUATION 21 The SchwingerDyson Equations Quantum Electrodynamic Theory (QED) deals with the description of the interaction of light and matter. Such interaction in its most elementary stage studies the electronphoton interaction. This description is best handled mathe matically using the concept of a propagator for each of the involved "particles" and an interaction site called a vertex. The relationships between these propagators (Green's functions) and the vertex are expressed in a hierarchy of equations known as the SchwingerDyson equations. To express formally these equations it is nec essary to define the meaning of a propagator for each of the "particles" and the vertex function. The probability of finding an electron at some point in spacetime given that it was at a different point in spacetime can be computed from its Feynman propaga tor or Green's function. This propagator satisfies a differential equation analogous to the wave function equation. In coordinate space the wave function of a nonin teracting electron I' satisfies the differential equation (i mo)4_ = 0 (2.1) where W  y' &/&x, and y are Dirac matrices. The noninteracting photon wave equation is written as E12A" = 0 (2.2) where [2 is the D'Alambertian, defined as []2 = g AV (2.3) and +1 if V = 0; g9 = 0 if IL uV; (2.4) 1 if [I =V # 0. The electron propagator satisfies an analogous equation (i MO) So(7',) = 4(_ _) (2.5) Taking the Fourier transform of Eq.(2.5) in momentum space it is found that the noninteracting electron propagator is given by sp) 1m0 (2.6) The noninteracting photon propagator satisfies a similar equation [2Do(x ;') = i64( ;,), (2.7) and the Fourier transform of the photon propagator yields 1 Do(q) j" (2.8) These two solutions provide a complete description of the electron and photon when there exists no interaction between them. When interaction is allowed, an infinite hierarchy of nonhomogeneous integrodifferential coupled equations arises. The exact solutions of these equations were not known. This hierarchy of equations was formulated in their pioneering work by Schwinger6 and Dyson7 and was known thereafter as the SchwingerDyson hierarchy. Using the notation of Bjorken and Drell2"'26 these integral equations appear as s(P) = So + S(P) E(P) !S(p) (2.9) or equivalently _l(P) = so, (P) _Z05) (2.10) where :(p)_ =2r F(p, q)S(q)Dj,(3p q)_yI d4q (2.11) DIw(k 2) = (Do)g,(k2) + (Do)1,(k2) l1" D#,(k2) (2.12) ll"(k2) 2) Tr S(q) E '(q, q +) S(q + k) d4q (2.13) (27) I (p, q) + A A (p, q), (2.14) and A"(p5,q) ( / D I(k2)(pp k)S(p k) (2r)4JI xr_,(p k)_(q k)r,(q k, q) d'k k)IS( k Y"(2.15) x rjq k., k._j)S(p k.)r.( k, q,) x d4k"...d 4 k(27) 4n + . . The zero subscript follows all bare quantities, that is, those functions or con stants which are associated with noninteracting particles. An overbar is used to represent fourvectors and an underline to represent matrices. Here eoA_" repre sents the sum of all possible threeexternalpoint, nodeless Feynman diagrams. A node is a bare vertex that if removed, leaves the diagrams separated into at least two unconnected parts. It is possible to relate r'" to a fourpoint diagram to either a dressed electron, or a dressed photon lines. These diagrams can be separated out of the fourpoint nodeless diagram eoAv leading to the equation ie2 p+ k) = I' + ( D A(P 0)2"S) x { ( )(+k)( k)}I d4q (2.16) The next step in the hierarchy relates the fourpoint diagram to the fivepoint and so on. In addition to the SchwingerDyson integral equations, there exist some relations between amplitudes whose numbers of external points differ by one. The lowest one relates the vertex amplitude __ to the electron propagator S, and it is attributed to Ward and Takahashi.21 The one relating the fourpoint amplitude EAI and the threepoint amplitude h_ was derived by H.S. Green.22 Ward's identity can be expresses as follows: (pp qp)r_: (P, q)=_ 1() (2.17) or in terms of Alt (p, q) (pl, q/,)A,(, 0) = EM !_(P). (2.18) An analogous equation can be written for the fourpoint amplitude in terms of the vertex amplitude and is given by (pAk q )Ev(, ,) =+ k(q\ q) _,( k, + ) (2.19) Av( k, q) A"(p,p + k) and k, E Av(f, k, A pi q) rA(p,p + (2.20) Similar relationships exist for the remaining npoint diagrams. These identities exactly define the longitudinal components of the npoint diagram in terms of the (n1)point diagrams. If we orient the u ih axis along the nth direction of k, then it can be shown that E A, can be written as E'(p, k,q) (p, + kA k )EI (/,kq)(p" + kA qA) (2.21) + kt E.(pk, q) k + Zt(/5, ,q) where EAt' is transverse to both of the attached photon fourmomenta. Substituting equations (2.19) and (2.20) into (2.21) gives E v(P, k,4 = [1_ (P k, q) EA(5, + k)] (p + kv q V) + [A(i k,) rA(p, + k)] ky (2.22) +E v, k q) where kEt v (p, k, 4) 0. (2.23) Equation (2.20) can also be written in derivative form if we orient the coordinate axis so that the vth axis lies along k. Then dividing by k" and taking the limit as k' vanishes gives F.;0, (pq) ___ D(,q ) (2.24) Similarly, letting kx approach (qA pA) in equation (2.19) gives W )(p, q) Ar" (p5' q) (.5 Ef'v(/, q _/p, l)= op, Oq, 2.5 Using similar arguments, it is possible to separate _P into longitudinal and trans verse parts so that , )=1() S'(q)] V' + E"(P, q) (2.26) where 417' = 0 or explicitly F(+ k,i + k) = ['(q + k) i(!+ k)] W(A pA) (2.27) ( /p)2 Using equations (2.16), (2.21) and (2.27) leads to the following equation for the vertex amplitude: .2 + ()~ +k r('p+ k) =7+ (21 D ()_S( q, +)_s(+) (27r) )j q)() [r(( + 1, + k) + [() (P) )] Vq( _PV)] 2 x [(qqki3k p)2'p)s [Ip( +,p+,:v (q,p)] kA (2.28) + [r'(2q Pp+k) 2p +kq)] (p Vq) + V_ (q,. q, P + k)} d4q. 22 Approximations to the SchwingerDyson Equations Although equation (2.28) is exact, it involves another equation in the hierar chy through the fourpoint amplitude EA. It is possible to close the hierarchy of equations by noticing in equation (2.28) that the integrand has a maximum at the pole (1 q) 0 and therefore it is reasonable to approximate Etv by Ev(q,p q,p+ q) E t(q, 0,P + q) (2.29) Since A\ is transverse to k, and v is transverse to (q q), Etv can be written as Ev = EA k k' E" qn)(p qU) (2.30) k2 (p_ q)2 Substituting equations (2.24) and (2.29) into (2.30) gives A 0 X (p ' k.,k(p q" i9 ~+ k) "+ [g.,g. 2 g. g.7 (p q)2  (2.31) which can be used to eliminate E\' in equation (2.28). This equation coupled with equations (2.9) to (2.16) could be solved in principle for ,A S and D,,. It is possible at this stage to solve equation (2.28) for the component of pA(p, + k) transverse to kA and then use equation (2.26) to construct F. Tak ing the transverse components of rA will eliminate all terms proportional to kA. Substituting equation (2.31) into (2.28) gives xk) + (i( J, ) s(+k)(q k) [ (2q P, P+ k) _F'(q, 2P + kq)] (p"qe) + aO ( q+k)(p, q)(p" q") t( q+k) }d4q. (2.32) The final goal would be to solve this equation with Eqs.(2.9) to (2.13) self consistently. To this end, it will be advantageous to solve a much simpler equation obtained by making some seemingly sound approximation. The solutions to this equations will shed light on the form and behavior of the complete solution in both the perturbation and asymptotic limit. The last step in the procedure of reducing the SchwingerDyson equations to a more tractable form was to convert the integral equations into a set of differential equations with appropriate boundary conditions. This method was first developed by H. S. Green in connection with the BethSalpeter equation. It was first used in the study of the SchwingerDyson equations by Bose and Biswas.2" 23 The Model Equation It is possible to recast equation (2.16) using Ward and Green's identities for both terms in the square brackets. Replacing fi by q and making use of Ward's identity gives 1'(mq+ k)S(q + k:I(q + k:, p+ k) rx'(q,q + k)S(q k) 1 aq ) (2.33) 18 Substituting this equation and the expression for EAV( 0, + k) in equation (2.16) and taking the transverse component yields +) +( f J D q + k)d4q (2.34) where E_(, q+ k) =q){E(,*+ k).q+ k)} S'(4 k) (2.35) aqv I Using the renormalization group,26 arguments have been presented to show that the asymptotic form of D,,, is the same as that near the mass shell. One of these arguments is that the contribution of the vacuum polarization to the Lamb Shift is only 27 megahertz out of 1058. The vacuum polarization gives a measurement of the departure of the photon propagator from its mass shell form. It is also observed, once again, that the integrand of equation (2.34) is largest when the argument of D, vanishes. Hence, the photon propagator is approximated by D,() Z3g, + t pq (2.36) where Z3 is the photon renormalization constant. The gauge is chosen to be that found necessary to obtain a finite solution to the electron propagator equation20 with a vanishing bare mass for the electron. In the above reference the electron propagator is found to be given by S'(p) A(p2) + B (2)p (2.37) where 2 r 2 3(2p2)/4,rp( A(!) =m 1 ,(2.38) M2 B(p2) ~ ,(2.39) and a R 1/137 is the fine structure constant. With these approximations it is possible to transform equation (2.32) to a differential equation by taking the D'Alambertian in momentum space (see Eq.(2.3)) on both sides of Eq.(2.34). Us ing the identity a a ln(p ) 2 gv 4 (p' q,)(P. q,) (2.40) 9(p, q,) a(pv q) (p ) )2 (p q)4 it is possible to write Eq.(2.36) as S 4) 1 g t 1 Z3 9ln(5 )2 (2.41) 2 (p q)2 4 D(p, q,) (pv qv) Application of the D'Alambertian to Eq.(2.41) gives [12 DAV(P q1 ZA_2 Z3(9/,0,[]2 ln(p q)2 (2.42) D2, (I) 2~v2 (p q )2 4 Substituting the identities [ 2 q)2 = i(27r)264(p q), (2.43) [2 ln(p )2_ 4 (2.44) (p 2 into Eq.(2.42) gives 02 DAV(p q) = 272 z g 4(_ Z3 ( (9 1 (2.45) z9(pll qA) O9(p" q") (p5 q )2 Therefore, D2I~(p+5k) 27 { w.Z37,, S(P) [rt~ )( ) 1i e ie Z3/ aa1 (2)2 a(p qi) a(pv q ) (p q)2 (2.46) X SM [1,"(q, q+ k)s(q+ )]_S1(q + k) d4q. k ISq  In terms of the tensor FP, Eq.(2.43) becomes r k) e fFA ]t (5,P+ k) jF5(pp+k) (2r)4J a(P,qV) Y(jq)2F2((+))dq = L + GA (2.47) where (2r)2 a q (p q)2 FAd4q (2.48) Applying Y to GA and using Eq.