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 Permanent Link:
 https://ufdc.ufl.edu/UF00099351/00001
Material Information
 Title:
 A solution to the SchwingerDyson equations of quantum electrodynamics
 Added title page title:
 SchwingerDyson equations of quantum electrodynamics
 Creator:
 Cartier, Joan F., 1950 ( Dissertant )
Reid, Charles E. ( Thesis advisor )
Ohrn, Yngve ( Reviewer )
Broyles, Arthur ( Reviewer )
Monkhorst, Hendrick J. ( Reviewer )
Sigmon, Kermit ( Reviewer )
 Place of Publication:
 Gainesville, Fla.
 Publisher:
 University of Florida
 Publication Date:
 1983
 Copyright Date:
 1983
 Language:
 English
 Physical Description:
 v, 197 leaves : ill. ; 28 cm.
Subjects
 Subjects / Keywords:
 Approximation ( jstor )
Differential equations ( jstor ) Eggshells ( jstor ) Electrons ( jstor ) Error rates ( jstor ) Evaluation points ( jstor ) Momentum ( jstor ) Photons ( jstor ) Tensors ( jstor ) Vertices ( jstor ) Chemistry thesis Ph. D Dissertations, Academic  Chemistry  UF Integral equations ( lcsh ) Quantum electrodynamics ( lcsh )
 Genre:
 bibliography ( marcgt )
nonfiction ( marcgt )
Notes
 Abstract:
 A nonperturbative solution to the unformalized SchwingerDyson
equations of Quantum Electrodynamics was obtained by using combined
analytical and numerical techniques. The photon propagator is approximated
by its form near the mass shell. The vertex equation is cut off
at the second order in the coupling constant and the remainder is
approximated by H. S. Green's generalization of the Ward Identity for
higher order diagrams. Under these approximations a functional form
for the electron propagator, S(p), and the vertex function, r y (p,p+K)
was obtained for all magnitudes of momenta. Both functions were found
to be finite. No infinities were subtracted to obtain the solutions.
 Thesis:
 Thesis (Ph. D.)University of Florida, 1983.
 Bibliography:
 Bibliography: leaves 195196.
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by Joan F. Cartier.
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 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 029488013 ( AlephBibNum )
10249455 ( OCLC ) ACB8571 ( NOTIS )

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Full Text 
A SOLUTION TO THE SCHWINGERDYSON EQUATIONS
OF QUANTUM ELECTRODYNAMICS
BY
JOAN F. CARTIER
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1983
ACKNOWLEDGMENTS
I would like to express my sincere thanks to all of those who
have helped me. I would especially like to thank Charles Reid for
his constant support and interest, Arthur Broyles for his thoughtful
criticism without which no progress could have been made, H. S. Green
for his invaluable suggestions and guidance, Ruben Mendez Placito
for being a strong ally and good friend, and Robert Coldwell for
his resourceful presence which saw me through many computational
crises with wisdom, insight and humor. This work was substantially
assisted by the Northeast Regional Data Center which donated a MUSIC
account and valuable technical assistance,and the Division of Spon
sored Research which provided a Research Assistant Fellowship Award.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS . . . . . . . . ... . . . . ii
ABSTRACT . . . . . . . . ... . . . . . v
CHAPTER
I INTRODUCTION . . . . . . . . ... .. .. 1
II THE GENERAL PROCEDURE FOR THE SOLUTION TO THE SCHWINGER
DYSON EQUATIONS OF QUANTUM ELECTRODYNAMICS . . . 9
21 The SchwingerDyson Equations . . . . . 9
22 Initial Approximations . . . . . . .. 14
23 Approximating the Vertex Equation with
Green's Generalized Ward Identity . . . ... 19
24 Converting the Integral Equations into
Differential Equations . ... . . ..... 23
III THE ELECTRON PROPAGATOR EQUATION . . . . .. 27
IV THE VERTEX EQUATION . . . . . . . .... .33
41 Introduction . . . . . . . . . 33
42 The Main Computer Program . . . . . . 37
43 The LeftHand Sides of the Eight Equations .... . 39
44 The RightHand Sides of the Eight Equations . . 44
V CHECKING THE ALGEBRA . . . . . . . . . 57
VI THE MASS SHELL SOLUTION . . . . . . . .. 75
61 An Approximation Solution . . . . . . 75
62 The F and I Functions . . . . . . . 80
63 The Go, G1 and G2 Functions . . . . .... 82
64 The Ho, H1 and H2 Functions . . . . .... 84
65 Summary of the Mass Shell Solution . . . ... 86
VII VERIFICATION OF THE MASS SHELL SOLUTION . . . ... 91
71 The Mass Shell Program . . . . . . .. 91
72 Contributions to Error in the Main Program .... . 93
73 The Mass Shell Program . .. .. . . .. . 95
74 Summary of Results of Mass Shell Program . . .. 104
iii
TABLE OF CONTENTS (Continued)
CHAPTER Page
VIII EXTENDING THE MASS SHELL SOLUTION . . ..... . 108
81 A Scaling Symmetry .. . .......... 108
82 The Large p2 Region . . . . . . 112
83 The Large k2 Region .. . . . . . . 119
IX THE CONCLUSION ...... ........... . . . . 133
APPENDIXES
A DIRAC GAMMA MATRICES: DEFINITIONS AND PRODUCT RULES . 139
B DERIVATION OF GREEN'S MASS SHELL SOLUTION ...... 143
C FORTRAN PROGRAMS .. . ......... ..... 157
D CALCULATION OF FOUR DIMENSIONAL INTEGRALS .. . ... 190
REFERENCES .... .... .. . . .. .... ..... 195
BIOGRAPHICAL SKETCH . .. ..... .. . . 197
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
A SOLUTION TO THE SCHWINGERDYSON EQUATIONS
OF QUANTUM ELECTRODYNAMICS
By
Joan F. Cartier
Chairman: Charles E. Reid
Major Department: Chemistry
A nonperturbative solution to the unrenormalized SchwingerDyson
equations of Quantum Electrodynamics was obtained by using combined
analytical and numerical techniques. The photon propagator is approx
imated by its form near the mass shell. The vertex equation is cut off
at the second order in the coupling constant and the remainder is
approximated by H. S. Green's generalization of the Ward Identity for
higher order diagrams. Under these approximations a functional form
for the electron propagator, S(p), and the vertex function, Fr(p,p+k)
was obtained for all magnitudes of moment. Both functions were found
to be finite. No infinities were subtracted to obtain the solutions.
CHAPTER I
INTRODUCTION
A physical theory must furnish an accurate description of observable
quantities. Beyond this supreme requirement a physical theory is judged
by its simplicity and ease of comprehension. Another important measure
of a theory is how widely the approach could be applied. There is a
sense of a theory being more truthful if it is applicable to more than
a single subject. If the same approach can be applied to several diverse
problems then the problems lose their diversity. To be a really success
ful theory it needs to be one that unifies a broad spectrum of ideas.
Quantum Electrodynamics (Q.E.D.) has been a successful theory to
the extent that it enjoys a number of these characteristics. One of its
attributes is that the theory had its beginnings in the bringing together
of several separate fields of study. This unification formed Relativ
istic Quantum theory and then as a natural extension the study of dynam
ics of the interaction of particles with electromagnetic radiation
evolved.
Quantum Theory (Q.T.) was developed in the 1930's from the ideas of
wave mechanics. It accomplished the explanation of atomic structure,
molecular structure, the structure of solids and the symmetry and energy
bands of crystals. However, Q.T. was insulated from the effects of
special relativity in all these endeavors because the ordinary effects
of chemistry arise from interactions with only the outermost part of
1
the atom. The calculation of reaction rates, surface potentials or
scattering cross sections requires a detailed description of only the
outermost electrons which require relatively low energies to excite.
Relativistic corrections represent a very small fraction of these cal
culations.
Successful as Q.T. was at describing in terms of fundamental
principles the workings of chemistry, it represented a break in the
general pattern of explaining physical phenomena because it had not
incorporated the principles of special relativity. Quantum Theory
needed to be formulated in a Lorentz covariant form. The relativistic
descendant of Schrodinger's equation is the Klein Gordon equation.
This equation allows for the relativistic variation of mass with the
velocity of the particle. However such effects as the fine structure
of the hydrogen atom were only partly accounted for by the relativistic
Q.T. of Schrodinger. It took Dirac's work on the fully relativistic
theory of particles with spin 1/2 to complete the analysis of the fine
structure and to explain the Zeeman effect.
A complete description of the relativistic electron would have to
include the interaction of the electron with its own electromagnetic
field. Feynman1 and Schwinger2 formed calculational methods in Q.E.D.
in two separate mathematical languages. Dyson3 demonstrated these
languages were equivalent. As early as 1930, Waller, Weisskopf and
Oppenheimer4 had calculated the self energy of the electron and found
it to be disappointingly, quadratically divergent. Later Weisskopf
established that the divergencies were only logarithmic. These inex
plicable divergencies that occurred in the calculation of measurable
quantities (though at the time such measurements were not practically
feasible), held the theory in a quandary for quite a while.
Real impetus was given to Q.E.D. when Lamb and Retherford5 succeeded
in measuring the splitting between the 2S and the 2P energy levels
of the hydrogen atom. Acting on a suggestion of Lamb's, Bethe6 circum
vented the divergence problem by simply cutting off the range of inte
gration over the divergent integrals. Surprisingly, Bethe came up with
a very close calculation of the "Lamb shift," as it has come to be known.
Other attempts were made at trying to eliminate the divergencies in
a more rigorous manner. Schwinger and Tomonaga29 developed the first
Lorentz covariant scheme designed to make the elimination of the diver
gencies more acceptable. But, by whatever the justification, calcula
tions in Q.E.D. have enjoyed remarkable practical success. Because of
the small coupling constant for the electromagnetic interaction, per
turbation techniques have resulted in impressive calculations of exper
imental values of the Lamb shift, separation of the ground state doublet
of positronium, the hyperfine structure of the hydrogen atom, the line
shape of emitted radiation in atomic transitions and other relativistic
but measurable phenomena.
The road which connects Quantum Theory with special relativity
leads to the theory of Quantum Electrodynamics. The road continues on
today to connect Q.E.D. with further frontiers. Quantum Electrodynamics
has become a model for other field theories. An appreciation of the
special role of Q.E.D. is one way to provide for a better understand
ing of the mathematical structure of field theories in general. The
recent success of the NonAbelian gauge theories in unifying the
electromagnetic interaction with the weak interaction further motivates
efforts to understand the ambiguities of Q.E.D. The current theories of
electroweak and strong interactions have been shown to have the same
underlying structure based on assumptions about global and local invar
iance. The unification has its dark side since as a part of the bargain
comes the problem of the divergencies
It is apparent that Q.E.D. sits as a bridge between many well
travelled roads of thought. Thisis why it is particularly frustrating
that the theory should be flawed by unnatural infinities which rear up
in the evaluation of physically observable quantities. If it were a
complete and satisfactory theory they should never have occurred.
There are four different kinds of divergencies. The following
classification of them has been paraphrased from the text by Janch and
Rohrlich,8
(a) divergencies associated with the description of the vacuum
(b) infrared divergencies
(c) divergencies associated with closed loops
(d) serious divergencies.
The type (a) occurs only in the form of a phase factor multiplying a
particle amplitude. Since this does alter the probability density it
can have no observable effect. It is possible to ignore type (a)
divergencies. Type (b) is an artifact of the mathematical procedure.
An analysis of the problem has shown it can be eliminated by an improved
mathematical treatment. Type (c) is associated with the photon self
energy. This type of divergence has been handled by invoking the invar
iance of the theory under gauge transformations. Type (d), the
"Serious" divergencies, comes up in the calculation of the vacuum polar
ization, electron self energy and the vertex function. These serious
divergencies are the object of this whole discussion.
The redressing of these infinite quantities is called Renormaliza
tion. In this process the amplitudes are expanded in a power series of
the coupling constant. Many terms in the series may contain divergent
integrals. It is possible to remove these infinities in a relativisticly
covariant way by redefining the parameters of mass and charge. The
finite terms which remain in the series are taken to be the renormalized
expressions for the amplitudes. Renormalization is neither simple nor
straightforward. It brings in a new and complicated set of rules which
are not properly anticipated by the initial understanding of the problem.
Renormalization is an after the fact reaction to something unforeseen and
undesirable. This leaves two possible interpretations of the problem:
(1) There is something wrong with the basic theory of Q.E.D.
(2) Some mathematical procedure has been inappropriately applied.
A new method of calculation must be tried.
It is hard to argue that there is something wrong with the founda
tions of the theory. Q.E.D. has enjoyed a huge practical success in
calculating various physical measurements. For example, from renormal
ized perturbation calculations of the anomolous magnetic moment of the
electron the following results for the inverse of the fine structure
constant were derived:9
 = 137.03549(21).
Presently the best experimental value is10
Sxp = 137.03604(11).
a exptl
It would seem that there could be little wrong with a theory that makes
such accurate predictions.
The unqualified success of Q.E.D. in calculating (by whatever means)
precise experimental results, and the success Q.E.D. has enjoyed in
linking quantum theory to special relativity, and its position central
to general unifying theory of forces are forceful evidence that the
fundamentals of Q.E.D. are sound.
The implication therefore must be that some mathematical procedure
has been inappropriately applied. The response to this implication has
been a long search by many persons for a selfconsistent and finite
approach to Q.E.D. In 1954, GellMann and Lowl2 sought to demonstrate
that the renormalizing constants which relate the bare mass and charge
are infinite. They found they could not rule out the possibility of infin
ite coupling constants but they isolated a necessary condition for the
vacuum polarization to be finite. Johnson, Baker and Willeyl3 took up
the interesting problem in a long series of papers. Under a certain
set of approximations they solved for the renormalizing constants and
concluded that in order for the self energy of the electron to be finite,
the bare mass of the electron must be zero and a special gauge must be
used. These results spawned interest in a number of people14 for find
ing asymptotic but nonperturbative evaluations of the self energy of
the electron, the vertex function and the vacuum polarization.
Chapter III will describe the paper in which the author was involved in
some of the effort of repeating, in a new way, the determination of the
electron self energy. The work reaffirmed the results of Johnson,
Baker and Willey by using an inventive nonperturbative approach of
H. S. Green's with less restrictive approximations than had been used
before. This work extended the results of Johnson, Baker and Willey by
finding a complete solution instead of an asymptotic one.
These three basic functions, the electron propagator, the vertex
and the photon propagator are identified by their role in the three
linked nonlinear integral equations known as the SchwingerDyson equa
tions. It is of great interest if a method of solution could be found
which would yield no divergent function forms for the three basic func
tions. Encouraged by the success of Green's method in extracting a
finite solution for the electron propagator, we decided to see if the
same nonperturbative procedure of H. S. Green would work to provide
a vertex function.
The description of the project is arranged in the following way.
Chapter II is a general description of the overall method of taking the
nonlinear linked integral equations and transforming them into a set of
differential equations. An outline is given of the proposed method for
solving the equations for the electron propagator and the vertex.
Chapter III provides a description of the electron propagator solution.
