An approximate vertex amplitude from the Schwinger-Dyson equations of quantum electrodynamics

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An approximate vertex amplitude from the Schwinger-Dyson equations of quantum electrodynamics
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Table of Contents
    Title Page
        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
Full Text










AN APPROXIMATE VERTEX AMPLITUDE FROM THE
SCHWINGER-DYSON EQUATIONS OF
QUANTUM ELECTRODYNAMICS













By

RUBEN A. MENDEZ-PLACIDO


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 .........
2-1 The Schwinger-Dyson Equations ... ............
2-2 Approximations to the Schwinger-Dyson Equations . .
2-3 The Model Equation .......................
DERIVATION OF DIFFERENTIAL EQUATIONS .......
3-1 Definition of variables in asymptotic regions ........
3-2 Derivation of Differential Equations ..............
3-3 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 e-2 Q'(z) ...... ................ ...106
Behavior of P/(z) and Q"(z) at the Singularities ... ....... 108


Table 3-1
Table 4-1
Table 4-2
Table B-1
Table B-2


Table B-3


Table B-4


Table B-5


Table B-6


Table C-1
Table C-2
Table E- 1

Table E-2
Table E-3














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
SCHWINGER-DYSON EQUATIONS OF
QUANTUM ELECTRODYNAMICS

By

Ruben A. Mendez-Placido
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 Schwinger-Dyson 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 electron-positron pair

by a photon may be interpreted in the following way. The vacuum consists of

an infinite sea of negative-energy 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 electron-positron pair. Thus the vacuum may be considered as a sort of
"polarizable" medium, because it potentially contains electron-positron pairs. A

photon may now interact with this polarizable vacuum even if its energy is not

sufficient to create a real electron-positron 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 self-energy, 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 hydrogen-like 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 cut-off in the integration limits. The final results were,

therefore, dependent on a cut-off parameter. This dependency was logarithmic in








the cut-off 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 so-called "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 Schwinger-Dyson 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 energy-momentum

near zero and electron's square energy-momentum 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 so-called








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 non-Abelian
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 electro-weak 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 fine-structure 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 10-12 (1.3)

ath = 1 159 652 460 (127) x 10-12 (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, Gell-Mann 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,14-17 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 non-perturbative manner, the solutions to the Schwinger-Dyson 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 Schwinger-Dyson hierarchy of equations using

a non-perturbative 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 Schwinger-Dyson 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 mass-shell 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 so-called 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
Schwinger-Dyson Initial Approximations
hierarchy 1. Truncation of hierarchy
of equations 2. Photon Propagator


9
Go to next level
in Schwinger-Dyson
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 Schwinger-Dyson hierarchy of equations.








In order to determine the photon wave-function renormalization constant, as-

sociated with the photon propagator Z3, it is necessary to devise a more accurate

form for the photon propagator. The Schwinger-Dyson 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 self-consistent, 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 Schwinger-Dyson 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

2-1 The Schwinger-Dyson 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 electron-photon 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 Schwinger-Dyson 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 space-time given that it

was at a different point in space-time 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 non-homogeneous integro-differential 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 Schwinger-Dyson 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 four-vectors and an underline to represent matrices. Here eoA_" repre-

sents the sum of all possible three-external-point, 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 four-point diagram to either








a dressed electron, or a dressed photon lines. These diagrams can be separated out

of the four-point 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 four-point diagram to the five-point

and so on. In addition to the Schwinger-Dyson 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 four-point amplitude EAI

and the three-point 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 four-point 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 p-i q)- rA(p,p + (2.20)

Similar relationships exist for the remaining n-point diagrams. These identities
exactly define the longitudinal components of the n-point diagram in terms of the








(n-1)-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 four-momenta. 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 +k-q)] (p V-q)

+ V_ (q,. q, P + k)} d4q.