(2.43) we obtain G\ a a F v (2.49) p and hence V[2rA(jj,p~k) =E _FV (P,P + k) + a F"v(p,p +) (2.50) We have define e2 = e2Z3 and e = e2/(27r)2 = (a/7r) where a again is the fine structure constant. It is possible to write Eq.(2.50) in a more symmetrical form as E]2r(Pil,P2) F [ v(PIP2) + L v(,2)] (2.51) where FAv(pi,p2) = S(pl)L(PI,p2)_(p2)Y" + A"A(h, P2) (2.52) and (P (2.53) Here p, and p2 refer to the outgoing and incoming electron momenta respec tively. The tensor F kv has been written in such a way that it supports further approximations. To first order in 5 it is well known that near p2 p2 M2 (i.e. near the mass shell) rA C2 yAZ1 and therefore S(P2) If we as sume that this form is correct we can neglect the derivatives of the vertex function; i "e r 0. Notice also that to this order of approximation Z, Z 1.o Under these approximations it is reasonable to assume that A A 0. This leads us to a simpler, more compact equation which we called the model equation for A and is explicitly written as ]2A(P1,22) = V _A(Pl,2) + (2.54) where A=(Plp2) = _(pl)"A(j ,P2)..(P2) (2.55) We are interested in the solutions to this equation coupled with equations (2.36) and (2.37) in the asymptotic region. These solutions must join smoothly with solutions near the mass shell (i.e. perturbation limit). To solve these equations, it is convenient to write them in term of invariant functions. This can be done as follows: let us define the matrices (see Appendix A) 4', = [WY, 2A~z, = 2 {_A',2.zi} = iAv)u,p'Y (2.56) =5 i021 222 7(2.57) where 60123 = =1 Here the symbols [,] and {, } represent the commutator and anticommutator of Dirac's matrices. One can also define the following vectors and tensors CA Tr [ ] (2.58) A= 1 Tr FA1 A 4Tr 4A 1) (2.59) AV = Tv] (2.60) A V (2.61) / 'P =4 r lyV The tensor equations derived from Eq.(2.54) are D2CA r3 DA, (2.66) []2DX = 2 ADA 20D, (2.67) I2 CA = eDA 2c'0,DaD + 2r8VA2aPDA (2.68) Z2 CA T 2 e0[]LOD~ipp 2 0auTh a D p~ 2=Ap&2 D (2.69) If we write h'i + A1 s(2+ A (270 _Sp)(p5 A ) _() (/32 A ) (.0 where A = Aj (p ), A2 A2(p ), and also define D=(2 A)(2A A), (2.71) then equations (2.58) to (2.61) become D 0 2 Tr[ (1,2)(/2 A2)(1 A1)1/D  p A A P0., [(A1A2+ p2)Ca+ (Aip+ 22 )C PAO ( 2)CapF12,]/D \ (2.72) = Tr[FA(pip2)(2 + A2), (A + AA)]/D [(A2p, + Ajp2A)CA + (A1 A2 P1 "p2)CA (PlP2 + P2,Pl')Cv' + (A2Pl + AIp2)CA, C p A (plp2)C, vp]/D, (2.73) DAV =1 Tr [:A(I, 2) (P2 + A2)_V(A, + A1)]/D = [(PliP2v p21AP1,)CA + (A2plv AIp2v)C" (A2p1 AlP2A)C (+ + P1 "p2)CI , (PiP + P2vpl)C" + (PltP + p2,pi')C (A2P1 + A+ p2)C,,P] /D (2.74) DAVP =Tr [!:A (h5,hi) (P2 + A2)1_7_VP (i + A,)]/ID = [(P1iP2v P2pPli)CA + (PlvP2p p2,p1.)CA + (PlpP2A P2pP1,L)C + (A2p, + AIp2p)Cvp + (A2P1v + Alp2)CA + (A2pl. + Ap2.)CA + (AiA2 + i . p2)C,, + (PIAP' + P2pP')C\,,, + (PlP2 + P2,P')C\, + (,piP + )CA ] /D (2.75) In deriving the above expressions, several matrix identities were used (see Appendix A). It is possible to express these tensors in terms of scalar functions using the following reasoning. To construct the vector function CA there is only one vector at our disposal, 5" since the vector k' does not appear in the transverse part of fA. Therefore, the most general form of CA is CA = upAx (2.76) To construct the tensor CA we have at our disposal only the tensors, & p and P2 pPA. Hence, the most general form for CA is CA v6  (=2PpV +A UlPp)A (2.77) In constructing the general form of C we notice from Eq.(2.60) and the antisym metry property of '_y under the exchange of the indices that CA = CV The only tensors at out disposal to construct C" are: pj PlA'8, P2AS, P2vSA and Ppp2jA. Thus, the most general form of CA which satisfies the antisymmetry relation is CAV = (v2pI, + vlP2,)SA (V2PlA + v1P2A) + Pp2vP P P (2.78) Similar arguments can be used to conclude that CA p = V3 [(PlvP2p P2vPlp)5 + (PlpP21A P2pPlp)S + (PtdPiv P2pP1P)P With these definitions the vertex amplitude takes on the form t(Pl, P2) ._ A V(2,pfj2 i'lP2) A 2 +pU (pji, ,l. 2 [ 1 2 P2) ](p2 2 .l +[/I, A]Vl (pl2,' j2 p, E i A fPlvp2pv3( f2,f P2) (2.79) (2.80) A tilde is used over variables to specify transverse parts to k. Explicitly, the trans verse part of pA is given by p pA 9 kA. Also the tensors DA, D', DA, and DV~P can be written in terms of scalar functions as follows: DA = RA, (2.81) DAP = 6A + (R2Plp + Rlp21,)PA ,(2.82) D.= (P2vSA Pp)6,A)Sl + (PlvbA l)vS + R3(PluP2v P21APlv )PA (2.83) DA. = 3 [(PlvP2p P2v'Plp) + (PlpP2p P2pPl)SA + (PlAP2v P2pPlv)SA] (2.84) Substituting these expressions into equations (2.72) to (2.75) gives the following results for the scalar functions R,R1,... etc.: R ={(AIA2 Il. p2)u + (AIp2 + A2pl)" (UIP2 + U21) +(A, + A2)v+(p5 1 P2)v2 (pi2 h .2)v1 [Pp2 P1/)2]U3 /D, (2.85) R1 ={Aiu + AlA2U1 + pU2 + v + (A1 A2)vI (A2P, + Alf1 P2)U3 (12 PI" P2)v3}/D (2.86) R2 + 2U +p pu1 + A1A2u2 + v + (A1 A2)v2 (A152 + A2p1 + (P 1 "P2)V3}/D, (2.87) R3 ={u AlU2 A2u1 + (A1A2 +Pl "f'2)u3 + (v2 vI) + (A1 + A2)v3}/D (2.88) S ={(AIA2 +P1 P2)v + (A2P21 + A11. 2)v2 Alp2 + A2P/1 2)vI + (pIp2 (P5i1/2)2 )u3 /D, (2.89) S ={Alv + A1A2v1 5v2 (A2 I + A151 .P2)v3 }/D, (2.90) 26 S2 A2v + AA2v2 2 + (A1 + A215 .212)V3}/D, (2.91) S3 = vA2V+AV2+(AA2+ P A P2)v3}/ D. (2.92) CHAPTER III DERIVATION OF THE SCALAR DIFFERENTIAL EQUATIONS 31 Definition of Variables To obtain the scalar differential equations resulting from the model equation, let us define the following variables which will prove to be very useful in describing the asymptotic region. Let V2P2; (3.1) 21 z2 Y = 2l. ( Z zl (3.2) P22 z=y+ yZ1; 1 = y 2 (3.3) The inverse transformations are p2_ 2. 2 = X_ jj2 2 (3.4) 1 =f2 +2p P k+ =xz; T 2 2p k+k (3.4) so that p2 =xyk2 ; p.k =lIx Pl1fP2 p2 k2=xy2k2 (3.5) If both p and p2 are timelike or spacelike 4vectors, the variables x and y are real. With these restrictions, the variables y or z are always greater than one. Using this choice of variables, the momentum space is divided into three regions: perturbation region, x M2 y 1, k2 0; inner asymptotic region, X/k2 > m2 y 1 ; (3.6) outer asymptotic region, x/k2 > m2 y 1 28 In terms of these variables the differential operator a,, /Op can be written as a a ,1, = Lt, + M, a (3.7) Ox LO where LA = ~2{1y Au /y2 1 kA1 (3.8) M1j Vy21Ppty kj (3.9) If we have a vector function WA of the form W' = w(x, y)5', then O,LWA (Oaw)j5A + wS (3.10) and []2WA =pA[]2w + 26Aw, (3.11) where []2 = aP49,, is the D'Alambertian operator. To specify the derivatives of w(x, y) with respect to the x and y variables we use the notation Ow 02w W Ox' W = OX2 Ow a2w WY =y wyy =y2, O2w wY OxOy (3.12) Using these symbols, it is found that the D'Alambertian operator acting on a scalar function of the variables x and y gives k2 4k2(y2 1) LI'W(x' y) =4(xy +2~ +4(2y )wX + Y x X +4 Y2 (y2l)WY_4(y2 _1)wXY" (3.13) 4 Yx2 x I With the help of Eq.(3.10) it is easy to see that E 4(xy + 4(3y 2 4k2(y2 1) Eli2WA (x y) =4xk2)w* + (y )W + x y k2 2(y2 1) y2 1)W(3 +4 Y2 x WY4 4( w~ (3.14) when acting upon a vector function, since from Eqs.(3.7), (3.8) and (3.9) gives S=w 2Y (Y2_ 1)l ] (3.15) To distinguish between the two differential operators, we used the subscript s or v to indicate the operator used in Eq.(3.13) or (3.14), respectively. Explicitly, L12=4(xy k 252 + 4(2y X) 4k2(y2 1) a2 [ k2 2(y2_1)1a + x2 ayX2+ ly xJ Oy 4(y2 1) (3.16) axay and 32 2 S4(xy k) k X2 + 4(3y ) ~4k 2(Y_1) a2 k2 (y2 1)a + 2 y2 + 4 y x2 x 32 4(y2 1) 92 (3.17) To recast the nine coupled differential equations that arise from the model equation in terms of the new variables, let us define the operators h and 1 by L9 (y2) ( 2 Y x } (3.18) and also :x x (3.20) NoiX tay Notice that with this notation S= Ah. (3.21) It is easy to show that in terms of these differential operators i= paz+ + P2Az = pmh + k1 (3.22) since z1i,+ zf = h (3.23) z f+ zf = i . 32 Derivation of Differential Equations At this moment we turn to the problem of transforming Eqs.(2.66) to (2.69) into scalar differential equations. It is easy to see that Eq. (2.66) can be written in terms of the scalar functions u and R, as l1'u = 3ER (3.24) To obtain the scalar differential equations from Eq.(2.67) it is observed that this equation is equivalent to two coupled differential equations. To this end we define the vector function E A = ep. With this definition, equation (2.67) becomes []2C = 2e [DA 01E"] (3.25) coupled with E] 2EA = vD (3.26) It is important at this moment to realize that from Eq.(2.67) .CA =0. (3.27) This follows from a direct application of 0A to C,\. The proof follows: a0 2C C,=[]2ap.0 ,'= 2e [&UD A_ 0,[I20a"DA] = 2e [0,D 0"D] =, 0 A D/]=0. Assuming that cl CA is finite at j52 = 0, and p2 = cc, then the only choice possible is the result stated in Eq.(3.27). In addition to this argument, we find that the tensor CA is defined in terms of the scalar function v. In the asymptotic region, the largest contribution to CA comes from this function. It is found (see appendix D) that in the asymptotic region the function v approaches a constant, thus supporting the hypothesis that aACA 0 to leading order in the variable x. If we apply in general, a, to CA we obtain a~~~ eC = AaV+ 'UP + ulP2,) + (Ui + U2 )i3As + [lpo'U2 +p2,ao uii] (3.28) Evaluating Eq.(3.28) at a = y gives OLC91 =DA v + 5(ul + u2)pA5 + pA[l1.U2u +.P2.Ul]= 0 (3.29) or using Eqs.(3.21) h(v) + 5(ul + u2) + h .L1u2 +P2 "Elul = 0. (3.30) Applying O, to Eq.(3.28) gives 2 A ~A I [] el [_LJ pv + 2(ul + us)1 + 2 [p1lM&?22 + p2j&Aui1] E2 s + 0]72), [pI1 SU2 + p2 ] + 2(.u1 + .,U2)P\ (3.31) Using Eq.(3.21) we obtain []2C1 = SA [Dv + 2(u1 + u2)] + PI P [2U2 + 2h(U2) + 2z1f(ul + U2)] +P2P [2U, + 2h(ul)+ 2zf_(uI + U2)] (3.32) ~~j2 [02V ~] 2h(v) + 2(ui + U2)l "P, "P [LIVU2 + 2z'f+ (U 1 + U2)1 "P2AP[D1vul + 2zfi(Ui +U2)] (3.33) From Eq.(2.82), it follows that the righthand side of Eq.(3.25) becomes 2e [D OEA] = 2,E (S e)& + pAP [R2 zf+()] +p2.mP [R1 zfi(e)] (3.34) Also, from a direct application of aV to DA we obtain that = {5(R, + R2) + (S) + Pl IR +p2 LIR2} (3.35) Comparing Eqs.