Chapter IV provides a preparation of the differential equations for the
vertex function. The tensor equation is broken down into eight linked
thirdorder differential equations for the eight scalar functions which
comprise the transverse part of the vertex. In Chapter V an algebraic
technique is described which makes the product of the complicated gamma
8
matrix function easier to obtain. In Chapter VI, Green's method is used
to obtain an approximate solution to the eight linked differential
equations. In Chapter VII this solution is tested and its range of
applicability is defined. In Chapter VIII some alterations are made to
the solution which extends its viability. The last chapter summarizes
the solutions which were obtained to the SchwingerDyson equations.
It is felt that the work described here will contribute to the
idea that Q.E.D. is a complete and satisfactory theory; a theory which
is a faithful rendition of experimental results, a theory which lends
clarity by unifying several fields of study, and a theory which is
unambiguously expressed.
CHAPTER II
THE GENERAL PROCEDURE FOR THE SOLUTION OF THE SCHWINGER
DYSON EQUATIONS OF QUANTUM ELECTRODYNAMICS
21 SchwingerDyson Equations
Quantum Electrodynamics is a description of the interaction of
light with matter. A classical charged particle generates an electro
magnetic field around it with which it can self interact. Calculations
of this interaction have traditionally led to infinities. To demonstrate
how these infinities arise in Q.E.D., it is convenient to express the
theory in terms of the propagators of the particles.
The amplitude of an electron at some point in spacetime is related
to its amplitude at a different point in spacetime by its Feynman
propagator or Green's function. The Green's function is determined by
the equation of motion that governs the wave function. For the free
electron the differential equation is
(i0 m): = 0 (21)
where W is a shorthand notation for yu and yW is a Dirac gamma
matrix.t The photon wave equation is
[] A' = 0 (22)
where ] is the D'ALambertian,  The propagator for the
ax axin
tSee Appendix A for representations of the y1.
electron satisfies a corresponding equation,
(imo)S (x' ,x) = 64(x'x). (23)
The solution to this equation in momentum space determines the Fourier
transform of the free electron propagator to be
S(p) m (24)
Similarly the photon propagator satisfies
Do (xx') =i64(xx'), (25)
so that the Fourier transform of the photon progagator is
D(q2) = (26)
q
Thus the propagators for the free particles are explicitly known.
When it is allowed that a source term may be present, the inter
action between the electron and photon will lead to nonhomogeneous dif
ferential equations. The exact electron and photon propagators are then
determined by these nonhomogeneous differential equations but the solu
tions are not explicitly known. The equations which determine the Fourier
transform of the photon and electron propagators are an open set of inter
locked integral equations. This hierarchy of integral equations was
formulated by Dyson3 and Schwinger.16 Using the notational practices of
Bjorken and Drell,17 these integral equations appear as:
tThe 4 vectors are denoted by a bar over the symbol and matrices
are distinguished by a bar under the symbol.
(a) S(p) = So(p) + S (p)_(p)S(p) (27)
or equivalently
S (p) = S (p) _(p) (28)
where
ie2
(p) = p(pq)S(q)D (pq)yd4q. (29)
S (2T)4
(b) D (k2) = D (k2) +Do(k2)~R B(k2)D (k2) (210)
where
ie 2
t (k2 0e Tr [yS(q) (qq+)_q+)] d4q. (211)
(27Tr)
(c) r'(pq) rl+A.:p,q) (212)
where
A(p,q) =ie2 fD )(k2) (p pp)S(pk)
r (pk,q)S( (qk,,q)d4k
+ ... f... f (p p pk) ... ...
rn (qn l ,qn)S(q )r. (qvq)
d4k ... d4k (2 )4n + ... (213)
The zero subscript follows all bare quantities, that is, those functions
or constants which are associated with the free particles. The I'(p,q)
is called the vertex function. It represents the sum of all nodeless
diagrams which are connected to two external electron lines and one
external photon line. If a closed expressed for the vertex function
could be formed, then a complete knowledge of the interaction propaga
tors would depend only on a solution to the three linked nonlinear
equations; the electron propagator equation, Eq. (a), the photon propa
gator equation, Eq. (b), and the vertex equation, Eq. (c). However,
Eq. (c) gives the vertex equation in terms of an infinite series.
Therefore the SchwingerDyson equations, though simple in form, possess
no simple solution.
One method for attempting a solution is to assume the interaction
propagators differ from the bare propagators by only a small variation.
Perturbation theory might then lead to at least an asymptotically con
verging series. The first iteration would replace the propagators.on
the righthand side of Eqs. (a), (b) and (c) by the bare propagators.
When this is done the equation for the electron propagator function
Z(p) becomes
_eo 1 v gv1 d4k
S(p) 4 ()m 2
(2fl 0 k
e2 4(0+mo) ) 4
4 [(p 2m d k. (214)
By a power counting of k it can be seen that for k~ the integrand
behaves like k3dk/k4 which would yield a logarithmic divergence. A more
careful consideration of this integral would take into account the
hyperbolic metric. The fourdimensional integral can be performed by
transforming into hyperspherical coordinates but the logarithmic divergence
persists. A corresponding divergence appears in the vertex function
See Appendix D on the subject of calculation of fourdimensional
integrals.
and in the photon propagator function. These are of the type (d)
category of divergences of Janch and Rohrlich8the socalled "serious
divergences."
These divergences can be handled by any of a number of methods
grouped under the title of Renormalization methods. The first proof of
renormalizability was provided by Dyson, Salam and Ward.7 Basically the
idea is that although absolute calculations cannot be made with the for
malism as it stands, still relative calculations can be made. The param
eters of the theory such as mass and charge are redefined to absorb the
terms which contribute the infinite quantities. The renormalized mass
and charge are taken to have the experimentally observed values. Renor
malization, as such, was a huge but puzzling step forward. It provided
the tools to make impressive calculations of relativistic corrections
in the spectrum of the atom but the meaning and the value of the bare
quantities remained a mystery.
The very fact that renormalization works is an indication that there
ought to be a mathematically consistent way to solve the hierarchy
without encountering undefinable quantities. The practical success of
renormalized theory argues against a fundamental flaw in the theory.
For these reasons the following scheme was developed to solve the equa
tions without recourse to renormalization methods.
The procedure which has made possible an unrenormalized solution
of the SwingerDyson equations has three elements to it. The first is
the assumption of a reasonable starting point in terms of an approx
imate form for the photon propagator. The second is the generalization
of the Ward Identity which provides a neat formula for systematically
truncating the hierarchy of the vertex equation at increasing levels of
accuracy. The last element consists of transforming the electron and
vertex integral equations into linked differential equations. The dif
ferential equations are more tractable than the integral equations to
numerical and analytic approaches to the solution. A description of
these three elements is the object of the following three sections.
22 Initial Approximations
The question which must be raised first is whether it is best to
solve first for the electron propagator, S(p), or first for the photon
propagator, D (k2). This can be decided by a consideration of which
of the two is easiest to approximate. Since Lorentz30 first offered the
idea, as early as 1909, it has been a popular view to consider the mass
of the electron as mechanical in origin. This was based on the observa
tion that the electron, when accelerated by interaction with the electro
magnetic field, behaves as though it were gaining mass. It is appeal
ing to imagine that the bare mass of the electron may be zero and the
self interaction with its own electromagnetic field is what "dresses"
the electron in its apparent mass. If the bare mass tis zero or very
small the self energy of the electron will be of theorderof magnitude
of its rest energy. In units of inverse time, the rest energy of the
electron is 1.2x1014 megacycles. On the other hand, renormalized per
turbation type calculations of the Lamb shift show the vacuum polariza
tion contributes only about 27 megacycles. The disparity between these
tObviously the assumption the bare mass of the electron is zero is
predicated on the additional requirement that there be no other forms of
interaction which contribute to the mass of the electron.
two numbers argues that the self energy of the electron may be more
important than the vacuum polarization of the photon. Therefore, it
is reasonable to start with an approximate form of the photon propagator
and solve first equation (a) for the electron propagator.
The electron propagator is a function of the scalar I, (y pP), and
as such its most general form is expressible in terms of two functions,
one the coefficient of unity and the other the coefficient of 0.
Because of the simplicity of Eq. (28) relative to Eq. (27), it is of
interest to focus on the form of S(p),
S1(p) = A(p2) B(p2)O. (215)
The vertex function is a matrix function of the Dirac gamma matrices
and the four vectors, p and k, the electron and photon momentum respec
tively. Its most general form is
A G
( (p+k,p) = F +y G + pX l + 2
V % 0 pk 2
+ kJ + kX 2 K + kXK L + iBp ak kM (216)
2 2
where F, G Gl, G2, H H H, and I are functions of p2, k and u.
The u is the angle variable defined as
u = (217)
P2 Ik I2
They are coefficients of that part of r which is transverse to k.
Similarly J, K, L and M are functions of p2, k2 and u. They are coef
ficients of the longitudinal part of F.
The general form of the photon propagator is known from its relativ
istic covariant properties. Since D (k2) is a second rank tensor which
depends only on the fourvector R, the photon momentum, D (k2) can be
taken to have two components. One component is proportional to k k
and the other is proportional to g Thus
D (k2) = dl(k2) g /k2 + d2(k2) k k /k4. (218)
In addition to knowing the general forms of the three functions,
the electron propagator, the vertex and the photon propagator, the
limiting form of the functions is known on the mass shell. In the
2 2 2
limit as the moment approach the mass shell where p 2 m k2 0,
the functions approach the following limits,
S(p) Z (gm) (219)
X(p+kp) 1 X (220
D (k2) Z (221)
P'Ou 3 k2
The constants of proportionality to be determined by the theory are
Z2 and Z3. First order perturbation theory finds them to be zero or
infinite. They are called renormalization constants because the
standard procedure is to use them to redefine the series expansions
so that the divergences are absorbed.
The initial step of solving for the electron propagator requires
a reasonable first estimate of the dressed function D and iF. The
approximation that was used for the photon propagator was
S(k2)= Z3[ (1b) 1 ]. (222)
v k2 k4
13,14
This choice was motivated by the results of earlier investigations.1314
Studies of the asymptotic forms of the propagators which were not incon
sistent with finite renormalization constants found the photon propagator
to be in the Landau gauge for at least the lowest order in perturbation
theory. The Landau gauge is a special case (b=0) of the general form
given in Eq. (222).
The approximation that was used for the vertex function was obtained
from the Ward Identity which related the vertex to the inverse electron
propagator,
k rF(p+f,p) = S (+) S1p). (223)
In the limit as the photon momentum, k, grows small this can be expressed
in a differential form,
as'I (p)
_(p,p)  (224)
As illustrated by Eq. (214), when the bare form of the functions
is used to generate a first approximation to the dressed function, it
is found that "serious" divergences occur. It is desirable then to avoid
the substitution of bare quantities for dressed quantities. Therefore
if the general form of the electron propagator were used in the differ
ential Ward Identity,it would be possible to generate a general form for
the vertex at vanishing k,
S(p,p) = [A(p2) + jB(p2)]. (225)
To solve the electron propagator equation a knowledge of the vertex
function r (p+k,p) is needed. Equation (225) represents a very good
approximation to the vertex function where Ik21 is small. In consider
ing the electron propagator Eqs. (27) and (29), it can be seen that
this approximation is at its best when, as the argument of D vanishes,
the integrand is at a maximum. This coincidence of the region of best
approximation with the region of most importance argues that
F (p+k,p)  [A(p2) + 0B(p2)] (226)
is a reasonable first approximation for the vertex.
By using the approximations in equations (222) and (226) for the
photon propagator and vertex, a solution to the electron propagator
equation can be found. A description of the method of solution and its
conclusions are found in the next chapter. Once a solution to the elec
tron equation had yielded a functional form for the electron propagator
functions A(p2) and B(p2), a solution to the vertex equation was sought.
The next section describes the method by which the vertex equation
was approximated so that it no longer depended on the higher levels of
the hierarchy.
23 Approximating the Vertex Equation with
Green's Generalized Ward Identity
A construction of the infinite hierarchy of equations can be selected
so that a highly repetitive pattern emerges which can be used to system
atically separate the equations at any chosen level of complexity.
Instead of the infinite series which appears in Eq. (c) the vertex equa
tion can be written as
= +A(p,q)
where
2
.2 r
A(peq)=  e BD (k2)S(_pk)"(p_,p4__)
S(pqk)Fa(pqkp )d k
ie2 f 2
+ Jo DaB (k (p ja(p q)d4k. (227)
(211)
A new function has appeared, E"a, a function of three external
moment, which corresponds to four independent points in spacetime;
hence Ela is a fourpoint function. Just as the twopoint functions
S(p) and D (k2) were related to the threepoint function, r A ,p+R),
and as the threepoint vertex, r was related to the fourpoint func
tion Ea(pR,qqk), so the fourpoint function, El", can be related to
a fivepoint function and so forth. The greater the number of moment
involved the lower the contribution from such a cross section should be.
Each new npoint function is created by pulling a dressed photon and
dressed electron propagator out of an nl point diagram, creating in
this way a new vertex or point.
In Table 21 the equations for the twopoint electron propagator,
the threepoint vertex, and the fourpoint E1 are given to show the
pattern that is emerging.
The repeated structure in the relations that link an n point diagram
to an n+l point diagram suggests there should be a generalization of the
wellknown Ward Identity which states
q A1(pq,p) = z(Pq) + E(p). (228)
A generalization of this for the four and fivepoint diagrams would be
q EUV(p+,p,k) = AV(p+q,pR,k) A"(p,pk,R) (229)
q F'uv(p+q,p,k,) = E'a(p+q,R+ipq,kl) E'(p,k+7p,k,L). (230)
This generalization of the Ward Identity was first proven by H. S. Green18
in 1953. These identities exactly define the longitudinal components
of the n+l point diagram in terms of the difference of two npoint
diagrams. These identities can be used to truncate the hierarchy of
equations by approximating any given diagram by its longitudinal compo
nent. In this manner the vertex function could be solved for if EV was
approximated through an application of Eq. (229).
In conclusion, it has been proposed the electron propagator can be
found by approximating the threepoint vertex by Ward's Identity.
TABLE 21. RELATIONSHIP BETWEEN NPOINT DIAGRAMS
Two point diagram
S1
S1(p) = S (p) Z(p)
where So = v m
ie
= 4 0 )S(q)D C2)y vd4.
(2)v q q
Three point diagram
"(p,pq) = r + AP(p,pq)
where o = y
A = (pE,,pq)S(p )o D(k2 yvd4k
(2) TJ
Four point diagram
(p,,p = + E'l(pK,q,pq)
0
where D]a = yp(pqA()S(pqR)(pq,)
ie
E = 4 0~ O p,p (p) (Z2) yVd4
(27F) v
Five point diagram
(. ,E,,_) 01=00 B + FP ^,,q,q)
whereOopa 
0 2 P4()
ie2 f2
Fp 0 (pk_s,l,k,q,pq)S(ps)D v(s )y.d4s
(2)I s
10
1I
+
II
r\
1&
'4
I
a
i;L
II
'a
0.^
1
11o
a
I I
lO .I
\0
aV
Although the Ward Identity provides an exact relationship for the
longitudinal component of the vertex, it only yields the limit of the
transverse part for small photon momentum. To obtain the transverse part
for large values of the photon momentum a solution to the vertex equation
must be found. This can be done by using the firstorder generalized
Ward Identity to determine an approximation for the fourpoint diagram.