2-2 Approximations to the Schwinger-Dyson Equations


Although equation (2.28) is exact, it involves another equation in the hierar-
chy through the four-point 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 + k-q)] (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 Schwinger-Dyson 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 Beth-Salpeter equation. It was first used in

the study of the Schwinger-Dyson equations by Bose and Biswas.2"


2-3 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,() Z3-g, + 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(2-p2)/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+5-k) 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+ )]_S-1(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(j-q)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

]2-A(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 anti-commutator 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- 20-D, (2.67)

I2 CA = -eDA- 2c'0,DaD + 2r8VA-2aPDA (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)CapF1-2,]/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)

+ Pp2v-P 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 -I-l. 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 = v-A2V+AV2+(AA2+ P A P2)v3}/ D. (2.92)














CHAPTER III
DERIVATION OF THE SCALAR DIFFERENTIAL EQUATIONS

3-1 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 -z-l (3.2)
P22
z=y+ y-Z1; -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 =xy-k2 ; p.-k =lIx Pl1fP2 p2 k2=xy-2k2 (3.5)

If both p and p2 are time-like or space-like 4-vectors, 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 Vy21Ppt-y 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) =4x-k2)w* + (y -)W + x y

k2 2(y2 1) y2 1)W(3
+4 Y2 x WY-4 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
z-1i,+ zf = h
(3.23)
z f+ zf = i .
3-2 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,[-I-20a"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

[]2C-1 = SA [Dv + 2(u1 + u2)]

+ PI P [-2U2 + 2h(U2) + 2z-1f(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 right-hand side of Eq.(3.25) becomes

2e [D- OEA] = 2,E (S e)&- + pAP [R2 z-f+()]

+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 z-f+(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) ={-3z-1f+(V + v2) 2h(V2)

z-f+ [P1 -f2]- z-1[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(v2-v)]-/ .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)

2z-lf [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) 2z-1f+ [p1 1V2] 2z-f+ [5 IEv1 ]

-2z-f+(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 right-hand 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)

3-3 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(xz-1/m2)- (361)








where by Eq.(2.38), 71 = 3e/4 1.75 x 10-3. 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 right-hand side of all the differential equations can be approximated by the

expression

D-x (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 3-1.







Table 3-1 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 + 2z-1f+(ul + U2)= 2c[R2 z-1f (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) 2z-1 f+[i E]V2] 2z-1 f-+[P2 Elvi ]
2z-1 + (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 2-X7 ,V (3.81)
1
R2v-1 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 -Ar-z2- x(zA2 A)3 (3.85)
1

S2 -- "2[ A2v xz-v1 + x(z-'A1 + y A2)v3] (3.86)
S3-1
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 C-2

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 y-dependent. 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


X1-1
f+( vg7_ (Da az)j I

xa--1
()=+ 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 ] = 2x--1 y 1 [a52 y ba]g (


h[k. h1] =2h E() + p 1( h
h [k. = ai .nk (oh)


(4.16)









0-4 2xo-1[ I a (4.17)

F. ] 2x-1 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 C-2 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

v-function (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 right-hand side of the differential equations equal to zero

except for the v and e-functions. Substituting Eqs.(4.21) to (4.24) into Eqs.(3.71)

to (3.75) and setting a = -1 gives


4[D-1 + 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)

4D-1i3 = -2eV, (4.28)

and


4D1e = l0V+{ z(D1 y) 2} z'(Di y) 2 U+2Di(-yU) (4.29)


where

D = (1-y2) 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 x-dependency 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

y-dependent. To solve these equations, we start by adding Eqs.(4.26) and (4.27).

This gives the result

2f-1(u1 + ii2) = E(v D-I) (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,


4D-1[b-1 + y]v + 2F[yb-l (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 1-I
7, )-y 1)'- v + 2- (4.39)

and

V= (y2 -1)rv"+47y(y2_ 1)-lvF+ {27(y2 -1)r1 +47(7-- )y2(y2 )r-2V

(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

(y2-1) ="+2yv'+{2+[4T(T-1)-27(47-2)]y2(y2-1)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

(Y-1)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. (y2-I 2)-
yI+,F(v 1 -, 1 t; 1 y-2) (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:

2-7r 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, .- ;Y-2) (4.50)








The P-function 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 U-function differential equation in the right-hand

side. From Eq.(4.32) and the expression for b-1 given in Eq.(4.30), the differential
equation that the c-function must satisfy becomes


(y2 1)' y -cl(y2 12 "Pf(y).