(3.33) and (3.34), it becomes evident that we can extract four dif ferential equations for the functions v, u1, u2 and e. They are LIv 2h(v) + 2(ul + u2) = 2E(S e) (3.36) L2uj + 2zf_(u1 + U2) = 2e[Ri zf_(e)], (3.37) LU2 + 2z'+(u + U2) = 2E[R2 zf+(e)], (3.38) and 02]e = 5(R + R2) +/5i LIR2 +P2 []R1 + h(S). (3.39) To obtain the differential equations for the functions v1, v2 and u3, we must first simplify Eq.(2.68). This can be accomplished if we realize that D& IVc = EaDDA (3.40) which follows from a direct application of & to Eq. (2.68). This implies that E] 2a"DA, = aVCA(4 A .V1 (3.41) and therefore Eq.(2.68) becomes AV + 2(auaPC' CaPCA) =eD, (3.42) Using the definition of C\ given in Eq.(2.78), we can write CA as Cl = (P pV2 + P2pVi)S (p1.v2 + P2,v) P +U3(P1iaP2p P2 ap)1p (3.43) Applying &~ to Eq.(3.42) gives aPCOc = [3(v1 + V2) + 1v2 +P2 1V] X (plaOAv2 + P2,aavl) + [P1aP2 []?t3 P2aPl EL3] P + 4u3(Pla p2C)iiA (3.44) In general, if we apply ce to CA,, this results in the identity o~AV bc(V1 +V2) + (Pliv&0v2 +P2,aV)& P 6;(vl + v2) + (pI A Ov2 + P2 Avl)SA' + (PP2v p2p )pAcU3 + u3(plp 2v P2Ap )& + U3 [6'P2, + &VP1A 3ZPl, + 5P2u] (3.45) Applying O, to Eq.(3.45) and factoring all scalar functions multiplying the tensors (P1,S' Pl'), (P2,S' P2p ), and (PlpP2v P2P1L,)P gives A2 A A~ l u) ]v (V ) [2 CAu =(PluSA _z PIA  []2 AV(p2 A V ) V2 + 2z'f_(vl + V2) 2u3} Ap2 ){ i + f(V + v2) + 2U31 +(PltP2v P2pPlv)P" {W]u3 + 6h(u3)} (3.46) An analogous, but otherwise tedious calculation yields o c. c = (p1 8 pw5 )GI (viv2,u3) +(P2,p 21t.v)G2(v1, V2, U3) +pA(pltP2v p2/p16,)G3(vl,v2,u3) (3.47) where G1(v1,v2,u3) ={3z1f+(V + v2) 2h(V2) zf+ [P1 f2] z1[P2 ]Vl] + P2 "LIU3 +4U3} G2(V1)VU3)={ 3zf_(vi + v2)2h(vl) Z[pi p I.V2] zf_[p2 v] l P1 .IU3 4u3} Ga(vIv2,u3) ={1h(v2v)]/ .i [u3 7h(ua)} (3.48) Substituting Eqs.(3.46) and (3.47) into Eq.(3.42) gives [] 2 ap A _[  CA C ) =(Pl Avj )H ( I = plt)Hl(VlV2,U3) + (p ,,p I2v p2,,pI)H(v, v2, u) (3.49) where HI(Vl,V2,U3) 2 4z'f+(Vl + _V2) 4h(v2) 2zlf [Pl .Iv2] 2z'f [P2 Elv] + 2P2 .IU3 + 6U} H2(vi,v2,u3) 2Vi 4zf(v V2) 4h(vl) 2zf [PI .lv2] 2zf[P2 Ely] .3 6U} H3(vjV2,U3) ={IvU3 10h(U3) 2p U3 h[h(V2 V)]} (3.50) Comparing Eqs.(3.42), (3.50) with Eq.(2.83) and factoring all terms proportional to the tensors p (P2,\ P2 ,,) and Pi(Pl,,P2, P2pPl,) yields the desired scalar differential equations for the functions V1, v2 and u3. They are Vv, 4h(vi) 2zf_[Pl v2] 2zfp[52 .L]vx 2zf_(Vl + v2) 6U3 2,1 .mU3 = 2c Sl, (3.51) L]2v2 4h(v2) 2z1f+ [p1 1V2] 2zf+ [5 IEv1 ] 2zf+(v +v2) + 6u3 + 22.E]U3 = 2e S2, (3.52) LU3 10h(U3) 25 L (U3) + h[h(v2 v1)] = 13, (3.53) where the functions S1, S2 and R3 in the righthand side of Eqs. (3.51) to (3.53) are given by Eqs.(2.88), (2.90) and (2.91). To obtain the final differential equation for the function V3, we first simplify Eq.(2.69) by realizing that ]2 J lC,= 2eOPD, (3.54) This equation still contains all the information of the original equation. It is possible to reduce this equation to a vector equation by multiplying Eq.(3.54) with the tensor . Due to the antisymmetry of and D, under the interchange of two of its indices, and the fact that 0 if any two indices are equal, we can solve the equivalent equation Z [PPC,,P] = 2E& [p1tp"DIvp] (3.55) Using Eq.(2.79) and contracting C, with PAP' gives PI P2 Cvp = V3 {(P i P2P2p 2 p2P)P + (51 P2Pp P2p)P\ + [2 2 ,P 01(6 "[lP (Pl" f2)2]Sp } (3.56) Applying OP to Eq.(3.56) gives p p 2 "C ' ] 8k 2 v3PA (3.57) Similarly, from Eq.(2.84) it follows that (1= S3 {(PI i2P2p Pp) + (1 P2P1p PP2p)PA 1+J [fJp _z pfl )2 P2)P} 1 (3.58) and 1P 2 [ppDJ = 8k2 S3P" (359) Thus, the function V3 must satisfy the differential equation VV3 = 26S3 (3.60) 33 The Model Equation in the Asymptotic Region To obtain the solutions to the differential equations in the asymptotic region, we must be very careful how to approximate the functions R, R1,... S) 1, .... In the inner asymptotic region, we cannot neglect those terms multiplied by the electron propagator functions A1 and A2 unless they are added (substracted) from terms that are clearly larger. For example we can approximate A1 A2 P1 "P2 PI "p2 but cannot neglect the term (A, + A2)v against P1 "P2 u. If we do so, it will not be possible to join the asymptotic solutions to the perturbation solutions. These terms are not negligible since, in the inner asymptotic region, the form of the electron functions is A1 m( xz/m2), A2 m(xz1/m2) (361) where by Eq.(2.38), 71 = 3e/4 1.75 x 103. Although the functions A1 and A2 approach zero for large values of x, one has go to an extremely high momentum to see an appreciable difference. This problem does not arise in the first five since all terms containing the electron propagator functions are negligible compare to those retained. In the inner asymptotic region the following form for the functions R1, R2,S,S3 are correct: R, [V p2 j0U2  (p02 P1 Pl"J2)V3] ID ,(3.62) R2 [V + p2Ul ( + p, P2)v3] /D (3.63) S [ [_p1 .p2v + [] P2 (pl .p2)2]V3 /,D (3.64) S3 [V + Pl" .p2V3]/D (3.65) The remaining functions have the form R [l P2U + (AIP2+ A2Pl).(UlP2 + U2Pl) +(A, + A2)v + (p2 p :v f f)l 2_ +... pP( P2)V2 (P22 P P2)V 1 2 (l _2)2]u3]/D. (3.66) R3 [u AlU2 A2u1 +j1 P p2u3 +(V2 V v) (A1 + A2)v3]/D. (3.67) Sl [Alv p2v2 (A2p2 + Alp, .2)v3]/D (3.68) S2 [ A2v p2v1 + (Ap2 + A2p1 "P2)v3] /D. (3.69) At this point, it is important to realize that in the asymptotic region the system of nine coupled differential equations has decoupled into two set of equations. Notice that Eqs.(3.62) to (3.65) are coupled together but not to the remaining four. Therefore, it is possible to attempt to solve this set of five equations independently from the rest. This is the approach we will follow. It is possible to further simplify the differential equations. To this end an explicit form for the coefficient of the five functions v, Ul,U2,v3,e will be used. 38 From Eqs.(3.4), (3.5) and (2.71) it follows that the denominator D that appears in the righthand side of all the differential equations can be approximated by the expression Dx (3.70) This approximation is valid in both the inner and outer asymptotic regions provided that (x/m2) > y. If we realize that in the asymptotic region, p2 , Pl P2 xy, then the differential equations are somewhat simplified. For convenience, all nine asymptotic forms of the differential equations are presented in Table 31. Table 31 Asymptotic Forms of Differential Equations vv 2h(v) + 2(u, + u2) = 2e(S e) (3.71) v1 + 2zf(uI+u2)=2e[R1 zf_(e)] (3.72) ]vU2 + 2z1f+(ul + U2)= 2c[R2 z1f (e)] (3.73) 02V3 = 2eS3 (3.74) ve 5(R1 + R2) + P1 "R2 + P2 1R1 + +(S) (3.75) LIvu = 3eR (3.76) 2V 4h(vl) 2zf_ [pi ]V2] 2z_ [P2 []v,] 2zf(vl + v2) 6U3 2h IU3 = 2e S (3.77) ]V2 4h(V2) 2z1 f+[i E]V2] 2z1 f+[P2 Elvi ] 2z1 + (VI + V2) + 6U3 + 2L2 lu3 = 2e S2 (3.78) "vJU3 l0h(u3) 2p [L h(U3) + h [h(v2 VI)] = e R3 (3.79) where 1 12 [xy u + (A1152 + A2I). (uaP2 + u2P1) + (A1 + A2)V + xy2 1 (v + v2) + x2(y2 )u3] (3.80) R, ] +X 2X7 ,V (3.81) 1 R2v1 x[ u+ X I+XNI 3 (3.81) 2 R3 ".. [u du2 A2u1 + xy u3 + (v2 vi) + (A1 + A2)v3] (3.83) S [y v + x(y2 1) v3] (3.84) x Si,1 Arz2 x(zA2 A)3 (3.85) 1 S2  "2[ A2v xzv1 + x(z'A1 + y A2)v3] (3.86) S31 X2 [V + xy V3] (3.87) CHAPTER IV ASYMPTOTIC SOLUTIONS TO THE MODEL EQUATION We seek solutions to the differential equations given in Eqs.(3.61) to (3.69) which join smoothly with the perturbation solutions. In order to do this, we use as a guide, the asymptotic form of the perturbation solutions found in Table C2 of Appendix C. As can be seen from this table, all perturbative solutions can be written in the form of a power law in the variable x. With this in mind, we make the hypothesis that the asymptotic solutions to the differential equations are of the form W, = xa#(y)i, (4.1) where, at this stage, we will use a bar over functions to denote functions that are only ydependent. The justification for this assumption lies in the fact that the asymptotic expression for the perturbation solutions are of this form and they satisfy the model equation in this region (within .5%) with the exception of the v1,v2 and u3 functions. The latter are good only within 3%. When we apply the D'Alambertian operator to such functions, it is clear that []2W,\ { 4k2 2 d2g d# 4(a+2) dy 4( )[(Y2 j _.y 9 e (4.2) x dy ) It is evident from this equation that, if g(y) is a smooth function of the variable y, then for large values of x, it is possible to neglect the first term since it is one power of x smaller than the second one. This leads to the approximate form of Eq.(4.2) []2W xP:A{4(a + 2) )g}x(4.3) 40 where D=_2 d +ay (4.4) To simplify the differential equations in the asymptotic region, it is necessary to find the form of the operators P, L, P2 l, h, 1, etc., when acting upon functions of the form given by Eq.(4.1). It is easy to verify that when these operators are applied to functions of the form (D = xcg(y), the following relations are valid: i* .F11( = x [zb + a] g , p. xa [Y&' + g] (4) = 2 1z h(i) 2x_1 1 X11 f+( vg7_ (Da az)j I xa1 ()=+ 1~ (Do  (4.5) (4.6) (4.7) (4.8) (4.9) (4.10) (4.11) (4.12) Also, a combined application of the h and I operators upon these type of functions satisfies the following relations: (4.13) (4.14) (4.15) where h[k .El ] = 2x1 y 1 [a52 y ba]g ( h[k. h1] =2h E() + p 1( h h [k. = ai .nk (oh) (4.16) 04 2xo1[ I a (4.17) F. ] 2x1 j2  D (y2 + 2xa]4 (4.18) = 4 2 Y b2 [(i2 .lb (4.19) and the operator D2 is defined by the equation b2 =(Y2_ 12 d ( 1)) d [ )2 C ( y2 2(a 1). + (4.20) Using Table C2 of Appendix C, it is reasonable to assume the following forms for the functions v, u1, u2, v3 and e: v = cJx +l (4.21) uI = U2 = x0e (4.22) V3 = Xze (4.23) and e = exa (4.24) where cl is an arbitrary constant. Furthermore, the asymptotic behavior of the vfunction (see Appendix D) leads to the conclusion that the value of o must be equal to 1. Other solutions are possible with a 5 1 but they proved to vanish at large values of the variable x. Such solutions are not favored since they do not have an asymptotic behavior consistent with Appendix D and they do not join smoothly with the perturbation formulas found in Eqs.