This process has two very important aspects to it. One is that at all
levels in the solution to the SchwingerDyson equations the Ward Identity
is exactly preserved. The second is never are bare functions substituted
for dressed functionsa procedure which has always been associated with
divergences.
24 Converting the Integral Equations
into Differential Equations
Another essential step in the whole procedure of reducing these
equations to a tractable form without giving in to the drastic approxima
tions, which have characterized earlier attempts at a solution, was the
conversion of the integral equations into differential equations. This
method was developed by H. S. Green19 in connection with the Bethe
Saltpeter equation. It was first used for a study of the Schwinger
Dyson equations for the electron by Bose and Biswas.20
Whenever the photon propagator appears under the integral it can
be used to eliminate the integration. The photon propagator was taken
to have the form,
D (k2) = 3[ + (b k ]. (231)
k k
Because
2 g k k
S n (k2) 2 9 4 (232)
3k Tk 2 4
J v k k
the photon propagator can be put into the alternate form,
D (k2) = z23 (b) k n (k2) (b+1) (233)
Notice also that the D'Alemberian operator,
 (234)
k oku Sk '
will operate on L to yield a delta function,
k
k = 4T2i5(k2). (235)
This delta function can be used to trivially perform the integration
over integrals of the form,
S(k2) ie2 fDV((_)2)ylF\F (q2) d4q. (236)
Substituting Eq. (233) into Eq. (236) results in
( k ) = i e 2 kb ) pl q k 
in((kq)2) (b+1) y F(q2)d4q (237)
nK q )J
Apply the D'Alembertian
nI kl(k2)= ie2 (lb) (k)2
0 4bD k) 4i k6 (kq )2
(b+1)g 4Ti6 (kq )I1" FXV(q2)d4q
= 2e2(b+l)i2y F (k2)
+ ie2f (1b)  Fv(q2)d4q. (238)
v (kq)
There is one remaining part under the integral. This can be removed by
either of two ways. The first is operate on the equation with
Y  = f. This yields the following thirdorder differential equation
aka
lkIDk(k2) = 2e2 2(b+l ky FXv(k2)
+ 4e272(1b) F(k2). (239)
The second method is to identify the remaining integral as some func
tion GX(k2) such that
G (k2) = i (1b)  F (q2)q. (240)
v (kq)
Then the problem of solving Eq. (238) becomes the problem of solving
the pair of equations,
SkI (k2) = 2e2(b+1) 2 y F (k2) + e2(k2)
VGX(k2) = 42 (1b)  FX(k2). (241)
By either of these routes a solution to an integral equation of the
form (236) is equivalent to a solution to the differential Eq. (239)
or the pair of differential Eqs. (241) when the appropriate boundary
conditions are satisfied.
To briefly summarize this chapter, a scheme has been laid out by
which the first two unrenormalized SchwingerDyson equations could be
solved. The first step of the scheme involved solving for the electron
propagator. To solve the electron equation required using Eq. (222)
and (226) to initially approximate the vertex function and the photon
function. The second stage of the scheme involved solving the vertex
equation to determine the form of the transverse part of the vertex for
other than very small photon moment. To make this possible the higher
order terms were also approximated using a generalization of the Ward
Identity. Finally the integral equations relating these functions were
to be converted into differential equations to make an analytic solution
easier.
In the next chapter, Chapter III, a survey of the work that was
done on the electron equation is given. In Chapter IV, an introduction
to the details of solving the vertex equation is given.
CHAPTER III
THE ELECTRON PROPAGATOR EQUATION
Recall the form of the electron propagator previously given in
Eq. (27) where S the Fourier transform of the bare propagatator, is
given by
S = 1 (31)
The m0 is the bare mass of the electron, that is, the mass the electron
would have if the electromagnetic interaction could be turned off.
The z(p) equation was given as
ie2 r
S(p) o f ( ,)S(q()DV (ptq)yVd4q. (32)
(2Tr)J
On the basis of rationalizations detailed in section 22 two
approximations are invoked to sever the connection of the electron equa
tion from the vertex equation and the photon propagator equation. These
were,
D ) + (1b) p 1
(pq)2 (q)
and
rX p+q,p) a S (p). (34)
When these substitutions are made the electron propagator Eq. (31)
becomes
ie2 aSl1p)
S (p)= S(p) + (q)
0 (2)4Jf 3p
S (1b) ( 4 d] 4qyV. (35)
(pq) (pq)
Because the electron propagator is a scalar function of only the electron
momentum, its most general form is given by
S(p) = A(p2) + B(p2). (36)
So that
S (p)= 2pA'(p2) + y (p2) + 2pl B'(p2) (37)
P,
where
A' (p2) = and B' (p2) = d (38)
dp2 dp2
Substituting Eqs. (32), (37) and (38) into the electron propagator
Eq. (36) yields,
A(p2) + lB(p2) m + ie 2 J 2p4 A'(p2)
+ yvB(p2) + 2plB' (p2))(A(q2) + gB(q2))
 (1b) P d4q (39)
(pq) (pq)
The equation was converted into a differential equation by
application of the D'Alembertian operator, as described in section 24.
For the purpose of perfonning all of the needed matrix multiplications,
a table of products of gamma matrices was prepared. This table appears
in Appendix A. Equation (310) then separates into twolinearly inde
pendent differential equations, the coefficient equations for the unit
matrix and V.
The unit matrix equation is
A"2 + 2A e2 4b) AB (2+b)p (B'AA'B)
At p 2 2 4 22 
4 22 4 (A p2B2) 4 A p B
(310)
where
S 6 B'AA'BA (311)
The V equation is
B"p2 + 3B' e2 (2+b) (A'AB'B 2) (1b)B2 (312)
4n2 4 (A2p2B2) (A2_p2)j
where
A' = dA and B' =dB (313)
dp dp
SA" d2A and B" d2B (314)
d(p2)2 d(p2)2
Numerical solutions to these equations were formed using a fourth
order Runge Kutta method which generated the values for the functions
A, B and T. The functions A and B were found to be very slowly chang
ing functions. In fact, for most practical purposes, B is essentially
2
equal to 1. The function A very slowly declines as p
A description of the solution and a simple analytic determination
of the asymptotic behavior of these functions was presented in the paper21
on the electron propagator equation. In that paper a different approach
was taken. The integral equations were not converted into differential
equations by the action of the D'Alembertian operator. Instead the
variables of integration were converted to hyperspherical coordinates;
a Wick rotation was performed; then the (pq)2 factor was expanded in
terms of Gegenbaur polynomials, Cn(p). These polynomials have an orthog
onality condition which was used to simplify the integration. This
useful procedure is illustrated in detail in Appendix D.
The solutions to the differential equations had to be restricted
to be particular solutions of the integral equations by the boundary
conditions. The boundary conditions for the four momentum, p2, approach
ing the mass shell is known. There the electron propagator is propor
tional to the bare propagator. The asymptotic boundary conditions,where
the magnitude of p2 is indefinitely large, are not explicitly known.
This has been an object of study of a large number of papers.14 Inter
esting things can be determined about the asymptotic form of the solution
when the differential equations are substituted into the integral equa
tions. This was carried out in the third section of that same paper.
There it was found that the functions A and B, of the electron propa
gator, must approach constants for large p2. In order that they could
approach constants the gauge parameter b had to be set to zero. With
b equal to zero, the photon propagator was set in the Landau gauge.
At the same time it was demonstrated that a finite solution required
the bare mass to be zero.
An expression for the function A was fitted to the tabulated
numerical solution over a finite range of moment. The accuracy of
the fit was around 0.1 percent. To an even greater accuracy the func
tion B was observed to equal the constant one. Thus the functions A
and B appeared to be well represented by
A(p2) = jlp21(P)p2 2 (315)
B(p2) = 1.0 (316)
where e = 1.74517 x 103
The tabulated values of A then predicted an asymptotic form of
A = Ip21E (317)
and an asymptotic B of
B = 1. (318)
When these asymptotic expressions were substituted back into the inte
gral equation the power law for A was explicitly determined. It was
found that
e = (3a/4i) + (3a/4T)2 + 3(3a/4r)3 + ... (319)
where a is the fine structure constant. Baker and Johnson13 obtained
almost the same expression for the power law of A. They concluded
E = (3a/4n) + (3a/4)2 + ... (320)
Last of all it was possible to see by comparing the limit of the electron
propagator as the mass shell is approached, to the propagator expressed
in terms of the renormalized propagator that the renormalization constant,
Z2, was equal to unity.
All of these result were in agreement with the results given by
14
Johnson, Baker and Willey.4 They had also concluded that the bare
mass was zero and they had determined a very similar value for the
power law of the asymptotic expression for A.
This paper represented an extension to the understanding of the
electron propagator because it went beyond trying to determine an
asymptotic form of the propagator which was consistent with finite
renormalization constants. This paper actually determined an approx
imate expression for the electron propagator which was good for all
moment. The electron propagator was determined to be
S (p) = g + lp21 c(p2)/p2 (331)
where e = 1.74517 x 103.
In this manner it was shown that the electron propagator could be
determined using approximations of a far less drastic nature than had
been tried before. No infinite quantities were encountered. Encour
aged by the success of this first stage of the project, an attempt to
solve the vertex equation was ventured.
CHAPTER IV
THE VERTEX EQUATION
41 Introduction
The next step in the process of seeking a solution to the Schwinger
Dyson equations is solving the vertex equation, restated here,
S(p,q) = y + A (p,q).
where
2
Aie
AX(pq) 0o4 fyD1 (k2)S(pR)/(pkq')S(qk)r (qkq)d4k
(2n)4 a
2
+ le /DB(k)S(pK)E'A(pkqk,q)d4k. (41)
(2iT)
As with the electron propagator equation, the solution to this equation
is preceded by three elements of preparation.
The first, the photon propagator is taken to have the same form
as was utilized in the electron propagator equation. However, it is now
fixed in the Landau gauge so that
2gpv k k\7
DU(k2) Z3 (42)
The second element is to recognize that the fourpoint diagram,
E which is defined in terms of the fivepoint diagram, which in turn
is defined in terms of all higher order diagrams, an infinite progression,
must be expressed in some closed and approximate form. Just as the
vertex function could be related to the electron function through
Ward's Identity, so the fourpoint diagram can be related to the vertex
through a Generalized Ward's Identity. There are two possible longi
tudinal components of the fourpoint diagram, one is longitudinal with
respect to k and the other which is longitudinal with respect to q.
The relationships between the longitudinal components and the vertex
function as given by the generalized Ward Identity are
k EX(pk,q,pq) = A'(p,pq) _A(pk,pq) (43)
q E (pk,q,pq) = A (pk,p) A(p,pq). (44)
It is possible to substitute r for AX and Fr for Av in these relations
because the difference between F and A is a constant. Using this fact
and putting Eqs. (44) and (45) in the differential form it can be
seen that
EX(p ,qpq) r ( ppq) (45)
ap V
EX(pk,0,p) = (pV,p). (46)
3p
Thus in this procedure the complete E V will be approximated by these
two parts,
E (pk,q,pq)  F (p,pq) + V(_pk,p). (47)
,v
For the purposes of simplification it is observed that the expression
r (pk,pq)S(pqk)r(Pqk,pq) r(p,pq) (48)
Pv
can be reduced to
a [(p k,pq)(pqR)] S(pq). (49)
aP,
To obtain this simplification use is made of the fact that
[(pq) S pq) S(pq) q), (410)
) 3ap
and the vertex function was again approximated by
v a 1 
r (pqk pq) S (pq). (411)
pv 
ap
By substituting the expressions in Eqs. (48) and (49) into the vertex
Eq. (41), one obtains a simplified vertex equation,
2
p,q) = + ie YD(k2)S(pk) rV (pk,p)d4k
(an7T) f
2
+ 4 yD (k2) S(pc) 3 [r(pk,pqk)S(pqk)]
(2n) 4p4 
S (pqk)d4k. (412)
The last step in preparing the vertex equation was to operate with
the D'Alembertian,
02 = (413)
'v ap
The D'Alembertian has the desirable feature of operating on the Fourier
transform of photon propagator, here in the Landau gauge, to produce
a Dirac delta function,
2 D (pr) = 22i g, 6(pr)Z3
Z a 1 (414)
3 3(p~r) D(prV) (pr)2
Therefore,
2 2
l2 r(prl+ ie 24 f [2 2i g 6(pr)]S(r)
(27T
[ r(r,r+q)S(r+q) ]_S(F+q)d4r
ie2 _4 S()
(27T)4 a(p _r) (pvr ) (pr) 
[_ (r,r+q)S(r+q)] S(r+q)d r
S22 S(p) [X(p,p+)S(p+q)1] S(p+q
82 p S( )
ie2 f[ 3 1 ]S)
(2n S(p r") a(pvr",) (pr)2
a [rA,(+q)S(Fr+)1] S(Fr+)d4r. (415)
v
In the above the definition Z e = e2 was used.
At the cost of having to solve higher order differential equations
the last integral can be eliminated by the action of
S= ^ a p
Sp"
This yields the final form,
3 F(p,r+p)= c[ Y V FV(p,p+q) + FV(p,p+q)] (416)
where
3= C, E = e2/47r2
and
F~ = S(p) [Fr(p,p+q) S(p+q) ] S(p+q). (417)
PV
42 The Main Computer Program
It is only necessary to solve the vertex equation for the transverse
components of the vertex since by Ward's Identity the exact longitudinal
components are known in terms of the solution to the electron equation.
The main computer program is a realization of the equation
13 X =_ e 1 Xv
I tras [j V yv F + L FXans
transPR 4 v trans + trans
where
Fa = S(p+k) [r (p+k,p) S(p)1] S(p). (418)
trans p trans
v
Hereafter the subscript "trans" will be dropped and it will be under
stood that any X superscript is taken to be transverse. Thus, for any
general function Q ,
Q = kA k (419)
k
In the vertex equation there appear two independent four moment,
k, the photon momentum and p, the electron momentum. Scalar functions
therefore will be functions of the variables p2 and
(420)
pk
where p = lp21 and k = lk2 2
2 2
and the functions will be parametrized by k The range of p is from
negative to positive infinity. The variable u has the same range owing
to the indefinite metric. The general expression for the vertex func
tion in terms of such scalar functions is given by
2 H H
r(p+, F) =_ F(p2p) + poVX] H (p2,u) + [y,,l]1 )p2 u)
p p k p
+ [y,] ~ (p2 u) + Y Go(p2u) + p (pu)
+ p G (p2u) + ^amvuyympk bk. (p2*u). (421)
In addition to the general expression for the vertex, the general
expression of the Fourier transform of the electron propagator, as given
in Eq. (216), is needed. So that
S(p) = A(p ) B(p 2 (422)
A2 (p2) p2B2(p2)
and
S(p+) = A((p+k)2) ( 2+ )B((p+k)2) (423)
A ((p+k)2) (+)2B2((p+k)2)
For simplicity the following notation will be observed throughout,
Al = A(p2) A2 = A((p+k)2)
B1 = B(p2) B2 = B((p+k)2)
D A2 p2B2 D = A2 (p+k)28 (424)
D1 =1 2 = A2 2
Thus S(p) AI  (425)
D1
A2 ( +X)B2
S(p+k) = 2 (426)
D2
Equation (418) presents the differential vertex equation in a
straightforward and simple form. Unfortunately when substitution of
the scalar functions of Eqs. (421), (422) and (423) is made and the
products of the gamma matrices are taken, thousands of terms need to be
resolved. Once expanded in this way it is clear the matrix equation is
equivalent to eight equations which are coefficients of the linearly
independent matrices: y p p K, E Bamy5 pak, icy ka, iua p
iou3p k pp and p The description of the process of identification of
these eight equations will consume the next several sections. In
section 43 the lefthand side of the eight equations will be explicitly
given, and in section 44 the righthand side of the eight equations
will be given. These eight equations plus a set of boundary conditions
will be used to determine the eight transverse vertex functions F, GoGIG2'
HoH1H2 and I.