(4.52)


Let assume that this function has the same form as the 5-function, 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 P-1=-(v+l), (4.59)

are satisfied. It is evident from Eq.(E. 17) that the e-function 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) - (D-1 + z-l) (4.61)

Substituting Eq.(4.32) into this equation and simplifying gives

I zv +~
z(E+Vy2-1')= -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(p-3) + (y2 1) (/-1)(y2 1)u'


(4.66)








Then the left-hand side of the Eq. (4.64) becomes


(Y2 1)u + Yu1 = (y2 1)6( -1) [A y + (y2 1)ui1' (4.67)


The right-hand 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 right-hand 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 i3-function 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 e-function 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 v-function 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 P2-A2p2]
X2 fL 1

+(Al+A2)v +xT:I(vI+v2)+x2(y2-1)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)Vy-1P.-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)
v-V 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 u-function 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


X-14(3 + 2)bD,9i = -3,(1 1,7)x-2-(z? + z-) (4.90)
2

Matching the x-dependency 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 + t-2] 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 10-x):

(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 4-1.

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 4-1 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 (1-3)x-+){(z z-'7)+2rjz-'lnz} /Ny2-1 (4.111)








Table 4-2 Perturbation and Asymptotic Solutions in the Overlapping Region



Perturbation Solutions (x = 1020 and y = 1.1)


1t derivative
2nd derivative


V
9.957E-01
7.301E-04
-4.155E-04


e
-9.679E-01
3.083E-01
-2.352E-01


U1
1.640E-04
7.341E-04
-4.877E-03


V3
1.124E-03
-3.579E-04
2.731E-04


3.372E-03
-1.074E-03
8.193E-04


Asymptotic Solutions (x = 1020 and y = 1.1)


1t derivative
2nd derivative


9.992E-01
7.298E-04
-4.149E-04


-9.671E-01
3.078E-01
-2.348E-01


1.640E-04
7.338E-04
-4.875E-03


V3
1.123E-03
-3.574E-04
2.726E-04


ii
3.373E-03
-1.081E-03
8.574E-04


Perturbation Solutions (x = 1020 and y = 10)


It derivative
2nd derivative


1" derivative
2nd derivative


v
9.979E-01
1. 138E-04
-1.101E-05

Asymptotic
T)
1.001E+00
1.139E-04
-1.102E-05


e
-3.008E-01
2.029E-02
-3.109E-03


U1
2.344E-04
-1.224E-05
1.390E-06


V3
3.493E-04
-2.355E-05
3.609E-06


U
1.048E-03
-7.066E-05
1.083E-05


Solutions (x = 1020 and y = 10)


-3.008E-01
2.027E-02
-3.105E-03


ii,
2.345E-04
-1.223E-05
1.388E-06


i3
3.493E-04
-2.354E-05
3.606E-06


U
1.046E-03
-7.063E-05
1.083E-05


Perturbation


Solutions (x = 1020 and y = 100)


It derivative
2nd derivative


v
1.001E+00
1.161E-05
-1.160E-07


e
-5.299E-02
4.299E-04
-7. 599E-06


4.991E-05
-3.831E-07
6.504E-09


V3
6.152E-05
-4.992E-07
8.823E-09


1.846E-04
-1.497E-06
2.647E-08


Asymptotic Solutions (x = 1020 and y = 100)


It derivative
2nd derivative


v
1.004E+00
1.165E-05
-1.163E-07


e
-5.305E-02
4.301E-04
-7.601E-06


U1
5.OOOE-05
-3.835E-07
6.508E-09


V3
6.160E-05
-4.994E-07
8.825E-09


U
1.839E-04
-1.493E-06
2.641E-08








Table 4-1 (Continued)



Perturbation Solutions (x = 102" and y = 1000)


1t derivative
2nd derivative


V
1.003E+00
1.161E-06
-1.161E-09


-7.601E-03
6.601E-06
-1.220E-08


ii1
7.664E-06
-6.503E-09
1.185E-11


U3
8.825E-06
-7.664E-09
1.417E-11


U
2.648E-05
-2.299E-08
4.251E-11


Asymptotic Solutions (x = 1020 and y = 1000)


at derivative
2nd derivative


1.007E+00
1.169E-06
-1.168E-09


-7.620E-03
6.613E-06
-1.222E-08


7.687E-06
-6.518E-09
1.187E-11


U3
8.848E-06
-7.679E-09
1.419E-11


u
2.633E-05
-2.288E-08
4.232E-11














CHAPTER V
THE CONCLUSION


A solution for the vertex amplitude has been found to the Schwinger-Dyson

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(m2-P2)/4 ,p2
d(, (5.2)

B(2)=c_ 1 (5.3a)


In the asymptotic region, a simpler form of the A-function was used, namely


A(12) ,,_ (/2/2)-,, 7 -_ 1.75 x 10-3 (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 'U-j21 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 x-dependency 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

u-function. 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 v-function, vanishes at infinity as
rapidly as O(1/x) and therefore is negligible compared to the v-function. Also,

these functions are one order in the coupling parameter smaller than the v-function.