(C.41) to (C.46). To find the asymptotic solutions to equations (3.71) to (3.75) that join the perturbation forms, we set all functions in the righthand side of the differential equations equal to zero except for the v and efunctions. Substituting Eqs.(4.21) to (4.24) into Eqs.(3.71) to (3.75) and setting a = 1 gives 4[D1 + y]V = 2c[y + E]( (4.25) 43 4bIii = 2[D + z(e + i2 1 E')] (4.26) 4bju2 = 2[3 + z'( V 1 ')] (4.27) 4D1i3 = 2eV, (4.28) and 4D1e = l0V+{ z(D1 y) 2} z'(Di y) 2 U+2Di(yU) (4.29) where D = (1y2) y. (4.30) The primes denote derivatives with respect to the variable y. Notice that with the choice, a = 1, all five functions are in agreement with the xdependency of the perturbation solutions and in effect have reduced the problem to that of finding the solutions to Eqs.(4.25) to (4.30) which are only ydependent. To solve these equations, we start by adding Eqs.(4.26) and (4.27). This gives the result 2f1(u1 + ii2) = E(v DI) (4.31) Since we assume ii = ui2 in obtaining these differential equations, we conclude that, for consistency, /1 = (4.32) Applying /1 to Eq.(4.25) and substituting the previous expression gives 4D)1 [1 + y]ij 2e[b(yv) + U]. (4.33) Using the identity by=yb(Y2 1), (4.34) and moving all the terms to the left side of in Eq.(4.33) yields the result, 4D1[b1 + y]v + 2F[ybl (y2 1)TY] + 21v = 0 (4.35) In terms of the operator D_1 obtained from Eq.(4.20) by letting a 1, _1T + (1 + e)yDI + [(1 1) + e] = 0 (4.36) In terms of the derivatives of V with respect to y, Eq.(4.36) can be written as (y2_l )U,,+ (3_ !e)yU' 0. (4.37) It is possible to transform Eq.(4.37) with the substitution, S(y2 1)r, (4.38) (Y ) +Z Y2 1I 7, )y 1)' v + 2 (4.39) and V= (y2 1)rv"+47y(y2_ 1)lvF+ {27(y2 1)r1 +47(7 )y2(y2 )r2V (4.40) Substituting Eqs.(4.38) to (4.40) into Eq.(4.37) gives (Y2 1)r{(y2 1)v "+ [47 + (3 E)]yV + [27 +[47(7 1)+ 27(3 E)]y2(y2 1)' 61]} = 0. (4.41) To simplify this equation, we select the value of the constant 7 to satisfy the con dition 4,T+ (3 ) =2 (4.42) or solving for r (4.43) Making these substitutions in Eq.(4.41) gives (y21) ="+2yv'+{2+[4T(T1)27(472)]y2(y21)1 } =0, (4.44) or after some simplifications (y21)=" =' {(4y2 v+2v {(42 +)+(y2 :1)}=o (4.45) If we define the constant p such that [ = 27 and the constant v by the relation v(v + 1) = 4 2r + E, (4.46) then Eq.(4.45) can be written in the form (Y1)v +2yv jv(v+1)+ (+ 1) =0 (4.47) where = (1 + 1) and v= !(1 + 1,) .(4.48) The general solution to Eq.(4.47) can be expressed in terms of the Associated Leg endre functions PI (z)(See Appendix E).28'29 These, in turn, can be represented by the hypergeometric series F(a, b, c; y) by the identity, (y) = 21. (y2I 2) yI+,F(v 1 , 1 t; 1 y2) (4.49) Pr(y) y) Y)' 2) 2 2 P valid for y > 1. For computational purposes, however, this expresion is not conve nient due to the slow convergence of the hypergeometric series when its argument is near 1. For large values of y the following expression gives a rapid convergence of the series: 27r F( V) y 2v r(l + V) U 2 +.1 2. 2U' F(7 +) 1) +v/v r(1 + v p)Y(Y F( 1p 1v, 1 1v 1p, . ;Y2) (4.50) The Pfunction can then be written in the form V = cl(y2 1) 21Pf (y) (4.51) where M and v are defined in Eq.(4.48). We now turn to the problem of solving Eq.(4.32) with the solution to the Ufunction differential equation in the righthand side. From Eq.(4.32) and the expression for b1 given in Eq.(4.30), the differential equation that the cfunction must satisfy becomes (y2 1)' y cl(y2 12 "Pf(y). (4.52) Let assume that this function has the same form as the 5function, i.e., e = Cl(y2 1)Ap;(y) (4.53) and e,= 2Ay(y2 1)lp;(y) (y2 1)P '(y) . (4.54) Substituting these two equations into the preceding one gives (Y2 1)A{(2A + 1)yP(y) I (y2 1)P '(y)} = (y2 1) p"(y) , (4.55) from which it follows that A = p. With this substitution, this equation becomes (Yi + 1)y Pp,(y) + (y2 1)P; (y) = P,1(y) . (4.56) Using the identity, (y2 1)PX '(y) /y P;(y) (a + )P;_1(y), (4.57) yields the equation (P +/3 + 1)y P;(y) (Ce +3)P;_1(y) = P"(y). ( (4.58) 47 If we set a = and f = (1 + y) then it follows that the first term in this equation vanishes and the conditions, (z+P)=I and P1=(v+l), (4.59) are satisfied. It is evident from Eq.(E. 17) that the efunction is given by the equation c(Y2 j'IAP_ ('+)(y) = ci(y2 1) /I"p(y) (4.60) We now turn to the problem of solving Eq.(4.26) to find the function ii. To do this, we will transform this equation with the help of Eq.(4.12). From this equation, (with a = 1), it follows that z z(e y y2 1 E1)  (D1 + zl) (4.61) Substituting Eq.(4.32) into this equation and simplifying gives I zv +~ z(E+Vy21')= 1 (4.62) With this result, Eq.(4.26) yields the equation 2e [y + (4.63) Ny 2~ or in terms of the first derivative of iii, 21 1 (y2 1)u + yiil = C 1 [yv + e]. (4.64) 2 To solve this equation we make the substitution, U1 C1(y2 1)(1)1 (4.65) and fI = cI( 1)yul(y2 1)1(p3) + (y2 1) (/1)(y2 1)u' (4.66) Then the lefthand side of the Eq. (4.64) becomes (Y2 1)u + Yu1 = (y2 1)6( 1) [A y + (y2 1)ui1' (4.67) The righthand side of Eq.(4.62) can be written in the form, 2 [ = 2 f U [yP (Y) P,(Y)] (4.68) where we have substituted the expressions for U and g found in Eqs.(4.51) and (4.60). Equating the expressions given in Eqs.(4.67) and (4.68) yields the equation U1) + [YP i = 2 [yP'(y) P(y)1 (4.69) where p and v are defined in Eq.(4.48). Notice also, that from Eq.(4.48), v 1 = andyz + v = i.1 Therefore Eq.(4.69) can be rewritten in the form, (y2 1 '+ P y= = (P + V) [yp,(y) P,(y)] (4.70) It is possible to transform the righthand side of the last equation by using several relations satisfied by the Legendre functions (see Appendix E; Eqs.(E.43) to (E.51)). It is straightforward to show that = (y' 1)P" (y)' +/P p(Y). (4.71) It is easy to verify that the general solution to the homogeneous equation is given by the expression (Ui)h = c2P,I(Y) (4.72) where c2 is an arbitrary constant. It is evident from Eq.(4.71) that a particular solution to this equation is (U) = P"(y). ( (4.73) If we set the constant c2 = 1, then the general solution to Eq.(4.71) is u P (y) P,;_I (y) = (2y + 1) / 1PM(y). (4.74) Using Eqs.(4.65) and (4.66) we finally arrived at the desired solution il = 2 = Cl(2[ 1)(y2 1) P(y) (4.75) The i3function can be easily obtained if we realize that the differential equation that it must satisfy is very similar to the differential equation satisfied by the i function. In fact, we notice that the difference between Eq.(4.28) and Eq.(4.32) is a constant factor. With this in mind, the function v3 can be expressed in terms of the efunction in the form, V=(3 + ) ( +)ci(y2 1)1,P (y) (4.76) The arbitrary constant cl needed to completely define the solutions can be determined by using the requirement that the vfunction must approach the value v (1 ) (1 1 7) (4.77) as x * cc and y + 1. This result was derived in Appendix D. From Eq.(4.51) and (E.57) we have that lim P'(y) 2 (1 A)( 1) (4.78) and therefore lim V = (4.79) y1 f(1 /) Equating the previous expression with Eq.(4.75) yields the result 1 =(( 1P) (4.80) We now turn to the problem of solving Eq.(3.76) and (3.80) for the function ii. Its is possible to simplify the function R given in Eq.(3.80) in the following manner. If we expand the scalar product involving the functions ul and i2 and substitute the condition that ii2 = U1, this equation becomes R'L xy U + Ul [Alp2 + A2P1 P2 Al P2A2p2] X2 fL 1 +(Al+A2)v +xT:I(vI+v2)+x2(y21)u3}. (4.81) Using the asymptotic forms o p and pi 12 given in Eqs.(3.4) and (3.5) yields the result R= xy u + (v ,/ l u)(A1 + A2)v + xVy 1 (v1 + v2) + X2(Y2 1)u3 (4.82) The terms involving the functions v and u1 can be simplified further. Substituting the expressions for these functions found in Eqs.(4.51) and (4.75) gives v/2 ul = cl(y2 )2' [P1(y)(2pi+1)Vy1P.l(y)j (4.83) Using equation (E.46) found in Appendix E, it follows that the previous equation can be written as V V/y2 l = cl(y2 2) [v(Y) +;I() Pp+1 ()] (4.84) If we notice that p + 1 = v, this expression becomes v V 1 u cl(y 1) P_,(y) (4.85) Furthermore, using Eq.(E.41) it follows that (4.86) v %/ 1lul = cl 2,/F( y) . or, using Eq.(4.80) vV lul = (1 2 (4.87) If we substitute this expression in Eq.(4.83) and set all other functions equal to zero, we arrive at the equation 1 2 Vu = 3(1 r7)(A1 + A2)/x. (4.88) 2 Using the same approach used in solving the first five differential equtions, we assume that the ufunction can be represented by the form u = x'1ii(y) (4.89) Substituting the previous form into equation (4.88) and using the asymptotic form of the D'alambertian operator as well as the asymptotic forms of the functions A1 and A2 gives X14(3 + 2)bD,9i = 3,(1 1,7)x2(z? + z) (4.90) 2 Matching the xdependency on both sides of Eq.(4.90) implies that the parameter must be defined by the formula (1 + 7). (4.91) Substituting the form of the D given in Eq.(4.4) yields the equation (y2 1)Ut' + (1 + 7)yi, = A(z" + z) (4.92) where 3 (1 7) (4.93) To solve this equation, we notice that, for values of y > 10, the approximation z 2y is very good. This leads to a relative error of less than .2%. Since the perturbation solutions are good in this region, we will attempt to solve the approximate equation y2 U + (1 + 77)yU A [(2y)" + (2y)q] (4.94) In order to solve this equation, we make the change of variables 1 1 =(2y)', < = 277(1"7, and y = (4.95) Subtituting these expressions in Eq.(4.94) and simplifying yields the differential equation 0* + (1 + 1/,) u= 2(A/i7) [1 + 2], (4.96) where the symbol i* stands for the derivative of u with respect to 6. This equation can be solved by introducing the integrating factor 40 = exp(l+/17) d = '(1) (4.97) The general solution can be written in terms of the integrating factor in the form d 1 1~r + (2A/71) f C [1 + t2] I(t) dt, (4.98) where d is an arbitrary constant. The integration in the previous equation is ele mentary an yields the result =d('+7) + (2A/7)(1+?) [ln + 21 (4.99) In terms of the variable y, the solution can be written as Iyy {d(2y) + (A/72)(2y)1 + 2A(2y) ln(2y)} (4.100) To determine the constant d, we require that this solution join smoothly with the asymptotic form of the perturbation solution given in Eq.