43 LeftHand Sides of the Eight Equations
The formation of the lefthand sides of the eight equations which
comprise the matrix equation for the vertex is a relatively straight
forward operation. The operation of ? on any general function, f, of
p2 and u will yield
Sf(p2,u) = 2f+ (427)
p
The prime denotes a partial derivative with respect to the scalar p2
and an asterisk denotes a partial derivative with respect to u.
When i3 was applied to rxthe coefficients of the eight linearly
independent matrices were obtained. These are given in Table 41.
TABLE 41 LeftHand Side of the Eight Vertex Equations
Equation 1 Coefficient of p
** 2 2 ***
S2 3F + 10 u F u(lu2) 14
9F + 23u 4 44
p p p p p P
**I
18 u F + 2(1u2) F2 + 20 F" 4u F + 8 p 'F
p P
*
2 Ho u *** 2 2 2***
64 u H + (30 86 u2) 4 + 26 (1lu2)H* (1u2)2 H0
p p p p
H **
+64 Y4 H + (44 u216) H 4 (lu2)Ho 32 u H"
P P P
S(1u2)H 16 p2uHo
**
H1 H1 2 6H1 ***
+ 18 + 46 u + (20 u 6) 2 4(1u2)H
p p P p
H *' 4 2 **'
28  36 H1 + (lu )H1 + 40 H1 8 uH1
p P P
+ 16 pH1 p
(428)
Equation 2 Coefficient of p K
**
SF 7 ** + 8 7 + 4
P P P
8 4 H Ho 8 Ho 28
p p p
F* +ui ) F***
F + F
P
2 Ho + (1u2) Ho
p p
II 2 11
+ 64 H0 + 16 p Ho
+ H1 8 H*
p p p
+ 18 4 H2 + [10
P
**
2 2 H2
u24(1u2)] 
P
+ 48 H 8 uH + 16 p2
+ 48 H2 8 uH2 + 16 p H2
Equation 3 Coefficient of y
**
1 32 F +5 u F (1u 2
p p p
*
8 H0 + 4(1u2) + 8 uHo
p p
**
2 H1
12 uH1 + 4(1u2) + 20 H1
P
4 ,, 8u *
+ 16 p H1 2 1
P
**
2 H2 2 H
+ (18 u 6) 14 u(1u 2)
P P
***
22
 2u(1u ) 42
p
**'
+ 4(1u2 H22
P
2 *1
28 H2
P
(429)
8 F' + 4 p2F"
2 **u 2 H
+ 4(1u )H1 + 64 p Hl
; I I4pH
2 2 H2
+ 2(1u2) H2 2
p
628 2 + 28 p
+ (1628u )H2' + 4u(1u )H2 + 28 p uH2
+ p4 2 (2 *"
+ 16 up H2 + 8p (lu )H2 .
(430)
Equation 4 Coefficient of e Xa'5y yk pn
 8 H Ho + 2 (lu 2H ** 8 H +8 H
P P P P
8 10 ** +2 2*** 8 *' *"
4 H1 104 H1 (u2)H1 + H1 8 H1
P P P P
**
U 2. H2
 18 u H2 [10 u24(1u2) 4 +
p P
***
2u(1u2) H2
P
*1 **1
H2 *" 42 1 6u2 _H
+ 28 u  48 H2 + 8 uH2 16 p 4(u2 ) 2
P P
Equation 5 Coefficient of ioa Bpk p8
*** **
k ** 2 G1 G1 *"
p p4 + 7 (1u ) 4 4 G1
p 4 p 4 1p 4 p2 1
*2 G2 **
+ 9 + 23 u [3 10 (u2) i 
9 p P
** '
G 2 u 2 G2
14 2 18 G2 + 2(1u2) +20
P P P
***
u(1u2) G2
P
S *" 2
G24 uG2 + 8 p G2
** ***
+ 9 + 23 u [3 10 (u2) i (u
P P P P
 14 18 u +2(1u2 I +20 I 4 ul
p P P
+ 8 pI I].
(431)
(432)
Equation 6 Coefficient of ioa.p
** *** *1
Go G G Go
9 u (c7u2) uu2) G 14 u 
p p P P
G**' 2
+ 2(1u2) 2 + 24 GO 4 u GO + 8 p Go
P
**
G1 G1 G G1
+ 4 4 + 5 u (1u ) 4 4G
p P p P
**
9 u [23 (u2) 8] + 10(u2)u I4
p P P
***
(1u2)2 + 14 u  (818u2) 
P P P
2 I 2 (433
2u(lu2) 20 uI 4(1u2)1 pul (433)
P
p
Equation 7 Coefficient of io ka
** ***
G G G *' 2 *"
[3 + 5 u (1u2) G 4 p G
P P P
**
3 5 u + (1u2) + G2
P P P
2 **
+4 p2G 3 5 u + (1u2) 2
P P P
2 I + 10 u I 2(1u2) I 28 p21 8 p41". (434)
Equation 8 Coefficient of p
** *** *1
G 2 G0 G 2
9 u [2 7 (u2 ) u(lu2)  14 u 
P p p p
**I
2 Go 2 '
p
+ 2(1u2) ~2 + 24 G" 4 u Go + 8 p Go
P
** *I
G G G G' G1
12 15 u + 3(1u2)  + 12  10 u T
p P p p p
**'
G G2
+ 2(1u2) ~ + 36 G" + 8 p2 G"' + 9 u
P P
** ***
[8 23 u] 2 2 10 u(lu2) 2 + (1u2 2 2
*i 'l
G6 2 G2 2 G2
14 u +(818 u2) 6 + 2 u(1u ) 2
P P P
+ 20 u G" + 4(1u2)G2 + 8 p2u G". (435)
44 The RightHand Sides of the Eight Equations
The righthand side of the vertex equation,
y Fv + F ], (436)
is compiled through multiple layers of matrix multiplication. It is
best represented, not by an exhaustive itemization of each and every
term but by definition of the various layers.
The first layer is to define the matrix function F
FX = S(p+k) r (p+R,p) + S(p+k)r (p+k,p) S(p)S (p)
ap, aV
A A' 2 BB2 B2 2
B2
v A2B2 B2 + 2p B2 B2A2
Y 2 + o1 pa + 2p 2 2 D
2 2 2 2
= s(p+R)ZX
where
ZX = + X[2pPPD+ 2p OMPD yVABD+ yVBBD]
where
A2A2
PPD + 2
02
A2B2
MPD = 2
ABD B2
BBD = 2
B2
BBD = D
2
2 B2B2
02
B2A2
02
(437)
(438)
(439)
(440)
(441)
(442)
The Z is a tensor with twentyeight different linearly independent
combinations of the available matrices. These twentyeight form a group,
fourteen elements of which are odd in gamma matrices, fourteen elements
are even.
odd = gZ1 + 4P pZ2 + p XkZ3 + gkvZ4
+ VKppVZ5 + Vp k Z6 + p 7y Z7 + Xp Z8
+ yk Zg + eva PY YkpbpZ + Ea a y 5 kaPp Zl
+ EhvB4 + y kvZ12
+ E Sx y 5y k kakvZ14 (443)
even = p p Ze + p +2 + v 5k p B3
+ io Ze4 + gv Ze5 + io kc0P p p Ze6
+ PAkvioaSk e7 + ioa V
caSe7 a e8
+ ioa XkVZe + iok pZlo + i i ap v Z
a e a ei p P ell
+ ica p k Z + ic pk + ilaBkp gvZ (444)
a e12 a 13 el4
where
G H1
Z = G0BBD + + 2 ABD ul BBD (445)
P P
MPD G1 2F G G 2 2 G1 PPD
2 pk p 4 p 2 p 4 p 2
H4 H1
+ 4u MPD 4 MPD (446)
p p
kG1 k Ho k
Z3 3 p2 BBD 2 ABD + I BBD (447)
3 p p P
4 = G2 + 2 H2 ABD + pk I BBD (448)
k k k k k
Z5 3 G2 u G2 + G2 BBD + 2 G2 2 G PPD
H
Ho k
+ 2 ABD 4 H MPD 4 H2 MPD  I BBD (449)
p p
47
G,
Z 2 (450)
6 2
F G1
7 = ABD + G BBD + 8 + G BBD + uG2 BBD (451)
p
P
H2
Go
H1AD + 42P uH MPD + uI B8 (452)
Zg = k 2 H2 ABD pk I BBD (453)
10 = G BBD 2 ABD + I BBD (454)
S2p p
S= 4 H2 MPD I + 2 k I u 2 I PPD (455)
ll p2 3 p
p p p
H1
Z12 = G BBD 2 ABD + uI BBD (456)
p
13 = 2 H2 ABD + I + pk I BBD (457)
14 (458)
14 7
i *
F F F F F
Z = + 2 2 PPD + p BBD
e1 p3 p.k kp3 pk PT
kp kp
G1
+ 2 G MPD + 2 G1 MPD ABD + 2 uG2 MPD
p
H HI H
+ 2 u  BBD 2 T BBD + 2 uG2 MPD + 2 u p BBD
HI
+ 2 1 BBD (459)
pk
*
F k
Ze2 = F G2 ABD 2 H0 BBD 2 H2 BBD
P
k
Z = I ABD + 2 H BBD
e3 p 2
Ze4 = G ABD + 2 8 + 2 t H, BBD + 2 8 u H2 BBD
Z5 k G ABD + 2 H1 BBD + 2 uH2 BD
e5 pk k 1 kH2
H' *
k H H Ho
Z6= G2 MPD 4 + 4 2 u
p p p
H
Ho k
4 PPD + 2 I MPD
2 p
P
Ho
Ze = 2
e7 ~ p3k
i *
H1 H1 H1 H1
S = 2 G MPD + 2 + 2 u + 4 PPD
e8 p3k k p 3k pk
2 u I MPD
*
H2
Z = 2
eg pk
Zel0 P G2 ABD + 2 H + 2 H BBD + 2 H2 BBD
P
G H
Z1 BBD ABD + 2 _u H BBD 2 p BBD
el2 = 4 H2 2  H2 4 H2 PPD + 2 H2 BBD 2 pk I MPD
P
+ k I ABD
p
(460)
(461)
(462)
(463)
(464)
(465)
(466)
(467)
(468)
(469)
(470)
H1 k
S= 2 BBD + I ABD (471)
H
Ho k
e14 =2 2 2 H BBD k I ABD (472)
P
Since FXv = S(p2)ZX
A1 V B1 X
D1 Z D ( +K)ZV (473)
1 1
Even = P~P1 + pXkvW2 + XavE 5ka W3 + io W4
+ a 5 + p~P p Viaak 6W6 + P ciaBkapk W7
+ ioXa VW8 + ioaxk kVW9 + ioav pXka10
+ ip oa W + ioak VW12 + io Pak W13
+ ioaBkap g 14 (474)
odd = g9vV1 + p~p V2 + pkV3 + gKV4
+ KpXpVV5 + + pkV + pVV7 + ypVV8
+ yk V9 + va 5 y kagpp V10
+ E Yy aP p kpV + E y5 y P V12
+ Aav 5 , + F. p k k1V
+ E 5y kV13+ E 1y 5P kak V1V 4 (475)
where
W1 = AD Zel + BD[ (p2+pk u)Z2 (1 + )Z5 Z7 Z] (476)
W2 = AD Ze2 + BD[ (p2+pk u)Z3 (k2+pk u)Z6 k2 Z7 Z9 (477)
W3 = AD Ze3 + BD[Z13 + k2Z12] (478)
W4 = AD Ze4 + DB[(p2+pku)Z2 (1 +P u)Z13 (479)
Wg = AD Ze5 + BD[ (p2+pku)Z (1 + P u)Z4] (480)
W6 = AD Ze6 + BD[k2Z2 Z5  Z 11] (481)
W7 = AD Ze7 + BD[Z Z6 Z10 14] (482)
W8 = AD Ze + BD[ Z8 + ( +R U)Z1 + Z12] (483)
W9 = AD Zeg + BD[Z9 + (p2+pku)Z14 + Z13] (484)
W10= AD Zel0 + BD[k2Z7 + (p2+pku)Z10 Z13] (485)
W11 = AD Zel + BD[ Z7 + (1 + u)Zlo Z12] (486)
W12 = AD Zel2 + BD[k2Z8 + (p2pku)Z1 + Z13] (487)
W13 = AD Z13 + BD[ Z9 + (k2+pku)Z14 + k2Z12] (488)
W14 = AD Z e4 + BD[k 2Z Z4] (489)
and where
V1 = AD Z + BD[ Ze5 + u e4+ Z4] (490)
e5 k e14+ e14
V = AD Z + BD[ Z + ( u+1)Z + Z + Z ] (491)
2 2 el k e6 e ell
V3 = AD Z3 + BD[ Z2 + pku Z + Ze + ekZ e+ kZ (492)
4 = AD Z4 + BD[ p2Ze4 k2Ze5 pku Ze l4 (493)
V = AD Z + BD[ p2Z Z Z k2Z pku Z ] (494)
V6 = AD Z6 + BD[ (p2+pku)Ze7 Ze9 Ze2 Zel] (495)
V7 = AD Z + BD[ Z + u Z (p2+pku)Ze + Zel (496)
V8 = AD Z + BD[Z (p2pku)Ze8 + (1 + )Zel2] (497)
V9 = AD Z9 + BD[(k2+pku)Ze9 (p2+pku)Ze13 + k2Ze4] (498)
V10 = AD Z10 + BD[Ze3 + 0 + k2Z ell (499)
V11 = AD Z11 + BD[Z2 Ze3 + k2Ze8 (4100)
V12 = AD Z12 + BD[(1 + u)Ze + Ze] (4101)
12 k e3 e4
V13 = AD Z13 + BD[ k2Z + (2pku)Ze3 (4102)
V14 = AD Z14 + BD[Ze9 + Ze3 Ze3 (4103)
Another way to write the righthand side of the vertex equation is
to notice
F X+
1 r yvXF, aFX
V 9p' V
= a{ [ g 3 av]F (4104)
The next layer of definition is to choose
S 3 i Xv (4105)
T = [ ( g a ]F (4105)
where
T even = p Tel + pXkvTe2 + cFX 5k pBTe3
+ i T e4 + gXvTe5 + p p Te6 + kpp k Te7
+ ia pTe + ioakc kVTe + iocvpXTe
*a0 e8 a eg em0
+ ioa pa p ell + i+a v el2 + ioa apk Tel3
+ ioUakcP bg Tel4
odd + 0~ppvTo2 + ~pkVTo3 + g To4
+ vp pVT + p Xk To6 + p YVTo7 + A p To8
+ ykVT 09 + vat 5Y 7ka Pp xToi 0
+ \^c ^ ^12
+ Xa y YmkaP p Toll + cNyBy T 012
+ xav y k Tol3+ ya 5y ka PTol4
where the
Tel = W1 2k 6 27 8 2 W
T 1 + p2 1W 3 1 + +
Te= 7 2 2 6 + pku W7 + 10 + 12 2+ 14
T 3 W 1 N 1N
Te3 3 12 W13 + W14
T W 1 W 1 2 12MW
e4 2 4 2W 5 2 W9 2 k 127 8 7 k 13
Te5 =2 5 2 4 k12 2 p2W8 7 k 13 9
Te = W6
Te7 = W7
e8 8
T = W
e9 9
(4106)
(4107)
(4108)
(4109)
(4110)
(4111)
(4112)
(4113)
(4114)
(4115)
(4116)
Te =2 W10 + W2 + 26 + pku
ellO 11 2 W1 6
5 1 I 1
Te2 12 2 3 W14 2 13
T 3 1 1 1
Te3 2 13  3 W14 + 12
Te 3 1 w 1
Te14 = W14 2 W3 12 2 W13
W7 W14 + 2 W2
+2W3
To2 = V2
3 1 1
T3 = 3 2 V5 10 2 11
T o =2V V P2V1 V pku V14
3 1 1 2 1
04 2 4 2 9 2 i V13 2 pku V14
T 1
05 7 7 S 7 V +lo 2 ii1
S6= V6
To8 =IV85 l I k 11 127 14
To V9 V4 + p V1 V 13 + pku V14
3 1 1 1 v1+ V + v
T10 2 10 3 2 5 10
Toi Vl
011 1 +
Tol2 = 12 + V1 7 8 2 kV14
(4117)
(4118)
(4119)
(4120)
(4121)
(4122)
(4123)
(4124)
(4125)
(4126)
(4127)
(4128)
(4129)
(4130)
(4131)
(4132)
(4133)
1 12 1 1
T13 2 V3 V p + Ppk V11 + p V9 (4134)
To4 = V14 (4135)
The righthand side of the vertex equation,
E T (4136)
is collapsed into a group of eight linearly independent matrix terms
by the summation over a. It is clear that in order to obtain the first
order partial derivatives of T a type functions the firstorder partial
derivativesof the W and Z had to be known. Initial work on this
main program tried to avoid this confrontation by seeking the deriva
tives of Tha by a standard numerical process. This was found unsatis
factory for two reasons. One was the numerical calculation added sig
nificantly to the time parameter of the program. The other was the
precision which this time bought was inadequate. This method was
shortly abandoned in favor of compiling the algebraic forms of the
partial derivatives of Ta, W' and Z These are recorded in the
program included in Appendix C.