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 self-energy 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 four-dimensional 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 4-dimensional 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 4-rank 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 -2-y1
--p-A 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)

V-9p= -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 right-hand side becomes


[j2cA = 6FmpA
(p m2)(p2 m2)

Integrating Eq.(B.19) using Eq.(B.15) gives


CA 3 Jj\1 d
4 pJ-1 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
(p2-p2)


2


(B.23)


(B.24)


a
5,7-C" ,
PA








where C and C. satisfy the differential equations


1 P,L + P21 in
(p2 )


LI2C = 2c (2 l 22)
(j51-M2)(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


8-Cp) =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


- (i-t2


1


22
1 l


2k2 ln(Z 1)d
Y z )


( 2
m2


-1)


0j2CA = 2e


+ OE,


(B.25)


(B.26)


( - m2
Ap2-m )


(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 6-integral 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 Pf-Z


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)]
A-1) 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 i-integral 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 1-me 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~p-PpP21/)&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
4-vector 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 4-vectors 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 u-function 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 B-1. Numerical values

for the functions u, u1, u2, v3 and e are presented in Tables B-2 to B-6. The values

of the variable x cover the perturbation and inner asymptotic regions.








Table B-1 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


1p-k 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)


An-2
In-U2


-- ){I


Z


I1- 1 ln(Z 1)}


2
Z 1] ln(Z, 1)1








Table B-1 (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 B-2 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 B-3 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 B-4


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 B-5


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 B-6


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 x-dependency of the c-integral 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( 2--X) (C.6)


To obtain a good approximation of the i-integrals, in the asymptotic re-

gion, it is necessary to keep only those terms in the integrand that have the

smallest power in x-1. 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 1-2Xtan-(1/X)+ ylnz (C.10)
2 L/ -12

Notice that the largest term in v(per) is x-independent; 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 C-1), and hence


u (Pert) (pert) 1 y i( z 3
1 2 2 (2-1) 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 C-1 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 C-1, 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 Vy2-1z [- (z+ vy2-1 )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 Y-1 123 y2-1, (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 tan-1(l/X) r1] (C.33)

where 7-1 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) -2--72yVy2- 1 in(2k. (C.35)
Cn-m2/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 -2Xtan-l(l/X) 2-1 (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 C-2.








Table C-1 Table of Useful Integrals.


Notations:


a = xy; b= xVy2 -1;


S=(xz-M 2) R = (xz-
(xz' m2)


_d 2inz
(a b XVy2-1

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 [lnS-21nz]


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)(xz-l-_M2)/







Table C-1 (Continued)


ln(a + #6)d6
(a + b6)


10./1
11./


1 [ln(xz/m2)]2 [ln(xz-/m2)]2
2 x V/y2-1


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) + 4-r2 X tan-, (l/x)






90

Table C-2 Asymptotic Forms of the Perturbation Solutions


Notation:
ylnz k2
1 y2- 1 72= m2(y2-1) +2

2y2 {ln(m2/ k2)/72 + 2Xtan1(1/X)- 2TI}j
y 1
H(x,y) = 1 ln(x/m2 + 2/ [7-1 X ta'(~1 "2 -



v(pert) 1+C=l+ 1 4 2-4Xtan-(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 y2-1 (C.45)


U(pert) ,1 3(A1 + A2)/E inz (C.46)
4 ~X Vy 2-l
pert) eH(x, y) x- C.7
1 4 1

(p ert) z (048)
(rt +-EH(x, y)- (0.49)

u (pert) 1-H(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 4-vectors 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 F--AL/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 f-dependency appears only in
the form (f;- q)-2. Notice that from Eq.(D.8),

V!=- [D]2q,'ln(i- q)2] FLd4q
qq
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 do-u 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)









[L2p-2 = 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,

EPE-v= [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,

6-V [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(p-VFvdo
--- (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