(C.46) of Appendix C. In order to join the solution just found, we notice that for values y not too large the following expansions are valid due to the small magnitude of the constant 77 (,7 ; 1.75 x 10x): (2y)l = exp{ 77 ln(2y)} = 1 + jln(2y) + . (4.101) (2y) = exp{ 7ln(2y)} = 1 r/ln(2y) + If we select the value of the constant d to be equal to A/7 then Eq.(4.97) becomes TY 7 {[(2y)'1 (2y)'?] + 2(2y)'1 ln(2y) (4.102) or because of the expansion given in Eq.(4.101), 2A A ln(2y)[1 ,7 ln(2y)] (4.103) Y Substituting the expression for A given in Eq.(4.90) gives 3 ln(2y) + O(E2), (4.104) 2 y which has the same form as that given in Eq.(C.46). Therefore, the solution is found to be given by the expression = 2(1 !E) {(2y)" (2y)" + 2,q(2y)" ln(2y)} (4.105) It is not difficult to construct a "solution" that is accurate near y 1. From the form of the perturbation solution near y = 1 and the fact that we can replace all terms involving z by (2y), it is reasonable to assume that this replacement will improve the solution in this region. With this argument the improved solution is given by the expression = 1(1 3e) {(z" z,) + 2rz" lnz} /V/y2 1 (4.106) 54 For reference purpose, the asymptotic solutions are presented in the Table 41. Numerical values for the functions U, e, iil, v3 and ii are presented in Table 4 2. The variable x inside the overlapping region was chosen to be 1020. Their derivatives are also presented. The values of the perturbations solutions are also shown for comparison. Table 41 Asymptotic Solutions to the Model Equation Notation: 4 1 2) v = (1 2 q)(y 1)2'p,'(y) (4.107) e = (1 1 )(y2 2 (4.108) 2 UI U2 = (1 2 TI)(2p + 1)(y' 1) "P,1(y)/x. (4.109) V3 6(1 1)2pP(y)/x. (4.110) 2 8 (13)x+){(z z'7)+2rjz'lnz} /Ny21 (4.111) Table 42 Perturbation and Asymptotic Solutions in the Overlapping Region Perturbation Solutions (x = 1020 and y = 1.1) 1t derivative 2nd derivative V 9.957E01 7.301E04 4.155E04 e 9.679E01 3.083E01 2.352E01 U1 1.640E04 7.341E04 4.877E03 V3 1.124E03 3.579E04 2.731E04 3.372E03 1.074E03 8.193E04 Asymptotic Solutions (x = 1020 and y = 1.1) 1t derivative 2nd derivative 9.992E01 7.298E04 4.149E04 9.671E01 3.078E01 2.348E01 1.640E04 7.338E04 4.875E03 V3 1.123E03 3.574E04 2.726E04 ii 3.373E03 1.081E03 8.574E04 Perturbation Solutions (x = 1020 and y = 10) It derivative 2nd derivative 1" derivative 2nd derivative v 9.979E01 1. 138E04 1.101E05 Asymptotic T) 1.001E+00 1.139E04 1.102E05 e 3.008E01 2.029E02 3.109E03 U1 2.344E04 1.224E05 1.390E06 V3 3.493E04 2.355E05 3.609E06 U 1.048E03 7.066E05 1.083E05 Solutions (x = 1020 and y = 10) 3.008E01 2.027E02 3.105E03 ii, 2.345E04 1.223E05 1.388E06 i3 3.493E04 2.354E05 3.606E06 U 1.046E03 7.063E05 1.083E05 Perturbation Solutions (x = 1020 and y = 100) It derivative 2nd derivative v 1.001E+00 1.161E05 1.160E07 e 5.299E02 4.299E04 7. 599E06 4.991E05 3.831E07 6.504E09 V3 6.152E05 4.992E07 8.823E09 1.846E04 1.497E06 2.647E08 Asymptotic Solutions (x = 1020 and y = 100) It derivative 2nd derivative v 1.004E+00 1.165E05 1.163E07 e 5.305E02 4.301E04 7.601E06 U1 5.OOOE05 3.835E07 6.508E09 V3 6.160E05 4.994E07 8.825E09 U 1.839E04 1.493E06 2.641E08 Table 41 (Continued) Perturbation Solutions (x = 102" and y = 1000) 1t derivative 2nd derivative V 1.003E+00 1.161E06 1.161E09 7.601E03 6.601E06 1.220E08 ii1 7.664E06 6.503E09 1.185E11 U3 8.825E06 7.664E09 1.417E11 U 2.648E05 2.299E08 4.251E11 Asymptotic Solutions (x = 1020 and y = 1000) at derivative 2nd derivative 1.007E+00 1.169E06 1.168E09 7.620E03 6.613E06 1.222E08 7.687E06 6.518E09 1.187E11 U3 8.848E06 7.679E09 1.419E11 u 2.633E05 2.288E08 4.232E11 CHAPTER V THE CONCLUSION A solution for the vertex amplitude has been found to the SchwingerDyson equations based on an approximation scheme which is characterized by the follow ing: (1) the photon propagator is approximated by its form near the mass shell, (2) the infinite hierarchy of the vertex is cut off at the second order in the coupling constant and the remainder is approximated by Green's generalization of the Ward Identity for higher order contributions. The gauge is chosen to be that found necessary to obtain a finite solution to the electron propagator equation20 with a vanishing bare mass for the electron. In the above reference, the electron propagator is found to be given by 5'(P) c A(p2) + B(p2)p (5.1) where 2 c(m2P2)/4 ,p2 d(, (5.2) B(2)=c_ 1 (5.3a) In the asymptotic region, a simpler form of the Afunction was used, namely A(12) ,,_ (/2/2),, 7 _ 1.75 x 103 (5.4) A simplified approximation of the vertex equation (the model equation) was found. This tensor and matrix equation can be reduced to a set of 8 coupled differential equations of third order in two variables. The order of the equations was reduced by the introduction of an additional function e. The set of 8 basic scalar functions is needed to define the vertex completely as required by the transformation properties of the vertex amplitude. In general, the transverse part of the vertex amplitude can be written in the form __ (P1,2) V(pl, P2, PI P2) +l Au( p2, P, 2) u( p2, p, p2) /2P 'Uj21 2[, ]u ( , ./ 1,, ,]v (2 2, Pl"/2 + P2) + [V61JA]Vl(pf2~.~ 2 ~ ~ 2 f2fAv( i~l P2) Attvp 2 + Zi& 7A P1 p2pV3(P ,Pi2,Pl" P2) (5.5) The longitudinal part of the vertex is given by Ward's identity (see Eq.(2.27)). The analytical approximate solution to this simpler equation has been found in the vicinity of the four momenta square equal to the square of the experimental mass.23'24 The validity of these solutions was tested by direct substitution into the differential equations (see Eqs.(3.71) to (3.79) with Eqs.(2.85) to (2.92)). A measurement of their accuracy is given by the difference between the left and right hand side of the differential equations divided by their average. These solutions were found to be valid within .5% for 6 of the equations (v, u1, U21 v3, e, and u) and a much higher error (3%) for the remaining functions (VI, v2, u3). The region of validity of these solutions was found to be x > 1 and 1 < y < 104. The introduction of the variables, 2 and z / (5.6) made it possible to separate the system of 9 coupled differential equations into three separate systems of five, three and one differential equations in the asymptotic region of large momenta. Using these variables, the solutions factor into functions of x and functions of z in this region, and the xdependency of all the solutions can be written in a simple power law form. This reduces the problem to finding functions of only one variable. The goal of this work was to find the solutions to the set of differential equa tions containing the largest component of the vertex amplitude, namely the v function. The differential equations for the functions v, u1, u2, v3, e, are linked together but do not involve the other dependent variables (see Eqs.(3.71) to (3.75) ). Solutions were found for these five functions that join smoothly with the asymptotic forms of the perturbation solutions. The percent difference between the perturba tion and asymptotic solutions is less than .1% in the region x/k2 > 1010 and 1.1 > y > 106 with a similar result for their first and second derivatives. This confirms the results24 already found that the perturbation solutions gives a good representation of the vertex amplitude in a very large region of the electron mo mentum. This conclusion follows from the fact that the largest part of the vertex amplitude (i.e. v _A) probably has a maximum at the mass shell. The smallness of the fine structure constant and the slowness of the decay of the electron propagator function A in the asymptotic region are also responsible for this result. These results were encouraging so that an attempt was made in solving the ufunction. The techniques and general insight gained in the solution of the v function proved useful. This attempt was successful and a solution was found that also joined smoothly to the perturbation. The form of these functions, except the vfunction, vanishes at infinity as rapidly as O(1/x) and therefore is negligible compared to the vfunction. Also, these functions are one order in the coupling parameter smaller than the vfunction. We are lead to conclude that the most important function in defining the vertex amplitude is v. Although the perturbation solution gives a good representation of this function in the intermediate region, this form may not be useful in determining the convergence of several interesting quantities. The analytical properties of the vacuum polarization integral and charge renormalization constant Z3 are strongly dependent of the analytical properties of the vertex amplitude. The convergence of this integral may well depend on the way the vertex amplitude behaves at infinity. The perturbation expression given in equation (C.41) shows a logarithmically diver gent form for this function as the variable z growth without bounds. The present work has demonstrated that the behavior is much less rapid (v z ). This result alone encourages the calculation of the vertex amplitude to higher accuracy to determine if it is constant everywhere or even decreases for large values of the variables. To address this question, a recalculation of the photon propagator is needed. The form of the solutions found in this work may lead to the calculation of a correction to the photon propagator. This in turn will provide a way to calculate the renormalization constant Z3 and the bare charge of the electron. It cannot be said that the present solutions, described in this work, consti tute a complete resolution of the problems in the theory of QED. However, this method has been successful in finding these solutions without the unreasonable in finite quantities that plague renormalization theory. Its felt that within the level of approximations made, the present solutions give evidence that the infinite quantities that occur in the usual perturbation calculations of the selfenergy of the electron and the vertex amplitude are not essential to the theory. Furthermore, the present 61 work sets the foundations for the subsequent calculation of the photon propagator and the closing of the iteration procedure of the project. APPENDIX A DIRAC MATRICES: DEFINITION AND IDENTITIES The relativistically invariant equations which in the case of the electron play the same role as the Maxwell's equations do in the case of the photon, were obtain by P.A.M. Dirac3" in 1927. In order to describe these equations in a relativisti cally invariant form, Dirac introduced a set of fourdimensional matrices (the Dirac matrices) given by = 0 1 ;  J 0 ;j 1,2,3 (A. 1) where __ represents the Pauli spin matrices. The Pauli spin matrices are two dimensional matrices defined by = 1 0 ; = i 0 ; = 0 1 The Dirac gamma matrices satisfy the anticommutator relation, 2,2, + 2"_,2 = 2gpv (A.3) The multiple products of gamma matrices form a group of 16 linearly independent 4 x 4 matrices. One possible representation of this group is given by the linear combination of products defined by the matrices I, y _, V where 1 [_ I _' ] (A.4) and 1 ::: 1 1 Y } PP 2 2 '~ (A.5) and I is the 4dimensional unit matrix. We have used the symbols [,] and {, } to represent the commutator and anticommutator respectively. Another useful matrix, used in defining the model equation is defined as 5 0123 1 W17211 (A.6) Equation (A.6) may be used to write Eq.