In conclusion the righthand side of the vertex equation is
comprised of eight linearly independent matrix terms. The coefficients
of each of these terms are given in Table 42.
TABLE 42 RightHand Side of the Vertex Equations
Equation 1 Coefficient of pAO
E[2 T T2 + 6 To + 2 p 2P + 2 u T3
101 p z702 T2 0+ p 2 k o3
+ (1 T + 2 T T ]
pk T3 07 p2 o7
Equation 2 Coefficient of pY
u 2
E[To3 + 2 To4 p u4 To4+ 5 T + 2 p To5
T 2k k *
+ 2 pku T + (1u ) T + T ]
06 P 06 P 07
Equation 3 Coefficient of yA
01 07 0o 08 k 09
+ (1u2) *
pk To]
Equation 4 Coefficient of Fasay k p
2 k* *
E[T + 4 T + 2 p T + T12 + 2 T13
010 011 011 p o12 ol3
Su T 2 pku T + k (1u2) T ].
p2 013 014 P 014
Equation 5 Coefficient of io3Ska PpA
2' k u2) *
E[6 T + 2 p2 T + 2 pku T + (1u ) T
+2 T T + kT + 2 T T ].
e10 2 e10 P ell ep4 p e14
P P
Equation 6 Coefficient of il pa
u 2'
[2 T Te4 5 Te8 2 p Te8 Tell
p
2 Pu T' (1u2 1 T ].
k el13 pk el 3
Equation 7 Coefficient of ioL'k
k 2k*
E + 2 + T
p e4 e9 p eg T 10
2
+ 4 Te2 + 2 pT T + T 14].
el2 el2 el3 el4
Equation 8 Coefficient of p
E[5 T 2 p2 T 2 P T (1u ) T 2 T
el 2 pk e2 e5
 u Te5].
CHAPTER V
CHECKING THE ALGEBRA
In performing all the myriad matrix operations necessary to
express the vertex equation we found that working out the algebra was
extensive, repetitive and subject to error whenever the practitioner's
strictest attention lapsed. After an unoriginal effort was made at
wading through the whole expression term by term,it was necessary to
confirm the results. Faced with the prospect of having to repeat the
monumental process, we drew the conclusion that this kind of work is
better done by machine. Efforts were made to solicit the use of a
computer language capable of symbolic algebra.
The software chosen was a programming language from the University
of Utah called Reduce. The program offered a great variety of general
algebraic calculating facilities. Amongst those of interest were
symbolic differentiation, automatic and user controlled simplification of
expressions, calculations with gamma matrices and tensor operations.
With all of these capabilities it offered a very promising approach to
the unwieldy task.
It was found that Reduce provided a direct and easily acquired check
on the lefthand side of the equation. The partial differentiation was
performed, including all nine possible mixed derivatives with respect to
p2 and u up to third order. A quick and accurate check of the original
lefthand side was achieved. Furthermore, this confirmation was achieved
with only a reasonable expenditure of time invested in becoming familiar
with the language.
However Reduce was less easy to make use of on the righthand side
where, in addition to first and second order mixed partial derivatives,
there were three layers of matrix operations to undergo. The size of
the arrays quickly outgrew the allotted workspace in the machine. All
of the calculations had to be performed in steps and then the results
were summed afterwards. A further investment of time would have been
necessary to learn how to design the output to be displayed in a form
more amenable to easy checking. Despite this user related ineptitude,
a complete expansion of the righthand side was obtained.
One of the reasons Reduce was found to be less useful than expected
on the righthand side was the operations, though limited in variety,
involved a proliferation of terms. It was almost a waste to bring all
of the ingenious operational flexibility of Reduce to bear on what was
only a problem of tensor and matrix multiplication and largescale sort
ing of terms.
The righthand side could be written down in a way that was better
designed for checking by giving up the luxury of exhibiting the right
hand side in terms of the basic functions A, B, F, G G2, Ho, H1, H2
and I. The final form settled upon for the righthand side,as it
appeared in the previous chapter, was expressed in a hierarchy of defin
itions. The righthand side was given in terms of the 28 components of
TXa tensor. The 28 components of T a were given in terms of the 28
components of the F v tensor. The F v tensor was defined in terms of
the 28 components of the Z v tensor. Finally the 28 components of the
Z v tensor were defined in terms of the basic functions, A, B, F, G G1,
G2, H H1, H2 and I. Each layer of redefinition represents the execu
tion of another tensor operation.
Table 51 sunmarizes the five steps taken to define the right
hand side. The first step was to define a tensor,
Z = 2PPD FAp ABD r y + BBD Xyv + 2MPD FXp (51)
The abbreviations PPD, ABD, BBD and MPD, represent combinations of the
electron functions, A and B, which were defined in Eqs. (439), (440),
(441) and (442). Step 2 forms a new tensor,
WX = (0 + b)Z (52)
Step 3 forms yet another tensor out of the former two tensors,
A B
F =DZ D W (53)
where D1 = (A p2B2)
Step 4 forms the last tensor,
T [g Y + OYy]F. (54)
Finally, in Step 5, then index is contracted by a differentiation with
respect to pa. The rightside equals
c a TAa. (55)
apC0
TABLE 51 Formation of the RightHand Side of the Vertex Equation
Step 1.T Z ' = 2PPD FpV ABD FXy + BBD p 0yv
+ 2MPD rFip)
Step 2. W X = (0l + g)Z
A B1
Step 3. F =A Z B W
Step 4. Tr = [ga + Ya Y,]F
Step 5. Righthand side = e TXa
apa
The abbreviations PPD, BBD, ABD, and MPD were defined in
Eqs. (439) through (442). The abbreviations Al and 01 were
defined in Eq. (424)
After the righthand side is separated into layers so that the full
length and breadth of the righthand side is disguised, a large number of
operations still have to be performed. Even a simple multiplication
like rA or (g+I)ZAV involves a large number of steps. Consider the
procedure necessary to perform the multiplication of a single element
of r say c 5 ky with p.
AaBQ 5 i I M# n 5 I
E y5k p6 p = ka p an pkpk
S kai 6i n r 55i+ TI 1I 3
= TW kpp[g +* Y ] 
= [2g Ba+ 2gal + 2g 1a]k pp I k
= [iokappXP + ipku o ap+ ip2oa ka I (56)
Table A1 was used to obtain the product y5 n in the second line.
Equation A12 was used to go from the third to fourth line.
It is easy to forget a sign or reverse the order of two indices
when a large number of such operations are performed. However it is
possible to express the multiplication rules for matrix operations in
a very simple way that allows the practitioner to do the same calculation
in his head without ever consulting Table A1. This simplified method
can be used to supply an algorithm to enable a computer to do the same
kinds of manipulations in a common language like Fortran which does not
possess symbolic capabilities. The basis of this method was designed by
H. S. Green.
To explain the simple multiplication method it is necessary to alter
slightly the form of the definitions of vector functions like r and
tensor functions like Z v. The expressions used in Chapter IV were
evolved through a historical process that did not necessarily produce
the most symmetric arrangement. In this chapter it will be shown that
when some small changes are made, things become very much easier.
In Chapter IV,
= A F A A G1 G2
p= A p + 6 G + p 2 + p 2
P
Ho a Hla2 H
+ 2p ioa p k p 22 + 2ia P p + 2ia'xk p
a p 2k2 F k a pk
+ Eca Y 5 Y p k (57)
Now a slightly altered form will be used.
px = p + 2 + PA3 + PA4
+ p'ioa0kaB B5 + iojapc 6 + ioykLa 7
+ ca y kaB 8 (58)
wF 2Ho
where pk 5 k2
2H1
2 = 6 pk
G1 2H2
3 p 7 k
P
(59)
I
p8kl .
The new form of rF is equivalent but notice that the order of the indices
in the sixth and seventh components has been changed. Throughout this
chapter it will be assumed that any general vector, say V is expanded
in terms of these same components.
V = pv1 + yv2 + pv3 + p v4
+ p ioR kapv5 + ioApav6 + ioakv7
+ aB5Sy kppv8" (510)
Similarly it will be required that any general tensor, TV, will be
expressed in terms of the following components only.
TX = pp t1 + p t2 + t3 + pioaBk t4
+ pv t5 + p iap t6 + pio akat7
+ P EXa Y 5 k t8 + pYt9 + p ioVCptlO
+ p k Ctll + pX VpyySk kP Bt12
+ i~Xtl3 + EA y5 yp t14 + :AcvtyS k atl5
+ vaBy 5ka Pt6 + p k tl7 + p kvotl8 + pAkvtl9
+ p k io BkPapt20 + y k t21 + kVioap t22
+ kVi akat23 + kvE ay5y kapg t24
+ gAVt25 + g t26 + gAVt27 + gAioaBka t28. (511)
Now that the groups of matrices have been carefully selected for
any general vector V or tensor T the following notation will prove
extremely useful. The components of a vector will be denoted by the
following brackets.
p = (p 0, 0, 0)
pg = (p X, 0, 0, 0)
p = (p, 0, 0)
pioa ka p = (pAI, 0, 0)
Y = (0, 0, y 0)
iAopc = (0, O, Y 0)
io Ak = (0, y, y O0)
ragSy ka B = (g, l, 0). (512)
The components of a tensor, T will be denoted by the following
brackets.
ppV = (p V, 0, 0, 0)
pp = (pp 0, 0)
pXp = (p p V, 0, 0)
p i aBkp B = (ppV, J, 0, 0)
pV = (p l 0, y, 0)
p ioh p = (pV~, A, y a 0)
pV Xaka = (p V y O)
pVCEXa6p 5y k p = (pVo i, Y O)
p y = (p 0, 0,
pXiovp = (p p, 0,
piovca = (pX,
PvE 'YkaP =
io = (0, 0,
v y 5pa =
Xav5y 5ka =
EXAvaBykSkap
iV = (0, 0,
ur5 c =
Sav45yok =
E vay5k p =
SV)
0, YV)
i:va,)V
O, yA, V)
O, 4A, 0V)
~, yA, yV)
0, ', Y)
y V)
l y X, yV)
y vA, yv)
pXk = (p kV, 0, 0)
pXkV = (p V, kv, 0, 0)
p k\ = (p, kV', 0, 0)
pkviok pBka = (pV, k4, 0, 0)
ykv = (0, kv, 0)
kvia = ( X, k 0, y, )
kiX a yk a = (k0 y, 0), O )
kv k = (0, )g, 0)
g V = (0, 0, gA, 0)
gXV = (0, 0, gV, 0)
g ViaBao p = (g, i, gXV, 0). (513)
The rule for the mapping of each component into the bracket notation
is straightforward. The first position in each bracket holds all electron
moment terms; p p and V. The second position holds all photon moment
terms, kv and k. (The kX term does not appear here because only the trans
verse part of the vertex equation has been considered. However it is
possible to extend the bracket notation to include longitudinal components
without any additional complications.) The third position holds yX and
gAv. The fourth and last position holds y.
Now to convert the component into its bracket merely tabulate the
occurrence of yP yk, y, y, p, kU and gWV and put them into
their appropriate "home" positions. For example, consider p iai k a. t
contains pA, y and k yQ so its bracket is (p /, 0, yV). Notice that
although there is a unique bracket notation for each component, it is not
necessarily possible to guess from the bracket notation what the original
choice was for the component.
( i, 0, 0) = ioapka
or
( 0, , 0) = iPk a g?
The originally selected meaning of (0, , 0, 0) must be preserved so it is
the latter relationship which is the correct one.
Now it will be shown that multiplying the components of the vector
V or tensor TAV by a term like ~, ,y p ,k etc. is simply done by
observing a few rules. To multiply by a unit matrix vector like p or
k merely add the vector to its correct position.
(~, O)pV = (pUV, /, 0) (514)
To multiply by a matrix like y V or i from the right, move the matrix
across the bracket from right to left. Each time the matrix crosses
another matrix take the scalar product of the two. Remove the scalar
products to the outside of the brackets.
(p g, p, 0, 0)g = p2(p 0, 0, 0).
(515)
One term will occur for each scalar product and one term will occur
when the multiplying gamma matrix reaches its "home" position,
(p kV, 0, 0) = (pX, k 0, 0) + pku(p kI, 0, 0). (516)
Finally, reverse the sign of alternate scalar products,
(p, k, 0, y, ) = (p kA, 0, 0) k2p2 W 0, 0, y")
+ pku(p A, 0, y"). (517)
In order to multiply a tensor or vector from the left the rules
remain the same except that the multiplying term is moved across the
bracket from left to right.
Recall the example given earlier of the multiplication of
EC Y kPa (h by 0 from the right. This example can now be writtenas
(0, X A, Y 0)  0 = [(pV Ji, 0, 0) pku(g, O, y 0)
+ p2(0, X, yX, 0)] p1 (518)
Translating the bracket notation back to the original components it is
found that
C By5y kp p [pxi [ kap pku ioXpa
+ p2icAak k 1 (519)
This is the same result which was achieved in Eq. (56), yet it was
achieved without using Table A1. With little practice the products
can be arrived at as quickly as they can be written down. This process
of bracket manipulation is not only many times easier for the human
mind, but it also lends itself to the writing of an algorithm for exe
cuting similar products in Fortran.
It is also possible to predict easily the products of a contraction
of a vector Va or tensor T a with another vector like pa or gamma matrix
like y To contract with a momentum, merely perform the implied con
traction. To contract with a gamma matrix it is necessary to first
move the gamma matrix across the bracket, taking alternate signs of
each possible scalar product. While doing so, perform the implied con
tractions on each term in the product.
A summary of the rules for bracket operations is given in Table 52.
Returning to Table 51 it is easy to see that a knowledge of the
following operations is all that is needed to generate the righthand
side, V~ VXp, VAyV, y, V, k T', Tv and y J where V stands
for any general vector function and TXV stands for any general tensor
function. Using the expressions for VA and T v given in Eqs. (510)
and (511) and using the method of bracket operations, these seven products
are easily written down.
(1) V = p (p2V2 + pku v3 + v5) + (p (v1 + pku v4 + v6)
+ p k( p2v4 + v7) + pioBkp'(v8 v3)
+ y p2v6 pku v7) + ioa% ( v5 pku v8)
+ io'ak (p2v8) + EXB5Y kaYpB(v7)
(520)
TABLE 52 Bracket Operations
A. Multiplying by a momentum: Add the momentum to its
"home" position.
B. Contracting with a momentum: Merely contract the momentum
with its proper complement and move the scalar product
to the outside of the bracket.
C. Multiplying by a gamma matrix:
(i) Move the matrix across the bracket from right to
left in order to multiply from the right. Reverse
directions to multiply from the left.
(ii) Take alternate signs of each scalar product that
can be formed.
(iii) If there is no matrix in "home" position, include a
term with the multiplying matrix in the "home"
position.
D. Contracting with a gamma matrix:
(i) Perform steps (i), (ii).
(ii) Perform the implied contraction as the gamma matrix
is moved across the bracket.
(2) VpV = pAp(v ) + pVp (v2) + pApN(v3) + p pvica kap(v4)
+ ppx(v5) + p"io'apa(v6) + P'ioak (v8) + PV y y pB(v8)
(521)
(3) V y = pAp(v2) + p A (v4) + Py(v6) + p)ioaak(v8)
+ p yV(vl) + pxiop (v2) + pAi a k(v3)
+ pP VaySy k P (v4) + ioa(v5) + cavy5y oa(v6)
+ pXkV(v3) + pkAk(v4) + p kV(v7) + p Akv iak (v8)
+ yXkV(v7) + kioXap (v8) + g9(v5) + gX\)(v6)
+ gAV(V7) + gAiok ap(v8) (522)
(4) y VV = p (v2) + ppv (v4 () + p ) + pioj k a(v8)
+ p yV(vl) + p ioGpca(v2) + p i(~k (v3)
+ p A v Styyk p (v4) + io (v5) + Yav5pa(v6)
+ xav y5yk a(v7) + Av SpBk (v8) + p k(v3)
+ pAk (v4) + yXk (v7) + kvioXpaa(v8) + g (v5)
+ gA (v6) + gA (v7) + gX9ioABk p(v8N) (523)
(5) 4TXv = pp (pku t2+ k2t3) + pXpV (k2t4)
+ pXp (tl+ pku t4) + pAP)ioaka p(t2)
+ pVy (pku t6 + k2t7) + p iop p(k2t8)
+ pvioak (t5 + pku t8) + pv cS y5ykas (t6)
+ p yv(pku t10 + k2 t11) + p iovpa (k t12)
+ p iovk (t9 + pku tl2) + pxE ySyk B(tO0)
+ iov (pku t19 + k2 t5) + avYyS p(k2t16)
+ FavtSy ka(tl3 + pku tl6) + X 5p k BY (t4)
+ p kv(tg+pku tl8 + k2t19) + pXkvO(t0k2t20)
+ p kv (tl7 + pku t20 tl) + p k i6 k (t12ti8)
+ ykv(tl3+pku t22 + k2t23) + k'iopa(t14k2t24)
+ kviToXk (tl5+t21+pku t24)+ kVE AB Y5kaPB(516t22)
+ g (pku t26 + k2t27) + g (k2t28)
+ gx(t25 + pku t28) + g AvioBk PB(t26) (524)
(6) gTAv = p (p2t2 + t + t9 + pku t3)
+ p pV (tI pku t4 t6 t10)
+ pApVK(p2t4 t 11)
+ pXpVioa"kBkp(t3 + t + t12)
+ pV (p2t6 + pku t7 t3)
+ pvioXa (t5 pku tg + t14)
+ pioAak (p2t8 + t15)
+ p 5'yaB kp5(t7 t16)
+ pA (p2 t10 + t13 + pku tl)
+ piovpa (t9 pku t12 t14)
+ p iovak (p2t2 + t15)
+ pA EBay5y kap(tl + t16)
+ io (p2t14 + pku t15)
+ E vy5 pa(tl3 pku t16)
+ XavySy ka(p2 t16)
+ y5p k a(tl 5)
+ Pkv(p2tl8 + pku t19 + t21)
+ pxkv)tl7 pku t20 t22)
+ pxkV(p2t20 t23)
+ pAk ia kaV (t19 + t24)
+ yk"(p2t22 + pku t23)
+ kio p o(t21 pku t24)
+ kvioaak (p2t24)
+ k ea 5y k y pB(t23)
+ g~(p2t26 + pku t27)
+ gXV(t25 pku t28)
+ gxk(p2t28)
+ gXioaBka P(t27) (525)
(7) yT = pX(p2t2 + pku t3 + t5 + 4t9 + pku t18 + k2 t19 + t26)
+ p (tI pku t4 t5 3t0 p2t20)
+ pX(p2t4 t7 3tl + t17 + pku t20 + t28)
+ p ioBkap(t3 + t8 + 4t12 t18)
+ (p2t6 + pku t7 3t13 + pku t22 + k2t23 + t25)
+ ioP (t5 pku tg + 4t14 k2t24 + t26)
+ ioak (p2t8 + 4t15 + t21 pku t24 t27)
+ cxasSy k 6 (t7 3t16 + t2 +t28) (526)
These seven equations complete the work necessary to execute a
check on the correctness of the righthand side as it was described
in Chapter IV.
CHAPTER VI
THE MASS SHELL SOLUTION
61 An Approximate Solution
In Chapter IV the transverse part of the vertex equation was
decomposed into a set of eight linked differential equations out of
the original integral expression for the matrix function. The eight
differential equations established relations for the eight unknown
scalar functions, F, G G G2, H, H2 and I. These scalar func
tions exactly describe the transverse vertex for a given photon momentum
squared, k2
X(p+k,p) = F(p2,u) + p[ _,K] (p2u)
p kp
+ EYgl] (p2,u) + [y yK] 2 (p2 u) + yG(p2
S x 1 1 2 + 2 2 + v 5 1 2( 6 
+ p (p ,u) + pKT (p ,u) + E k V (p ,u). (61)
The circumflex signifies the transverse component. The scalar functions
are shown as functions of the electron momentumsquared, p2,and the
relative angle function, u, where u = 2 The scalar functions
(ip ilk )I
are parametrized by the value of k2. The eight equations were checked
by a process described in Chapter V. The eight linked equations which
are third order in derivatives including all nine possible mixed
75
derivatives with respect to p2 and u, were complicated to write down,
difficult to check and promised to be significantly more difficult to
solve. Therefore it was important to start with a good guess of the
correct solution. From the experience of solving the electron equation,
where it was learned that the mass shell solution dominated a wide region
around the mass shell, it was hoped that the vertex equation would
similarly be a slowly changing function. If this were so, then a solu
tion to the eight differential equations which worked near the mass shell
might be extended into the asymptotic region by simple variations of the
eight scalar functions.
In 1981 H. S. Green communicated an approximate solution to the
second order tensor equation,
V2FA = 1[ Fy + 3 VFv]. (62)
The third order tensor equation, Eq. (416), is obtained from this equa
tion by the operation of 4. The tensor functions Green found to be
approximate solutions to Eq. (62) were used to identify the approximate
form of the eight scalar functions. The approximations employed were
commensurate with the known behavior of the vertex and the electron
propagator near the mass shell.
In Eq. (62) the tensor
F V= S(p) [r(pX 2)S(p2)]S1 (p2) (63)
Av
was approximated by A v where
A 1 S( )F (Pl,2)S(p2).
(64)
This amounts to neglecting
S(p1)[ r pA(pl2) (Pp2)S(P2)(P2)( S 2) y)]. (65)
This term approaches zero as r approaches its mass shell limit of y.
Near the mass shell the appropriate form of the second order vertex
equation is given by
V 2 1 A V (6l6)
V2 = c[ y y + 1]. (66)
This is a far simpler equation than (62).
This equation can be decomposed into a set of four linearly inde
pendent matrix equations by the following device. Take the product of
Eq. (66) with 1, y y and Y then take the trace of each product.
To facilitate this the following definitions weremade:
C = tr [rF ] (67a)
C = tr [rh y] (67b)
CA 1 tr [ryV ] (67c)
C = 1 tr [rF ] (67d)
Wvp 4 L vvp
D = tr [AA] (67e)
D = tr [A F] (67f)
D tr [AyI ] (67g)
DA 1 tr [Ay ] (67h)
Pvp 4 'vp
where Yv = [Y X'Y
and y { 'Yv (68)
and Yp 2 Y >YI*
Now it can be seen that the set of four equations below is equivalent
to Eq. (66).
2C = 3cDx (69a)
V2Cx = 2 E D 2 e V23DAX (69b)
S p pvo
The D D, DO and D tensors are evaluated by using the
= [(PI'P2+A1A2) (A1Pe+A2P3 )C p pp2C v]/DX (610a)
D = 4 tr [r(#2+A2)Y( 1+A1)]/D12
S[(A2~IP 2 OD) x + A2P P2)C + (p DP2pp
2 D VC 2 Dx ( 619d)
d+ A2P 2 q. (4)+ (610b) (6 ) (6 )
[ (p p 2v 2p +P I) C +(A2 P1 vA1P2G)C (A2 VPUA1P2P)C (Pl P2+AA2)C
S ( 02 2Vp 2 l p 1)2
+APlVI2 PPlV P P)C(llA VI xApl VI .10a)
(Ap VA ^c +ip PCA PI/D (61Oc)
DA = tr [FXA(2+A2)y (~p +Al)]/D12
=vv 4 02 2 2
= (Pl2P2Pl) + (PlvP2pPvPl)p + (PlpP2pP2pP P
+ (A2PI +AIP2 )C + (A2Piv+AIP2v)C + (A2Plp+A P2)Cv
+(A1A2P'P2)Cp + (Pip+P22pPo)Cpo + (plvP2+P2vP)Cpo
+ (PlP2+P2pP)Cu/ 2. (610d)
These definitions are somewhat intimidating in appearance but the
full effect of assuming the vertex function will be dominated by y in
the region of the mass shell has not yet been fully explored. If P is
well expressed by yX for some range of values of p2 and u, then this
means C is well expressed by 6 and the other tensors C C and
X A
CA have negligible effect relative to C Furthermore, the electron
propagator has the simple form Z2/0m. Under these assumptions the
D D D and D tensors take on the greatly simplified appearance of
D = 2mpA (61la)
D12
D = [(m2 _P 2)6A + (plp+P2p)pA]/Dl2 (611b)
DXA X X ]/D (611c)
D = m[[(lp 2)6 (PlpP2)6/D12 (6 )
Dp = [(plvP2P2vPlv)6 + (P1vP2pP2vPl)p 6
+ X]/D (611d)
+ (PlpP2pP2pP1p) ]/D12 (611d)
where D12 = (pm 2(p2m ).
5 6A is defined in Appendix A.
11
Green's work using these definitions of the D type tensors to solve
the four equations, (69a) through (69d), for the four tensors C C,
C and C is reproduced in Appendix B. In the next three sections
his expressions for C C C and C are used to determine the near
themassshell forms of the eight scalar functions F, Go, G1, G2, Ho, H1
H2 and I.
62 The F and I Functions
In Appendix B, Eq (B32) gives the expression for the C tensor
where
S= Emp 1 [e In ( ) + ]
x = [N + (8+1) ]2 and u = m2 (182)k2.
(612)
(613)
Furthermore, from the definition of C Eq. (67a), the scalar function F
is determined.
C tr (') = p F
S t .
(614)
Combining Eqs. (612) and (614) the scalar function is identified in
terms of an integration over beta.
3 cmp/I d u x
F = p [ In(I6 ) + 1]. (615)
Sl 1 X B
Obtaining the I function is slightly more involved but in the end
it will be seen that the expression for I is simply related to the
expression for F. The defining equation for C0 is given in Appendix B,
Eq. (B59) and it states,
V2 AP 1
( PipC ) k2 1 d (616)
Using the fact that when
U0 X
S (1 ) In (15) (617)
4 x u
then
V2= 1 (618)
x uB
so that Eq. (B16) can be reexpressed as
V2 (pP p C ) k2B V2 de. (619)
It follows that
Pp aC k2 J k dR. (620)
From the definition of CX the scalar function I is defined.
pvp
C =4 tr [y ]vp] = {6 [kp k p ]
+ [k p k p ]+ 6[k pk p k]} k (621)
From the above it can be shown that it follows that
P p C =2 I (622)
From Eq. (617) it can be shown that
2
3 fD dB= 2 2p { 1n (1 A) + }dB. (623)
x 25 X
By substituting Eqs. (622) and (623) into Eq. (620) the function I
is identified,
l _iI u x x
I = pk [ In (1 ) + ]dp. (624)
xu x
It is now apparent that a simple relationship has emerged. That is,
I k F. (625)
63 The G G1, and G2 Functions
The C tensor is defined in the appendix by the following statement.