(A.5) in the form 1Pols ZEVpa= z _7 (A.7) The symbol eltpo stands for the 4rank antisymmetric tensor (Levi Civita) defined by the rules 6pL = 0 &1234 1 +1 E14VPO 1 unless y, v, p, a are all different , for even permutation of the indices 1,2,3,4; for odd permutation of the indices 1,2,3,4; for even permutation of the indices 1,2,3,4; for odd permutation of the indices 1,2,3,4. It is simple to verify that the gamma matrices satisfy also the following identities: A 'A1 =y/Z 2y1 pA 0 A( y 2 V (A.12) (A.13) (A.14) (A.15) and (A.8) (A.9) (A.10) (A.11) 64 By definition, the operator Y/ ,), and therefore, using Eqs.(A.12) to (A.15) it is possible to show that 3t7 =T1 + 2&a, (A.16) Wi= 7_I, 5 + 2(,9t,7y O7/), (A.17) V9p= 2y 3t +2(o9,,y,+ &,y +&(pv). (A. 18) APPENDIX B DERIVATION OF GREEN'S PERTURBATION SOLUTIONS To obtain the perturbation solutions to the model equation, one sets C A CA CV CA 0 and A1 A2 m as the zeroth order ap proximation in equations (2.52) to (2.55). Therefore, the D tensors become D' = 2mi5/D (B.1) D'\ [(im2 P1 P2)&A + (Plg + P21,)PA] /D, (B.2) DA = m [(m2 51 P2)P (Plt P2p)]/D, (B.3) DAv; = m{[(PlgP2v PlP2u)' +(PlvP2p PlpP2v) '\ + (PlpP2p PlpP2p)6;] /D, (B.4) where/51 = p + k and 2 P k. To solve the model equation under these approxi mations one must substitute equations (B.1) to (B.4) into equations (2.66) through (2.69) and integrate. This can be accomplished using the following functional rela tion. Let Xq ( + )2 = (.2 + 2xt + q2). Then it follows that E]2 [x qF(xq)] = 4F" G a(xq) (B.5) and therefore, Ffq) = j x 2 dx"G(x") (B.6) Using this relation, one can easily show that, if xVI(Xq/m) = Xq = 2 (B.7) then Also, if one defines [k, = (in2 2) + k262 then 1 I<[cd (p2 ,2)(2 m') =2 ..,(Ze 1) With this definition it follows directly that if, (p2 M2)(F) m2) ! in(z 1) It is easily shown that one can write In .P 2 2 2 P2 A 1 2 I XC  E](6=_ n( 2n~ 2  p1 p2 (B.14) then (B.15) Similarly if 021 d(m 2 p) In(Z 1), 1 1 { ] L2(Z ) ( )ln(Z 1)' 4c _1d ( )(XI/ =1 ) ln(2 xq (B.8) and Z =X /p (B.9) (B.10) then (B.11) (B.12) so that, if (B.13) then (B.16) (B.17) X$ = (p + 6k)2, (a 8 1 2f 8j_ Z where L2 = f ln(1 z') d' is the dilogarithm. The dilogarithm satisfies the integral relation L2(z')dz' = zL2(z) + (z 1) ln(1 z) z. (B.18) With these relations, it is possible to solve equations (2.66) to (2.69) as follows: Equation (2.66) with (B.1) on the righthand side becomes [j2cA = 6FmpA (p m2)(p2 m2) Integrating Eq.(B.19) using Eq.(B.15) gives CA 3 Jj\1 d 4 pJ1 PZ 3cm a 2ln p23 p p2 1+ ln(Z 1) } Next, substituting Eq.(B.2) into Eq.(2.67) yields ]2CA = 2e (Mn2  P2) A + (PIP + P2Au)Pa IC= (p2 m2)(P 2 m2) where EA satisfies the differential equation E\ = ( { (9pE, (rn2 P2 2) (P, p2)pa One can easily verify that the following relation holds true: (in2 p2) + (Pip + P2p) (p2 M2)(p rn2) 1 m2 + { (M2 p 2) I(p2 M)(: ; 1 / 2 2 2n P1 P2 2 ) Therefore, C, can be written in the form CA = C + p, (B.19) (B.20) EA} (B.21) (B.22) {PipA P2A In (p2p2) 2 (B.23) (B.24) a 5,7C" , PA where C and C. satisfy the differential equations 1 P,L + P21 in (p2 ) LI2C = 2c (2 l 22) (j51M2)(p2 M2) 1 +52 _/21 n ( 2 m p2 n) 1l 1 4k2 + 2 (2 m2) (p m2) (p2 m2)(p2 m2) 2 2 + l In2 + 2 2 and 2m2 2 []2E, = (P22 2 2In (p,m2)(p2 m2) 1p Using Eqs. (B.25) to (B.27) it is found that D2 (c 8Cp) =0 which implies that C = a This also imply that AC = 0 AL This relation is valid in general for the model equation as can be seen from a direct application of 91 to Eq.(2.67). From equations (B.8) and (B.12), one obtains (P2~ 4& .j)a + 21EIb  (it2 1 22 1 l 2k2 ln(Z 1)d Y z ) ( 2 m2 1) 0j2CA = 2e + OE, (B.25) (B.26) (  m2 Ap2m ) (B.27) (B.28) (B.29) P2 (B.30) /52 M2) In 2A5 (m2 C 2M, ) + and from equation (B.16) with Cj = CAL k&Cck/k2, one finds ]2D 2, (p2 PA p2 in S1 2) 2 M) 2 a { J ln(Z 1)d 1 + E, 2e { d1 (M2 ')ln(Z 1)} This follows from equations (B.12),(B.15) and (B.22) from which E = 2m2.'a 2b= I Z + 4 Z m ) Z 1) In (Z 1) + constant If we define the function e such that E = eP5A, then it follows that e =20E1/OX and thus 1 d + (M 2 2 1 Z[Z 1) Using Eq.(B.16) one can calculate im = E j d (m2 .){L2(Z ) + (1 ln(Z 1)} 2 0a5" I 1 P d_ (m 2 ) d Z 1 (I ln(Z 1) It also follows from Eqs. (B.16) and (B.18) that [2 k C A = 2e 1 In A 2 + k. E} and 12(kACA ep "kE1) = Eln (2 M2 2 JM 2 (p2 _p2 ( 1 2: ) 4 (p2 M p2 n2) Therefore (B.37) 0 +8 E} IP (B.31) . (B.32) (B.34) (B.35) (B.36) lZ 1)] .(B.33) lnZ kA C/ 5p " k  f Z, _L M2 u,) ln(Z, 1) Using Eq.(B.34) with (B.37) gives 1 P1 d 2 4 =21Ei]1 P Zm  1 (p) kA  ) {1 (1 A ) ln(Z 1)} (B.38) __( 2 )in(Z 1) Using Eqs. (B.24) and (B.38) and the fact that the function v multiplies P, in the definition of CA, we finally find that v=C j f_ 1 (1  1)} (B.39) It also follows from Eqs.(B.38) and (2.77) that if CkP = kAC1= k (u2p1 + ulP2)PA (B.40) then 1) Iln(Z 1)}. PJ Zi 1) /P Z The corresponding equation for Cp is CpPA = PAC, = [v +P(u2p1 + U2)]j PA (B.41) (B.42) {(m2 t)(m2 k2) [1 ln(Z 1)] k 2 ln(Z 1) Using Eqs. (B.41) and (B.43) it is possible to solve for the functions u, and u2 simultaneously leading to the expressions U (CP v)P, U1 [ * k CkP1 p (p. k)2] 2 C2 [ P (Cp V)P2 k U2 2 2[p2k2 (j5. k)2] (B.45) and thus 1 f 1d 1 RZ (B.43) and (B.44) 1 ln(Z  Ck = 2: cpm k r2_U) 1P Z or, in terms of the 6integral transform, U1 =f d(m ul = 2 22 m  1 e k 1 +4 k2 f 2( 2 _ 2) (Z 1) +2ln(Z 1) 17 U2 = 4  l k 4 k2 < 2 ) 22(m 1 d Z + [1] ln(Z 1)} 2ln(Z 1) Z 2" Let us now solve for the functions v1 ,v2 and u3. To do this, we substitute Eq.(B.3) into Eq.(2.68) which gives 2CA = P 20,C 2mn(k,2 PC"1 2&,. = 2(p2 m2)(p in2) (B.48) where we have used the relation, JA = c = 2m e k k AVA)(p2 m 2)(p M 2) 12 (B.49) From Eqs.(B.12) and (B.49) it follows that (B.50) 1 d O'C'I'I =2me(Pk .I f'd) l(Z 1) aAd Aence, cZ 1) and hence, we can define the vector functions ' k" 9vCA k AV' 1 me2 d [ 1 _2 1i Z [ 1) 1 ln(Z 1) Z FA = pILO"C I 4Mek2A j( 2 = A' 2 f1 PfZ and (B.46) (B.47) and (B.51) a (z 2) i (Z 1) (B.52) +{ + [ 2 1] ln(Z 1)} [Z1 1 ln(Z 1)] A1) Z From Eqs.(B.48) and (B.50), we find that Me D (~k k~bA)&,, ( rl4 k1P&A)a,1 1 ln(Z, 1) 2m.(kS  k.A) 1 (B53) 1 2zZ and thus, using Eq.(B.17), one obtains MeX [(~k LI 5~a (S~k.E Li, Aa / 1 x d {L2(Z ) + (1 Z) ln(Z 1)} + 1mc(SAk, k,) d ln(Z 1). (B.54) Notice also that ( Ak.[]  k 6A)l d {L2(Z ) + (1 Z ) ln(Z 1)} = dC [SAk.(P + k) pk] {1 1 1)ln(Z 1)} (B.55) Therefore, the function u3 is given by U3 d (1  ln(Z 1) 2 (B.56) This expression, together with the scalar functions fk and fp allows the defini tions, FkA = fk3A and FA = fp4A, in Eqs.(B.51) and (B.52) and are sufficient to determined the functions v, and v2. The calculation is straightforward although somewhat tedious. The final form of the functions v, and v2 expressed in terms of the iintegral transform can be written as follows: let us define three subsidiary functions of Z ( 1,, 12, 13) 2ln(Z _)] (B.57) z2 2 and 73 13 = 1n(Z 1) (B.59) With these definitions, the functions vi and V2 are given by 1 f' d Vl m E [21 + 13] +t 1me J  [. k(1 6)12 k2(1 _)2] (B.60) and 1 f_' d V2 1me I [21, 13] 8 _I~ / +4me  [p k(1 + )i2 + k2 (1 + )12] (B.61) Finally, the equation for CAvp is []2A = 2eOPDA /11/p P11/ = 2 6x [ PIpP2, P1vP2/1 1 (P~~vP~pPpP21/)&A + (PlpP2A Pl/1P2p)SVl OP[(PiP~~ (~M2)(p2) J }M(2)2 Since only one function needs to be evaluated, one can try to solve the the following 4vector equation without any loss in generality, namely, 2E(Pip cp.)= (B.63) p 1.. 2 i.,t p ) obtained from Eq.(B.62) by contracting with the 4vectors p" and p Equation (B.63) can now be written as y (plp~a c,) = 2.'~ I (p (l. P2)) L (p2 M2)(p2 m2) 10 1 ( 22 M ap (I P2 )P2 p p~Plp + (pP2Pl PJP2p] (p M,2)(p M2) 5= _eO[(.p2 4k2p52 ap 41k2!p~p 1642 pA = 4J26 1 d. (B 64) M2)(p m2) 1) Using Eq. (B.12) one has that p,,P PkZ2aj d,Z 1)ln(Z 1) (B.65) If one compares this equation with Eq.(B.20), it follows that pipt OC, = 8 P CA (B.66) 3m The scalar function v3 is given by Pl p {V3 [fP2 (p1 p2)2] f "{V3 [(Pi P2)(PIp + P2p) (P2p + 2P1p)]} = 8k2 v3P5A (B.67) as terms involving derivatives of v3 disappear. Thus the function v3 is related to the ufunction by the equation, v3 U (B.68) 3m or explicitly V3 E1 + ln(Z 1)} (B.69) For reference purposes, all functions are presented in Table B1. Numerical values for the functions u, u1, u2, v3 and e are presented in Tables B2 to B6. The values of the variable x cover the perturbation and inner asymptotic regions. Table B1 Perturbation Solutions to the Model Equations Notation: x = (p+ k)2, 1 = (m2 k2) + k 2 , I, = 1  2 12 T2_ (1 1 )In(Z: 1)] Z 1 2 )ln(Z 1) 13 = I ln(Z 1) c Ej1 4 4 2 2k2) ln(Z 1)d z ) in(4 2m m2) v= C j d ( i2  4 P z 21 1 1 PJZ 1) + (i2 _ 2 IL 1 [Z (Z 1) = (+ 2 Z2 1 p.k f' d 2 2z (m  41fi 1 2(zn U 2 6 2g (M2 )2 + 42 2 1pk j' (m 4 k2 Pe : V3 = I PA d 4 1 Y , U e m <_ Z _ 1+ ,{ [1] 2ln(Z 1) Z ln(Z 1)} (Z 2) +2 ln(Z 1) (Z 1) Z + ln(Z 1) Z  in(Z 1) } (B.70) (B.71) (B.72) ln(Z 1) Z2 (Z 2) (Z 1) + (B.73) (B.74) (B.75) (B.76) An2 InU2  ){I Z I1 1 ln(Z 1)} 2 Z 1] ln(Z, 1)1 Table B1 (Continued) 1= V1  me]1 ck [2I1 + 13] + f 'd v2 = me +[2I1+13] 8 1  [ ( + )i 1 4 ' U23 = Ir~  + k2 (1 + )I2] d Z {(1  n )l(Z 1)2}  6)2] (B.77) (B.78) (B.79) [pb k(1 )I2 k 2 (1 Table B2 The Perturbation Functions u, u1, u2, v3 and e as Functions of the Variable y for x = 1 and r2 = 0.1 y u (xl03 ) u1(x104 ) u2(X104 ) v3(X103 ) e (x100 1.1 3.0219 5.4450 5.1248 1.0073 0.6688 1.2 1.9098 5.4118 5.5385 0.6366 0.6167 1.3 1.3252 5.1836 5.5005 0.4417 0.5797 1.4 0.9509 4.9085 5.3192 0.3170 0.5500 1.5 0.6881 4.6286 5.0860 0.2294 0.5249 1.6 0.4933 4.3587 4.8372 0.1644 0.5031 1.7 1.8 1.9 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0 800.0 900.0 1000.0 0.3436 0.2254 0.1304 0.0527 0.2891 0.3686 0.3847 0.3802 0.3684 0.3543 0.3398 0.3256 0.2270 0.1755 0.1430 0.1200 0.1027 0.0891 0.0782 0.0693 0.0620 0.0276 0.0165 0.0112 0.0082 0.0064 0.0051 0.0042 0.0035 0.0030 4.1044 3.8671 3.6469 3.4428 2.0523 1.3236 0.8929 0.6161 0.4275 0.2934 0.1948 0.1205 0.1363 0.1795 0.1907 0.1888 0.1789 0.1648 0.1489 0.1331 0.1182 0.0313 0.0024 0.0093 0.0145 0.0171 0.0182 0.0187 0.0188 0.0186 4.5889 4.3486 4.1197 3.9037 2.3685 1.5404 1.0481 0.7318 0.5166 0.3637 0.2515 0.1670 0.1252 0.1752 0.1885 0.1877 0.1783 0.1644 0.1488 0.1331 0.1183 0.0315 0.0025 0.0091 0.0144 0.0170 0.0182 0.0186 0.0187 0.0186 0.1145 0.0751 0.0435 0.0176 0.0964 0.1229 0.1282 0.1267 0.1228 0.1181 0.1133 0.1085 0.0757 0.0585 0.0477 0.0400 0.0342 0.0297 0.0261 0.0231 0.0207 0.0092 0.0055 0.0037 0.0027 0.0021 0.0017 0.0014 0.0012 0.0010 0.4838 0.4665 0.4508 0.4365 0.3387 0.2821 0.2441 0.2164 0.1951 0.1781 0.1643 0.1527 0.0927 0.0684 0.0548 0.0460 0.0399 0.0353 0.0317 0.0288 0.0265 0.0150 0.0107 0.0084 0.0069 0.0059 0.0052 0.0046 0.0042 0.0038 Table B3 The Perturbation Functions u, U1, u2, v3 of the Variable y for x = 10 and 12 = 0.1 and e as Functions y U (x10 ) u1(X105 ) U2(xlo' ) V&(103 ) e (x100 ) 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0 800.0 900.0 1000.0 4.1158 3.9866 3.8658 3.7525 3.6458 3.5452 3.4500 3.3597 3.2739 3.1922 2.5381 2.0595 1.6313 1.2907 1.1057 0.9760 0.8780 0.8004 0.4456 0.3175 0.2482 0.2036 0.1721 0.1484 0.1301 0.1155 0.1036 0.0489 0.0307 0.0218 0.0165 0.0130 0.0105 0.0086 0.0072 0.0061 2.2057 2.9492 3.4246 3.7571 3.9982 4.1754 4.3058 4.4006 4.4675 4.5123 4.3514 3.7814 2.9726 2.2549 1.9353 1.7255 1.5713 1.4506 0.8852 0.6699 0.5547 0.4782 0.4193 0.3705 0.3290 0.2935 0.2629 0.1068 0.0535 0.0284 0.0141 0.0051 0.0011 0.0055 0.0087 0.0112 2.1843 2.9287 3.4049 3.7383 3.9802 4.1582 4.2893 4.3848 4.4524 4.4979 4.3420 3.7753 2.9689 2.2533 1.9349 1.7259 1.5722 1.4518 0.8864 0.6708 0.5553 0.4786 0.4196 0.3708 0.3293 0.2937 0.2631 0.1068 0.0536 0.0284 0.0141 0.0051 0.0011 0.0055 0.0087 0.0112 1.3719 1.3289 1.2886 1.2508 1.2153 1.1817 1.1500 1.1199 1.0913 1.0641 0.8460 0.6865 0.5438 0.4302 0.3686 0.3253 0.2927 0.2668 0.1485 0.1058 0.0827 0.0679 0.0574 0.0495 0.0434 0.0385 0.0345 0.0163 0.0102 0.0073 0.0055 0.0043 0.0035 0.0029 0.0024 0.0020 0.1073 0.1045 0.1020 0.0996 0.0974 0.0953 0.0934 0.0916 0.0899 0.0883 0.0772 0.0730 0.0924 0.0557 0.0447 0.0381 0.0335 0.0301 0.0157 0.0110 0.0085 0.0070 0.0060 0.0052 0.0046 0.0042 0.0038 0.0021 0.0015 0.0011 0.0009 0.0008 0.0007 0.0006 0.0005 0.0005 The Perturbation Functions u, u1, u2, v3 of the Variable y for x = 103 and 2 = 0.1 and e as Functions y u (x106 ) u1(x106 ) 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0 800.0 900.0 1000.0 3.3953 3.2917 3.1954 3.1055 3.0215 2.9428 2.8687 2.7990 2.7331 2.6709 2.1921 1.8762 1.6496 1.4780 1.3427 1.2331 1.1420 1.0651 0.6579 0.4882 0.3913 0.3269 0.2805 0.2452 0.2176 0.1953 0.1770 0.0904 0.0603 0.0452 0.0361 0.1990 0.2031 0.2057 0.2071 0.1942 0.1657 0.2229 0.2600 0.2865 0.3062 0.3212 0.3328 0.3417 0.3486 0.3538 0.3625 0.3448 0.3233 0.3027 0.2841 0.2676 0.2529 0.2398 0.1614 0.1248 0.1034 0.0889 0.0781 0.0696 0.0627 0.0569 0.0520 0.0273 0.0183 0.0137 0.0110 0.0179 0.0381 0.0536 0.0672 21.3921 u2(x106 ) v3(x106 ) e (x104 ') 0.1665 0.2235 0.2606 0.2871 0.3068 0.3217 0.3332 0.3421 0.3490 0.3542 0.3628 0.3450 0.3234 0.3028 0.2842 0.2676 0.2529 0.2399 0.1614 0.1248 0.1034 0.0890 0.0782 0.0696 0.0627 0.0569 0.0520 0.0273 0.0183 0.0137 0.0110 0.0179 0.0381 0.0536 0.0672 21.3921 1.1318 1.0972 1.0651 1.0352 1.0072 0.9809 0.9562 0.9330 0.9110 0.8903 0.7307 0.6254 0.5499 0.4927 0.4476 0.4110 0.3807 0.3550 0.2193 0.1627 0.1304 0.1090 0.0935 0.0817 0.0725 0.0651 0.0590 0.0301 0.0201 0.0151 0.0120 0.0663 0.0677 0.0686 0.0690 0.0647 9.6858 9.3891 9.1132 8.8560 8.6154 8.3899 8.1779 7.9782 7.7897 7.6114 6.2410 5.3369 4.6888 4.1979 3.8114 3.4979 3.2379 3.0182 1.8564 1.3738 1.0994 0.9179 0.7874 0.6886 0.6112 0.5490 0.4980 0.2556 0.1712 0.1286 0.1029 0.1316 0.1100 0.0954 0.0845 0.0760 Table B4 U2( X 106 ) V3( X 106 ) e (X104 ) The Perturbation Functions u, u1, u2, v3 and e as Functions of the Variable y for x = 10'0 and V = 0.1 y U (X1013 ) 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 200.0 300.0 400.0 5OO.0 600.0 700.0 800.0 900.0 1000.0 3.3725 3.2691 3.1729 3.0832 2.9993 2.9206 2.8467 2.7771 2.7114 2.6492 2.1715 1.8563 1.6304 1.4593 1.3246 1.2154 1.1247 1.0482 0.6433 0.4752 0.3798 0.3170 0.2719 0.2379 0.2112 0.1898 0.1722 0.0887 0.0595 0.0447 0.0358 0.0299 0.0256 0.0224 0.0199 0.0179 u1(xl014 ) 1.6408 2.2043 2.5703 2.8312 3.0253 3.1727 3.2859 3.3731 3.4403 3.4914 3.5711 3.3919 3.1761 2.9703 2.7848 2.6200 2.4740 2.3442 1.5661 1.2033 0.9914 0.8491 0.7442 0.6622 0.5961 0.5413 0.4954 0.2634 0.1779 0.1341 0.1075 0.0897 0.0769 0.0674 0.0599 0.0539 Table B5 U2( X 1014 1.6408 2.2043 2.5703 2.8312 3.0253 3.1727 3.2859 3.3731 3.4403 3.4914 3.5711 3.3919 3.1761 2.9703 2.7848 2.6200 2.4740 2.3442 1.5661 1.2033 0.9914 0.8491 0.7442 0.6622 0.5961 0.5413 0.4954 0.2634 0.1779 0.1341 0.1075 0.0897 0.0769 0.0674 0.0599 0.0539 v3( XlO13 1.1242 1.0897 1.0576 1.0277 0.9998 0.9735 0.9489 0.9257 0.9038 0.8831 0.7238 0.6188 0.5435 0.4864 0.4415 0.4051 0.3749 0.3494 0.2144 0.1584 0.1266 0.1057 0.0906 0.0793 0.0704 0.0633 0.0574 0.0296 0.0198 0.0149 0.0119 0.0100 0.0085 0.0075 0.0066 0.0060 e (xlO") 9.6795 9.3825 9.1064 8.8489 8.6082 8.3825 8.1703 7.9705 7.7818 7.6035 6.2323 5.3278 4.6794 4.1884 3.8017 3.4882 3.2281 3.0083 1.8463 1.3639 1.0902 0.9098 0.7804 0.6827 0.6062 0.5448 0.4943 0.2545 0.1707 0.1283 0.1028 0.0857 0.0735 0.0643 0.0572 0.0514 The Perturbation Functions u, u1, u2, v3 and e as Functions of the Variable y for x = 1020 and I2 = 0.1 U u (x 1023 ) 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0 800.0 900.0 1000.0 3.3725 3.2691 3.1729 3.0832 2.9993 2.9206 2.8467 2.7771 2.7114 2.6492 2.1715 1.8563 1.6304 1.4593 1.3246 1.2154 1.1247 1.0482 0.6433 0.4758 0.3818 0.3210 0.2780 0.2460 0.2211 0.2010 0.1846 0.1044 0.0743 0.0582 0.0481 0.0412 0.0361 0.0321 0.0290 0.0265 u1(x1024 ) 1.6408 2.2043 2.5703 2.8312 3.0253 3.1727 3.2859 3.3731 3.4403 3.4914 3.5711 3.3919 3.1761 2.9703 2.7848 2.6200 2.4740 2.3442 1.5657 1.1994 0.9826 0.8378 0.7334 0.6541 0.5917 0.5412 0.4993 0.2899 0.2089 0.1651 0.1372 0.1179 0.1036 0.0926 0.0838 0.0767 u9(x1024 ~ 1.6408 2.2043 2.5703 2.8312 3.0253 3.1727 3.2859 3.3731 3.4403 3.4914 3.5711 3.3919 3.1761 2.9703 2.7848 2.6200 2.4740 2.3442 1.5657 1.1994 0.9826 0.8378 0.7334 0.6541 0.5917 0.5412 0.4993 0.2899 0.2089 0.1651 0.1372 0.1179 0.1036 0.0926 0.0838 0.0767 v3 (x1023 ) 1.1242 1.0897 1.0576 1.0277 0.9998 0.9735 0.9489 0.9257 0.9038 0.8831 0.7238 0.6188 0.5435 0.4864 0.4415 0.4051 0.3749 0.3494 0.2144 0.1586 0.1273 0.1070 0.0927 0.0820 0.0737 0.0670 0.0615 0.0348 0.0248 0.0194 0.0160 0.0137 0.0120 0.0107 0.0097 0.0088 e (x1021 ) 9.6795 9.3825 9.1064 8.8489 8.6082 8.3825 8.1703 7.9705 7.7818 7.6035 6.2323 5.3278 4.6794 4.1884 3.8017 3.4882 3.2281 3.0083 1.8464 1.3654 1.0958 0.9212 0.7980 0.7060 0.6344 0.5770 0.5299 0.2996 0.2132 0.1671 0.1382 0.1182 0.1035 0.0922 0.0833 0.0760 Table B6 U2( X1014 ) APPENDIX C ASYMPTOTIC FORMS OF GREEN'S PERTURBATION SOLUTIONS In order to obtain the asymptotic forms of the perturbation solutions derived in Appendix B, we must observe that the xdependency of the cintegral transform occurs in the function X = f;2 + 2P. k + k2. In the asymptotic region, 2 k = xv/y2 (C.1) P ~.xy; 2~~ Consequently, X can be approximated by the expression x XO(6), (C.2) where o( ) = y + 6 i (C.3) Notice also that Z 1 can be written in the form where X= 1, (C.5) and thus ln(Z, 1)' in f + In 0(6) In( 2X) (C.6) To obtain a good approximation of the iintegrals, in the asymptotic re gion, it is necessary to keep only those terms in the integrand that have the smallest power in x1. We must be careful, not to neglect terms such as Z I against Z, I In Z due to the slow variation of the logarithm. To avoid confu sion between the perturbation solutions in the asymptotic region and the actual asymptotic solutions, we will use the superscript "pert" when we refer to the former solutions. Using Eq.(B.20), (See also Integral Table at the end of the chapter) it follows that (t) 3 3 1 d6 = 3 In z upe ~ me m.7 4 1 X 2 x vy2 1( This expression is not accurate enough in the asymptotic region due to the fact that in obtaining the perturbation solutions we set A1 A2 m. However, by Eq.(3.61), A1 A2 x". It has been shown23,24 that an extended solution can be constructed if we replace the mass m by the average of A1 and A2. Assuming this to be valid, then u(pert) = 3(A1 + A2)E /nz (C.8) The form of the function v(pert) in the asymptotic region can be obtained from Eqs.(B.30) and (B.39). From Eq.(B.30), we see that it is possible to approximate the function C by the form C e 1ln(Z, 1)d6 !F [ln(M))n(2 (C.9) and therefore, v(pert) 1+C=l+ I e 12Xtan(1/X)+ ylnz (C.10) 2 L/ 12 Notice that the largest term in v(per) is xindependent; all other terms being of order O(1/x). From Eq.