C C*6 xC* (B48)
where
C = [1 ] n (1 U)dB
{( In (1p2) + (1 ) In (1 p2)} (B40)
Pi P2
and where
Spk2 1 1
k ( + 21) (m2u6 In (1 B) d. (B47)
4 2 B XU
For greater ease in manipulating these quantities, the following defin
itions are made:
2d = 1 i) n (1 ) d (626)
l X 2x
fl3dB = u ) [( 1) In (1 ) + 1]dB (627)
14dB = (1 u) n (1 ) (628)
CK= (1) n ( p) + (1 ) In (1p). (629)
Pl P2
Then
C5= 1 6J dSB + Ei x 3 dx
k k x
6 k E[p k + 4d k [p +  I4 dB
(630)
where
Sx apk pk
S2p ; Ix 31 ; = 0; and p = p k
ax' A 92
Then
t k pk
2 3 2 3 2 4
and
C = J2d5 CK (632)
so that
Cx = C 6 6 C*
=h 6 2d CK /13d]
+ pXp[ f13ddg
+ p k[E (2 + 1)(JI3dB + I dB) fE I d]. (633)
From the definition of C Eq. (67b), the identification of G G1, and
G2 can be made.
C= 1 tr [FAy]
G1 A 62
+ pp 2+ p k (634)
From a comparison of Eqs. (633) and (634) the final expressions for the
three G functions are formed.
G = 12d  CK I 3d6 (635)
2
G1 = E I 3d (636)
G2 = (2p2u + pk)( fI~ d + f/ d)
/Ix pkd6. (637)
64 The Ho, H1, H2 Functions
The defining equation for CX is shown in the appendix to be
CA EM [6Ak Pk D A)% (6 Ak Dk 3A) ]JfI5d
1cv 16 va v va app/
+ m (Ak 6 k )f/I d (B55)
where
1 0 x6
d 1 [k2L ) + I x6) In (1 x)]d, (638)
5 ) + (1(638)
l 2D
I6de= 1 n (1 )]de (639)
L (z) = In(1z) d (640)
In order to perform the implied operations first notice that
a = I(x) x a (2p + (B+l)k )Ix (641)
where
Ix =_ I (642)
Dx
and
a23 I = 26a x + k ka(B+1)2 xx + pk 2(B+1)Ixx
+ 2p ka(B+1)Ixx + 4pnpaIXX. (643)
Completing the operations implied in Eq. (B55) yields a new expression.
CX = C ((p p k pxpk) 2 fxx d5
+ (6k 6k )[2f 1dB + 2k2fI XB2dB
+ (2pku+2k2) fIx OdB + 2pkuJIxxdB + I6gdo]
+ (6 p6 A )[4k2 /fxxBd + 4pku Ixxd6]}. (644)
The relation between the tensor function CA and the three scalar H
ui
functions is given by the definition
C = 1 tr [FA Y ]. (67c)
This can be expanded to reveal the relation between C and the three
H functions.
A A 2H
C =(pp k ppk ) 0
S2H
+ (6 k 6 k ) 2
2H
+ (6 6X p ) 22 (646)
By comparison of Eqs. (644) and (646) the three H functions are
now expressible in the following way,
Ho kp2JfIxdB (647)
H1 = [ pk2 I xxBdB 2(p2ku+ k)I xxd ] (648)
2 = d + 5 fI6dJ
2
mk [(pku ixxdB]
8 2 4%/55
+ k3J B 2d + (pku + ) JIxxda. (649)
65 Summary of the Mass Shell Solution
All of the preceding description of the eight scalar functions, F,
G Gl, G2, H H2 and I has established the transverse part of the
vertex function in any region which is typified by the satisfaction of
two constraints. One constraint is that the electron propagator be well
represented by
_ m
where m is the experimental mass of the electron. This means that the
2 2
function A(p2) or A(p2) must be essentially constant and equal to the
experimental mass. The second constraint is that r must be dominated
by the contribution from y so that the transverse vertex is expressible
in a series expansion,
a A ()a2 A 3k
A = / + ~ L + 2 + ) L + ... (650)
rl 1 T 2 IT 3
where a is the fine structure constant. If it is possible to assume
that the first term in the expansion dominates and if the electron propa
gator is on the mass shell then it is expected that the functions F
through I will satisfy the eight differential equations which have been
written into the Main Program. These eight function solutions have been
collected together for ease of reference in Table 61.
By using y as the first order contribution to r a solution was
found to the vertex equation up to second order. In effect the solu
tion which was determined represents an identification of the L[ term in
Eq. (650). If this new improved version of r = y + L were put into
the vertex equation the vertex could be determined up to third order.
There is a practical limit to how far this process should be carried
toward self consistency. The coupling constant, is a very small
number; 2.32 x 103. Where it is true that r is dominated by y the
third order corrections would make little difference to the solution.
An even more important consideration; the determination of the vertex
function is only one step in a larger iterative procedure which seeks
to reevaluate the electron and photon propagators to self consistency.
TABLE 61, THE MASS SHELL FUNCTIONS
F = ap I ld
4 f j
I = pkfI d
Go 1 I 2dB  CK jI 3d
G2 = pk (,2Pku + 1 )[I 3d + xd ] pkf IXdd
Ho kp2m If d1
H1 [ pk2/fIX xdB 2(p2ku + k)JIxXdB]
H =  [Id + 1 d ] + (Pu + )[i d/+ I xdB
+ i 1ixx 2dO.
1 in (1 ) +1]
2 2
S(1 k2) In (1 x,)
2 p 2 p2
C = (1 u) I (1  ) In (1 X) 1
P m px m
14 x
2 2
2 p1 2 P2
CK C(1 L) in (1 2) + (I %) in (1 )
P1 m P2 m
TABLE 61 (Continued)
2u
13 = (m2u )[ 2 In
x
x = (m2 u) In (1
XB
x5
I5 =L2 u)
B
+ U
x
X
UB
1 1
UBXB X1
(1 )
u
"B
In (1 
u
L2(z) = in (1z)
2 Jo
I 1 In (1
6 Xg
u
I 1 i_ n
2
3B
2u
In (1
x
X
(1 )
UB
XB
(1 )
u
6
 B
UB
2
uB
+ In
XB
u
+ x 1
XB Xp
(1 x
x (1 , )
2 P
UB 1
7 l
XB (1 A)
UB
XB =[p + ( 1 k]2
uB = m B2)k2
a
a = fine structure constant.
x
(1 )
xI
In (1 )
u
2
7]
x
X)
uB
x
Xp
x
where
2
us
x2
3
XB
This project involved an initial approximation for the photon propagator,
from that a calculation of the electron propagator, from that a calcula
tion of the vertex. Future plans for the project looked toward using
the information gained by this work to calculate the photon propagator.
From this new photon propagator a new electron propagator could be cal
culated. From these two a new vertex. Each cycle of calculation should
contribute a finer resolution of the exact solutions. There would be no
point in refining the vertex function much beyond the level to which
the electron propagator is known. The functional form of the electron
propagator is accurate to about 0.1% of the value of the function.
Where y is dominant, corrections to order (a )2 in Fr would be expected
to amount to less than a 0.1% difference.
A direct consequence of solving the vertex equation only up to the
L L1 term will be that the eighth equation, the coefficient of
Syxvp k p will not be solved. It happens that the righthand side
of the eighth equation is second order in Therefore to this level
of solution the eighth equation should equal zero on the right.hand side.
For this reason equation eight will not be used as a criterionin assess
ing the correctness of the solution.
In the next chapter a test of the viability of the mass shell
solution is made, and a description of the programming methods necessary
to enact it is given.
CHAPTER VII
VERIFICATION OF THE MASS SHELL SOLUTION
71 The Mass Shell Program
Chapter VI established the functional forms of the eight scalar
vertex functions. There is no single absolutely superior way to compile
these functions which were summarized in Table 61. The Mass Shell
Programthe Fortran Programwhich computes the values of the functions
at given points, assumed many strategies. The decision of how to com
pute the functions was influenced by concern for ease of assembly, the
demand for machine time, and the inherent error in each method. Three
basic categories of method were tested, and we became familiar with their
merits. These three categories are described in section 73.
Once the mass shell solution was computed by one of these methods,
it had to be interfaced with the Main Program by way of a matrix of
numbers. The data matrix contains the value of each of the eight scalar
functions and the nine possible partial derivatives of each function at
one or more points in the infinite plane of the variables p2 and u.
Also included as data are the simpler electron propagator functions A
and B (which were given in Eqs. (315) and (316)) and their derivatives.
The Main Program uses the data matrix to evaluate the left and right
hand sides (L.H.S. and R.H.S.) of each of the eight equations at each
point provided. The Main Program evaluates the relative error in each
equation at each point.
Relative Error of Equation i = (L.HS. R.H.S. of Equation i (71)
L.H.S. of Equation i
for i = ...8.
The relative error is reported as a percentage error in the output. An
optimal solution will have a minimal error over the entire plane defined
by p2 and u. The question of what constitutes an acceptable minimal
error invites an analysis of what are the possible sources of error which
contribute to the Main Program and the Mass Shell Program.
The relative error in the eight differential equations is the effect
of a number of contributory causes. There is inherent error in the data
matrix just because the Mass Shell Solution is an approximate solution
to the eight differential equations. There is inherent error in the
electron propagator functions A and B because they represent only approx
imate solutions to the electron equation. There are truncation errors
introduced by the numerical procedures used to perform integration
and differentiations. Finally there is roundoff errorthe inevitable
outcome of any calculation which is carried out to a fixed finite number
of figures. All of these sources of error had to be either maintained
below a preset, tolerable level or, where they could not be controlled,
they at least had to be understood well enough so that we could recognize
when they were contributing to a significant loss of information. The
next section will consider what were the sources of error in the Main
Program and how these were controlled. The following section will discuss
the three principal methods used to evaluate the vertex functions, F,
G Gi, G2, H HI, H2 and I, and how each method affected the overall
level of uncertainty.
72 Contributions to Error in the Main Program
For the purpose of analyzing the kind of error that is being
generated within the Main Program alone, we will assume for the time being
that the data matrix of the eight scalar functions, their derivatives, the
two scalar electron functions and their derivatives, could be supplied to
the Main Program without error. If this could be done what would be the
remaining sources of error and how would they propagate through the
Main Program?
One of the earliest versions of the Main Program defined the partial
derivatives of the twentyeight TAa components by taking firstorder
differences. (This tensor appears in the R.H.S. and it was defined in
Eqs. (4 95) through (4116).) This saved writing the analytic expres
sions for the partial derivatives of the twentyeight components of T a
which, in turn, had to be expressed in terms of the partial derivatives of
twentyeight components of F v which, in turn, had to be expressed in
terms of the partial derivatives of the twentyeight components of Z
and W which, in turn, were at last expressed in terms of the partial
derivatives of the eight vertex functions and two electron functions
provided by the data matrix. (See Figure 51 to be reminded of the
hierarchy of tensors which define the R.H.S.) The numerical procedure
for evaluating the partial derivatives of the twentyeight components
of TAa was quickly abandoned because the Tha components are very often
extremely large and slowly changing, and roundoff error eliminated most
useful details. It is a characteristic of the R.H.S. that, not only
are the T h components large, but that the R.H.S. is very much smaller
than its components. The R.H.S. is very sensitive to roundoff error
because it involves the difference of many large and almost equal terms.
In order to effectively calculate the R.H.S. analytic expressions of the
partial derivatives of the components of the T", F Z7 and W
tensors had to be supplied and the data matrix had to provide the func
tions and their derivatives to more than six figures.
Once the numerical means of calculating the derivatives of the
components of TXa was discarded, the only remaining source of error in
the Main Program is the accumulative effects of roundoff error. In
hexadecimal based arithmetic the roundoff error for a single operation
will be proportional to the 16t+, where t is the number of digits in
the mantissa when the number is expressed as a fraction times the base
raised the power of the exponent. For a calculation of standard pre
cision, the number of figures in the mantissa is only 8. This is not
enough to provide an accurate evaluation of the R.H.S. It was neces
sary to increase this precision to 16 significant figures. This meant
15
the roundoff error for a single operation was proportional to 161
a very small number. However as the hundreds of thousands of single
operations of multiplication and addition take place this error will
grow systematically through the program. As mentioned before, the
problem grows particularly severe on the R.H.S. of the vertex equation
when the differences of large and nearly equal quantities are taken.
It would be a huge task to draw a process graph to follow the propaga
tion of the approximate error throughout the program. Even if this was
done the projected error would be an upperbound with a large possible
deviation since the error would be assumed maximal at each individual
step. As an alternative measure to estimate the intrinsic roundoff error,
both the Main Program and the Mass Shell Program were converted to qua
druple precision. The stability of the program results under the change
from double precision to quadruple precision was taken to indicate that
the data matrix was being supplied with numbers of sufficient accuracy
that roundoff error was not a matter of principal concern. In this way
the Main Program was established to be working satisfactorily. The
really significant problem of error management lay in the design of the
Mass Shell Program.
73 The Mass Shell Program
The Mass Shell Program takes the functional form of the F, G Gi,
G2, Ho, H1, H2 and I functions and computes the value of the functions
and all the derivatives at selected points. The vertex functions are
parametrized by k2 and are dependent on the variables p2 and u, where
p k" p ka.
u P= a a (72)
pk (p2)(k2)1 "
Due to the indefinite metric the domain of p2 is from m to + m.
p2 = pp
= gpVp (73)
Therefore both (p2) and (k2)+ can take on imaginary values. This leaves
a dilemma in the interpretation of the meanings of the symbols p and k
used in the definition of u and in the definitions of the functions in

Full Text 
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A SOLUTION TO THE SCHWINGERDYSON EQUATIONS OF QUANTUM ELECTRODYNAMICS BY JOAN F. CARTIER A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DIVERSITY OF FLORIDA 1983
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ACKNOWLEDGMENTS I would like to express my sincere thanks to all of those who have helped me. I would especially like to thank Charles Reid for his constant support and interest, Arthur Broyles for his thoughtful criticism without which no progress could have been made, H. S. Green for his invaluable suggestions and guidance, Ruben Mendez Placito for being a strong ally and good friend, and Robert Coldwell for his resourceful presence which saw me through many computational crises with wisdom, insight and humor. This work was substantially assisted by the Northeast Regional Data Center which donated a MUSIC account and valuable technical assistance, and the Division of Sponsored Research which provided a Research Assistant Fellowship Award. n
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TABLE OF CONTENTS Page ACKNOWLEDGMENTS n ABSTRACT V CHAPTER , I INTRODUCTION ' II THE GENERAL PROCEDURE FOR THE SOLUTION TO THE SCHWINGERDYSON EQUATIONS OF QUANTUM ELECTRODYNAMICS 9 21 The SchwingerDyson Equations 9 22 Initial Approximations I4 23 Approximating the Vertex Equation with Green's Generalized Ward Identity y 24 Converting the Integral Equations into Differential Equations " III THE ELECTRON PROPAGATOR EQUATION 27 IV THE VERTEX EQUATION 33 41 Introduction 33 42 The Main Computer Program u 43 The LeftHand Sides of the Eight Equations jW 44 The RightHand Sides of the Eight Equations .... 44 V CHECKING THE ALGEBRA 57 VI THE MASS SHELL SOLUTION 75 61 An Approximation Solution 75 62 The F and I Functions Â«} 63 The G , G, and G 2 Functions Â« 64 The H , H, and W ? Functions 84 65 Summary of the Mass Shell Solution 86 VII VERIFICATION OF THE MASS SHELL SOLUTION 91 71 The Mass Shell Program 91 72 Contributions to Error in the Main Program w 73 The Mass Shell Program 95 74 Summary of Results of Mass Shell Program 1U4 ii i
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TABLE OF CONTENTS (Continued) CHAPTER Page VIII EXTENDING THE MASS SHELL SOLUTION 108 81 A Scaling Symmetry 108 82 The Large p Region 112 83 The Large k Region 119 IX THE CONCLUSION 133 APPENDIXES A DIRAC GAMMA MATRICES: DEFINITIONS AND PRODUCT RULES . . 139 B DERIVATION OF GREEN'S MASS SHELL SOLUTION 143 C FORTRAN PROGRAMS 157 D CALCULATION OF FOUR DIMENSIONAL INTEGRALS 190 n 95 REFERENCES BIOGRAPHICAL SKETCH 197
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Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy A SOLUTION TO THE SCHWINGERDYSON EQUATIONS OF QUANTUM ELECTRODYNAMICS By Joan F. Cartier Chairman: Charles E. Reid Major Department: Chemistry A nonperturbative solution to the unformalized SchwingerDyson equations of Quantum Electrodynamics was obtained by using combined analytical and numerical techniques. The photon propagator is approximated by its form near the mass shell. The vertex equation is cut off at the second order in the coupling constant and the remainder is approximated by H. S. Green's generalization of the Ward Identity for higher order diagrams. Under these approximations a functional form for the electron propagator, S(p), and the vertex function, r y (p,p+K) was obtained for all magnitudes of momenta. Both functions were found to be finite. No infinities were subtracted to obtain the solutions.