(B.33) it also follows that the function e(Pe"t) is given by Slnz(C.11) 2 X( xC/.2 The form of the functions u (pert) and (e) can be extracted from Eqs.(B.46) U 1 'U2 cnb xrce rmEs(. and (B.47). It is clear that the largest contribution to these functions comes from the integrals being multiplied by p k. From these equations, it follows that there is a simple relation between the two functions, i.e. U(pert) = (pert) 4 (M c. This integral is elementary (see Table C1), and hence u (Pert) (pert) 1 y i( z 3 1 2 2 (21) V 2(C. 13) We now turn to the problem of finding the form of the functions, v (pert) (pert) 1 2 (p ert) and u3 in the asymptotic region. These functions are particularly difficult to calculate since we must split the terms containing the function ln(Z 1) as pre scribed in Eq.(C.6). Also a test of these functions by substitution in their cor responding differential equations, show that there is not a simple assumption we can make regarding the transition of the mass m from its perturbation value to its asymptotic form. _(pert) The function u3 given in Eq.(B.56) has the following form in the asymp totic region, (pert) (1 'U3 4 1/Z 1 Z,, [2 +ln(Z (C114 Using Eq.(C.6) and the integrals found in Table C1 it is possible to write u(pert) in a somewhat complex form. To this end, let us define the functions 7'1 and T2 by the equations yIn z __ 2___ 2n and 72 = m2(y2 P1) + k2 (C.15) y 1 Using these definitions, Eq.(C.14) integrates into the form, 1e [1 ln(x/m2) + T1 + 272[x tan'(1/X) Ti]] IX2 (C.16) 3 2rr where X is defined in Eq.(C.5). The form of(p ert) thas Thet roin the asymptotic region can be found in an analogous manner from Eq.(B.60). In this region we have v(pert) 1M ln(Z 1)] 1 8 11 PZ 2p .(1 )[2 1n(Z* 1)] . (C.17) Using Eq.(C.6) and defining the integrals, I < j2 1 [ln(x/k 2) + In ()] k 2p. k +X2( + lnO(6)]} ]1 (1 2 = j d ln( 2 + 2) 12 x X ) [ln(xl k2) + 2f k X2(1 it is possible to write Eq.(C.17) in the form, v(pert) 1 1 8 MC(11 12) Using Table C1, it is found that the integral I, can be written in the form 11 = [111 2p. Ic(13 114)] [ln(x/rn2) 2] + 2P kI5 , I 21nz i X x  h12 = f 1In () 1 x 113 1 2 1X and 2Ilnz y X2 (y2 1 1 d6 _1 jQ 2 X2 2 X2 Vy21z [ (z+ vy21 )n zj 0O, (0.22) (C.23) and (C.18) (C.19) (C.20) (C.21) where j in 0(6) 1) ( )In 0(6)d6 115 X2 (C.24) Substituting Eqs.(C.22) to (C.24) into Eq.(C.21) gives 11=7{1+1 ln(x/k2) } (.5 rl= Iz1+ Tj (C.25) The integral 12 given in Eq.(C.19) can be split into three integrals of the form, 12 l [421 + I22 Y1 123 y21, (C.26) where 21 1=11 ln( 2 + X2) (0.27) 22 = I ln(62 + X2)d6 I22 =1 02 ) '(.28) and 123 = 6 ln( 2 + x2)d6 /23 =(C.29) fl 02( ) The integral in Eq.(C.27) cannot be expressed in terms of a finite number of ele mentary functions. Fortunately, it will not be necessary to evaluate this integral in order to define v, (or ) since the integral 123 can be expressed in terms of the integral 121. To show this, we must integrate by parts Eq.(C.29), i.e. {=+1 i1 123 U'22 I22d (C.30) In general, we have f ln(t2 + a2)dt 1 ln(t2 + a2) 2f tdt G t) (t +d)2 = c (ct +d)2 + J~ ,)td (0.31) (c +d2 c(t+d2 C (t2 + a2)(ct + d) and I t d 2(t a lnh(t2 + a 2) d ln(ct + d) + ac tan'(ta)} (t2 + a + d) 2+ d2{ af +a2)(2 (C.32) Setting a = X, c = Vy2 land d = y in Eq.(C.31) and evaluating the integral gives 122 = 21n(l + X2) + 472[X tan1(l/X) r1] (C.33) where 71 and r2 are defined in Eq.(C.14). To complete the evaluation of the integral 122, we must integrate once again Eq.(C.31). This integration is elementary, and yields the result, I'fd dln(t2 + a2)dt 121 d t (ct + d)2 Vy2 1 + 2y2 {ln(l x2) + 2Xtan'(l/X) 272} (C.34) Multiplying Eq.(C.31) by t and evaluating the expression at t = 1 gives tG(t) 272yVy2 1 in(2k. (C.35) Cnm2/k2 Using these results, we find that the integral 12 can be written in the form, 12 = V/2 1 [122 T3(y)] (C.36) x where the function 73 is defined by T3(Y) 2y2 {ln(m22)/.2 2Xtanl(l/X) 21 (C.37) V~ 1{nm/k)r+ Notice that the integral 121 has disappeared. Finally, substituting Eqs.(C.36) and (C.25) into Eq.(C.20) yields the desired result for the function vipert), namely (pert) 1 \Z H(x,zy) (C.38) where H(x, y) = 1 ln(x/m2) + ylnz + 272[Xtan'(1/X) 71] (C.39) A similar procedure can be used to show that the function v2 is given by the expression, =4pert) + 15H(x,y) Z (C.40) For convenience, all asymptotic forms are presented in Table C2. Table C1 Table of Useful Integrals. Notations: a = xy; b= xVy2 1; S=(xzM 2) R = (xz (xz' m2) _d 2inz (a b XVy21 2.1 d 2 2 (a + b )  f.1 (a+ b6) 4. 1 (a2bd) fi (a + b6) 2 x 2y x(y2 1) 1 ylnz E y2 [1 yl_] 5 d 2 5 1 (a + b )2 X2 6.1 __ 1 d 2 [lnz y] 1(a+ b)2 X2 (y2 1) 2 8.f 9.11 2d6 2 [ + 2y2 ylnz] (a+b6) x2(y2 1) d6 2y (a + b6)3 x3 d6 2 1 (a + b)( + #6) 7 + m 2x 1 [lnS21nz] 10. 1 ln(a +3 )d = 2 + 2ylnz + 2in(x/M2) 1 + 2 1n/m) 4 m 2 mr4 x = F 1; k2 7"2 =m2(y2 1)+k2 a  2 )  m2)(xzl_M2)/ Table C1 (Continued) ln(a + #6)d6 (a + b6) 10./1 11./ 1 [ln(xz/m2)]2 [ln(xz/m2)]2 2 x V/y21 ln(a+b )d 0 (a + b ) 12 [1 ln(a + b)d6 1 (a + b6)2 13. 1ln(a + b )d4 .i (a + b6)2 14. j 2[1~ ylnz 2y i/ 1+21nz 1+ 2 1i (2 + 2)d 2ylnz in(a + b )2 = 21n(1 + x2) + 4r2 X tan, (l/x) 90 Table C2 Asymptotic Forms of the Perturbation Solutions Notation: ylnz k2 1 y2 1 72= m2(y21) +2 2y2 {ln(m2/ k2)/72 + 2Xtan1(1/X) 2TI}j y 1 H(x,y) = 1 ln(x/m2 + 2/ [71 X ta'(~1 "2  v(pert) 1+C=l+ 1 4 24Xtan(l/X) + 2yinz (4 4xy y 12 S(pert) I in z (.42) (per) + 6 y n z I(C .43) U1 2 (~y2 ) y2 1 (pe rt) 1 [ylnz 1 (0.43) 2(y2 1) 1C.44 (pert) 1 In z 3 y21 (C.45) U(pert) ,1 3(A1 + A2)/E inz (C.46) 4 ~X Vy 2l pert) eH(x, y) x C.7 1 4 1 (p ert) z (048) (rt +EH(x, y) (0.49) u (pert) 1H(x, y)/x 2 (C.49) 3 2 APPENDIX D ASYMPTOTIC BOUNDARY CONDITIONS To obtain the boundary conditions to be used with the differential equa tions derived from the model equation, we use a technique developed and used by A. Broyles2' in connection with the determination of the electron propagator functions. This technique is equivalent to the use of Green's theorem by which it is possible to convert certain volume integrals into surface integrals. In order to find the boundary conditions for the vertex function, we start with the definition of the vertex integral equation as given in Eq.(2.34), namely, rA(l,2) = z3(2)2 JDrp q)_yPFA(ql, 12)d4q (D.1) where Z3eC 2 0 V (D.2) (27r)2 7r' a, /ap , F~r'(ql ( ) _.~ql)FAqlq2 S(t2) _u ,(D.3) and DIZV(k) = Z3 [guv + kikk2] k2 (D.4) We have also defined the 4vectors P, = p + k P2 = P k which represent the outgoing and incoming electron momenta respectively. Substituting Eq.(D.4) into Eq.(D.1) gives FA(p p2)= + [2 {(EA) (= 7 A(P qV)EAv }d4q (D.5) _27 (p )2q) Using the identity, 1 2 pp, = _&,cOvlnp2 + gpvp (D.6) the last term in Eq.(D.5) can be written as (/f)( q,)F d 4 1.F'vd 4 q 1 (p_ q)4 2= 7 9 I (p q)2 4 (D.7) where a, I n(p~ )2 "(q1, q2)d4q (D.8) Substituting Eq.(D.7) into (D.5) yields Aflf2 A 2 (27r)2 jpi . 4 (2 ')2 (D.9) We can apply W to Eq.(D.9) with the result, 1 ic f FAL/d4q 1i F(Pl,P2) 2 (27r)2 v f (p q)2 4 (27r)2 (D.10) We would like to expresss the integral a so that the fdependency appears only in the form (f; q)2. Notice that from Eq.(D.8), V!= [D]2q,'ln(i q)2] FLd4q q ] {[ In()2]Av} d4q + J {LI2 ln(P a v)2 } :q FA d4q (D.11) where []2 stands for the D'Alambertian operator in the momentum q (see also Eq.(2.35)) and q'  = /Oq,. In general, a subscript q will be used to denote derivatives with respect to q. The first integral in Eq.(D.11) can easily be obtained and is given by J 9q, { [D2 iln( q)2 1AvI} d4 q I2n(i q)2]F"}do, (D.12) where dou is the outward drawn normal to the volume d4q. The identity, 02 Inp2 = 4P 2 (D.13) can be used to show that Y70 = 4 1 (F dc2 (p q)2 +4 1 (p _)2 . Also, from the definition of V, it follows that F d4q _ (p_ ) y a V P(P = _, ,( q)2]FA "d4q It is possible to transform Eq.(D.15) using the identity, qC ( q) FE"v] = [Oq'(p _)]~ q)2]&qFAL+ Using Eq.(D.16) in Eq.(D.15) yields the result EJ''d 4q _,'Y J + 'd Ci a J(q AD'd 4 q s it q( ) i nto Eq)2 O we obt ain Substituting Eq.(n.17) and (D.14) into Eq.(D.10) we obtain 2 (27r)2 { ~ A I 4y I FAvdo Jqvf_ dq } or, grouping all integrals over the volume d4q, i= J[" jE + F q'A,] d4q + + (27r) 2 (p )2 FAdcv } (p q) 2f We can reduce Eq.(D.18) to a differential equation by using the identity (D.14) q)2]FAld4q (D.15) (D.16) (D.17) i+ + (2 (D.18) (D.19) [L2p2 = i(27r)264(p) (D.20) If we exclude an infinitesimal volume around the point f , there will be no contributions from the surface integrals in Eq.(D.19). Hence, if we apply the D'Alambertian []2 to Eq.(D.19), we obtain [I2V_ A(p1,p2) = 6(oF,, + (D.21) Notice that this is the same equation obtained in Chapter II by an independent method (see Eq.(2.50)). In order to obtain the boundary conditions to the differential equation given in Eq.(D.21), we integrate Eq.(D.8) by parts excluding the point (p = q) to obtain .= V /[Oq ln(p q)2]FAi/d4q fln( )2 F Avdu + f ln(5 q)2 aqE"d4q. (D.22) We can rearrange Eq.(D.21) to obtain the form, EPEv= [2,A 47 F A (D.23) Substituting Eq.(D.23) into Eq.(D.22) gives = =' JV ln(i q)2FA'"do, + W Jln(P )2 I7qAd4q + ln(P q)2 _qjFAvd 4q (D.24) Integrating by parts the last two terms in Eq.(D.24) gives = J ln(p q)2 F Azdav + V J ln(p q)2[]2 d !P V l J[[Pq ln(p q)2][IFAd4q ln(F q)2 d4fFA, 6V [7 ln(P q)2]7 _FAd4q (D.25) 2 J  Replacing 3/q with 5 and rearranging terms gives 6!= J02 ln(P o)2]02FAd4q 1 [02 n(p q)2 ]FAd4q c ln(p )2F"do,, + Jln(P 2 2 1Jln( q)2d7_ Fv. (D.26) We can now make use of Eq.(D.13) and substitute the previous expression into Eq.(D.9) to obtain 1(,2)= (2){ 4J Ad4q ln(pVFvdo  (27r) (p? in(_ 2) _F + fln(fi q)2 dE]F F } (D.27) At this stage, it is possible to partially integrate the first volume integral excluding the point ji = q and therefore q i qjr d o, i [ q,( .p q)2] r d J EVF'ddq P 0q [aFq)1drd (p q)2 A + I [[]2(.p q)2 4 rxd (D28 pj5O j~[~(P (3.28) Notice that the last integral in Eq.(D.28) vanishes as can be seen by looking at Eq.(D.20) and the fact that we are excluding the point j5 = q in the integration. Using Eq.(D.16) once again, and replacing Oq by a' we obtain, ([_]2 =i 4: +A (Aq) f Lq q /, r F d ,, fo F(D .29) (q)1(p q) p q) (D'9 