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CHAPTER I INTRODUCTION A physical theory must furnish an accurate description of observable quantities. Beyond this supreme requirement a physical theory is judged by its simplicity and ease of comprehension. Another important measure of a theory is how widely the approach could be applied. There is a sense of a theory being more truthful if it is applicable to more than a single subject. If the same approach can be applied to several diverse problems then the problems lose their diversity. To be a really successful theory it needs to be one that unifies a broad spectrum of ideas. Quantum Electrodynamics (Q.E.D.) has been a successful theory to the extent that it enjoys a number of these characteristics. One of its attributes is that the theory had its beginnings in the bringing together of several separate fields of study. This unification formed Relativists Quantum theory and then as a natural extension the study of dynamics of the interaction of particles with electromagnetic radiation evolved. Quantum Theory (Q.T.) was developed in the 1930's from the ideas of wave mechanics. It accomplished the explanation of atomic structure, molecular structure, the structure of solids and the synmetry and energy bands of crystals. However, Q.T. was insulated from the effects of special relativity in all these endeavors because the ordinary effects of chemistry arise from interactions with only the outermost part of 1
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the atom. The calculation of reaction rates, surface potentials or scattering cross sections requires a detailed description of only the outermost electrons which require relatively low energies to excite. Relativistic corrections represent a very small fraction of these calculations. Successful as Q.T. was at describing in terms of fundamental principles the workings of chemistry, it represented a break in the general pattern of explaining physical phenomena because it had not incorporated the principles of special relativity. Quantum Theory needed to be formulated in a Lorentz covariant form. The relativistic descendant of Schrodinger's equation is the Klein Gordon equation. This equation allows for the relativistic variation of mass with the velocity of the particle. However such effects as the fine structure of the hydrogen atom were only partly accounted for by the relativistic Q.T. of Schrodinger. It took Dirac's work on the fully relativistic theory of particles with spin 1/2 to complete the analysis of the fine structure and to explain the Zeeman effect. A complete description of the relativistic electron would have to include the interaction of the electron with its own electromagnetic 1 2 field. Feynman and Schwinger formed calculational methods in Q.E.D. 3 in two separate mathematical languages. Dyson demonstrated these languages were equivalent. As early as 1930, Waller, Weisskopf and Oppenheimer had calculated the self energy of the electron and found it to be disappointingly, quadratically divergent. Later Weisskopf established that the divergencies were only logarithmic. These inexplicable divergencies that occurred in the calculation of measurable
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quantities (though at the time such measurements were not practically feasible), held the theory in a quandary for quite a while. 5 Real impetus was given to Q.E.D. when Lamb and Retherford succeeded in measuring the splitting between the 2Sp and the 2?j energy levels of the hydrogen atom. Acting on a suggestion of Lamb's, Bethe circumvented the divergence problem by simply cutting off the range of integration over the divergent integrals. Surprisingly, Bethe came up with a wery close calculation of the "Lamb shift," as it has come to be known. Other attempts were made at trying to eliminate the divergencies in 29 a more rigorous manner. Schwinger and Tomonaga developed the first Lorentz covariant scheme designed to make the elimination of the divergencies more acceptable. But, by whatever the justification, calculations in Q.E.D. have enjoyed remarkable practical success. Because of the small coupling constant for the electromagnetic interaction, perturbation techniques have resulted in impressive calculations of experimental values of the Lamb shift, separation of the ground state doublet of positronium, the hyperfine structure of the hydrogen atom, the line shape of emitted radiation in atomic transitions and other relativistic but measurable phenomena. The road which connects Quantum Theory with special relativity leads to the theory of Quantum Electrodynamics. The road continues on today to connect Q.E.D. with further frontiers. Quantum Electrodynamics has become a model for other field theories. An appreciation of the special role of Q.E.D. is one way to provide for a better understanding of the mathematical structure of field theories in general. The recent success of the NonAbelian gauge theories in unifying the
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electromagnetic interaction with the weak interaction further motivates efforts to understand the ambiguities of Q.E.D. The current theories of electroweak and strong interactions have been shown to have the same underlying structure based on assumptions about global and local invariance. The unification has its dark side since as a part of the bargain comes the problem of the divergencies It is apparent that Q.E.D. sits as a bridge between many welltravelled roads of thought. This is why it is particularly frustrating that the theory should be flawed by unnatural infinities which rear up in the evaluation of physically observable quantities. If it were a complete and satisfactory theory they should never have occurred. There are four different kinds of divergencies. The following classification of them has been paraphrased from the text by Janch and Q Rohrlich, (a) divergencies associated with the description of the vacuum (b) infrared divergencies (c) divergencies associated with closed loops (d) serious divergencies. The type (a) occurs only in the form of a phase factor multiplying a particle amplitude. Since this does alter the probability density it can have no observable effect. It is possible to ignore type (a) divergencies. Type (b) is an artifact of the mathematical procedure. An analysis of the problem has shown it can be eliminated by an improved mathematical treatment. Type (c) is associated with the photon self energy. This type of divergence has been handled by invoking the invariance of the theory under gauge transformations. Type (d) , the
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"Serious" divergencies, comes up in the calculation of the vacuum polarization, electron self energy and the vertex function. These serious divergencies are the object of this whole discussion. The redressing of these infinite quantities is called Renormalization. In this process the amplitudes are expanded in a power series of the coupling constant. Many terms in the series may contain divergent integrals. It is possible to remove these infinities in a relativisticly covariant way by redefining the parameters of mass and charge. The finite terms which remain in the series are taken to be the renormalized expressions for the amplitudes. Renormalization is neither simple nor straightforward. It brings in a new and complicated set of rules which are not properly anticipated by the initial understanding of the problem. Renormalization is an after the fact reaction to something unforeseen and undesirable. This leaves two possible interpretations of the problem: (1) There is something wrong with the basic theory of Q.E.D. (2) Some mathematical procedure has been inappropriately applied. A new method of calculation must be tried. It is hard to argue that there is something wrong with the foundations of the theory. Q.E.D. has enjoyed a huge practical success in calculating various physical measurements. For example, from renormalized perturbation calculations of the anomolous magnetic moment of the electron the following results for the inverse of the fine structure 9 constant were derived: 137.03549(21).
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Â• TO Presently the best experimental value is ., = 137.03604(11). a exptl It would seem that there could be little wrong with a theory that makes such accurate predictions. The unqualified success of Q.E.D. in calculating (by whatever means) precise experimental results, and the success Q.E.D. has enjoyed in linking quantum theory to special relativity, and its position central to general unifying theory of forces are forceful evidence that the fundamentals of Q.E.D. are sound. The implication therefore must be that some mathematical procedure has been inappropriately applied. The response to this implication has been a long search by many persons for a selfconsistent and finite 1? approach to Q.E.D. In 1954, GellMann and Low sought to demonstrate that the renormalizing constants which relate the bare mass and charge are infinite. They found they couldnotrule out the possibility of infinite coupling constants but they isolated a necessary condition for the 13 vacuum polarization to be finite. Johnson, Baker and Willey took up the interesting problem in a long series of papers. Under a certain set of approximations they solved for the renormalizing constants and concluded that in order for the self energy of the electron to be finite, the bare mass of the electron must be zero and a special gauge must be used. These results spawned interest in a number of people for finding asymptotic but nonperturbative evaluations of the self energy of the electron, the vertex function and the vacuum polarization. Chapter III will describe the paper in which the author was involved in
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some of the effort of repeating, in a new way, the determination of the electron self energy. The work reaffirmed the results of Johnson, Baker and Willey by using an inventive nonperturbative approach of H. S. Green's with less restrictive approximations than had been used before. This work extended the results of Johnson, Baker and Willey by finding a complete solution instead of an asymptotic one. These three basic functions, the electron propagator, the vertex and the photon propagator are identified by their role in the threelinked nonlinear integral equations known as the SchwingerDyson equations. It is of great interest if a method of solution could be found which would yield no divergent function forms for the three basic functions. Encouraged by the success of Green's method in extracting a finite solution for the electron propagator, we decided to see if the same nonperturbative procedure of H. S. Green would work to provide a vertex function. The description of the project is arranged in the following way. Chapter II is a general description of the overall method of taking the nonlinear linked integral equations and transforming them into a set of differential equations. An outline is given of the proposed method for solving the equations for the electron propagator and the vertex. Chapter III provides a description of the electron propagator solution. Chapter IV provides a preparation of the differential equations for the vertex function. The tensor equation is broken down into eight linked thirdorder differential equations for the eight scalar functions which comprise the transverse part of the vertex. In Chapter V an algebraic technique is described which makes the product of the complicated gamma
PAGE 13
matrix function easier to obtain. In Chapter VI, Green's method is used to obtain an approximate solution to the eight linked differential equations. In Chapter VII this solution is tested and its range of applicability is defined. In Chapter VIII some alterations are made to the solution which extends its viability. The last chapter summarizes the solutions which were obtained to the SchwingerDyson equations. It is felt that the work described here will contribute to the idea that Q.E.D. is a complete and satisfactory theory; a theory which is a faithful rendition of experimental results, a theory which lends clarity by unifying several fields of study, and a theory which is unambiguously expressed.
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CHAPTER II THE GENERAL PROCEDURE FOR THE SOLUTION OF THE SCHWINGERDYSON EQUATIONS OF QUANTUM ELECTRODYNAMICS 21 SchwingerDyson Equations Quantum Electrodynamics is a description of the interaction of light with matter. A classical charged particle generates an electromagnetic field around it with which it can self interact. Calculations of this interaction have traditionally led to infinities. To demonstrate how these infinities arise in Q.E.D., it is convenient to express the theory in terms of the propagators of the particles. The amplitude of an electron at some point in spacetime is related to its amplitude at a different point in spacetime by its Feynman propagator or Green's function. The Green's function is determined by the equation of motion that governs the wave function. For the free electron the differential equation is (i? m)Y = (21) where 7 is a shorthand notation for y y ^T and y U i5 a Dirac 9amma matrix. + The photon wave equation is DA^0 (22) where Q is the D'ALambertian, Â— 9 ~ 9 Â— ' The propagator for the a x a x y ^See Appendix A for representations of the y v .
PAGE 15
10 electron satisfies a corresponding equation, (i>m )S (x',x) = 6 4 (x'x). (23) The solution to this equation in momentum space determines the Fourier transform of the free electron propagator to be S (p) = J. (24) Similarly the photon propagator satisfies DD (xx') = i6 4 (xx'), (25) so that the Fourier transform of the photon progagator is D(q 2 ) =4 Â• q (26) Thus the propagators for the free particles are explicitly known. When it is allowed that a source term may be present, the interaction between the electron and photon will lead to nonhomogeneous differential equations. The exact electron and photon propagators are then determined by these nonhomogeneous differential equations but the solutions are not explicitly known. The equations which determine the Fourier transform of the photon and electron propagators are an open set of interlocked integral equations. This hierarchy of integral equations was formulated by Dyson 3 and Schwinger. 15 Using the notational practices of Bjorken and Drell, 17 these integral equations appear as: T The 4 vectors are denoted by a bar over the symbol and matrices are distinguished by a bar under the symbol.
PAGE 16
11 (a) S(p) = S (p) + S (p)E(p)S(p) (27) o vr/ o or equivalently S'Vp) "^(p) E(p) (28) where ie 2 f I(P) =Â—^4 I y (PÂ»q)i(q)D yv (pq)Y V d 4 q. (29) (2tt) J Â») v* 2 * v 2) + %y '" aB t k2) v k2 > < 2 10 ' where H (k 2 =^\ Tr [ Y a S(q)r 6 (q,q+^)S(q + R)] d 4 q. (211) (c) ^(p.qJlX + A^P.q) (212) where A u (p,q) = ie 2 /D VT1 (k 2 )r v (p,pk)S(pk) 6 (Zir)' r y (pR,qR)S(qk)Â£ (qk,q)^ + ... J... J r 6 (p,pR)s(pR) ... i y ... d 4 k n ... d 4 kl (2Tr)4n + ... . (213) The zero subscript follows all bare quantities, that is, those functions or constants which are associated with the free particles. The r y (p,q) is called the vertex function. It represents the sum of all nodeless diagrams which are connected to two external electron lines and one
PAGE 17
12 external photon line. If a closed expressed for the vertex function could be formed, then a complete knowledge of the interaction propagators would depend only on a solution to the three linked nonlinear equations; the electron propagator equation, Eq. (a), the photon propagator equation, Eq. (b), and the vertex equation, Eq. (c). However, Eq. (c) gives the vertex equation in terms of an infinite series. Therefore the SchwingerDyson equations, though simple in form, possess no simple solution. One method for attempting a solution is to assume the interaction propagators differ from the bare propagators by only a small variation. Perturbation theory might then lead to at least an asymptotically. converging series. The first iteration would replace the propagators .on the righthand side of Eqs. (a), (b) and (c) by the bare propagators. When this is done the equation for the electron propagator function Zjp) becomes 2 ,\ 3 o I U I V "UV jt. 2(p) = Â— t r /^m , y T dk e o fW+K* 1 ,4 4 d^k. (214) ~7^.V[(p+k) 2 %] k 2 By a power counting of k it can be seen that for k>~ the integrand behaves like k 3 dk/k 4 which would yield a logarithmic divergence. A more careful consideration of this integral would take into account the f hyperbolic metric. The four dimensional integral can be performed by transforming into hyperspherical coordinates but the logarithmic divergence persists. A corresponding divergence appears in the vertex function + See Appendix D on the subject of calculation of fourdimensional integrals.
PAGE 18
13 and in the photon propagator function. These are of the type (d) category of divergences of Janch and Rohrlich 8 the socalled "serious divergences." These divergences can be handled by any of a number of methods grouped under the title of Renormalization methods. The first proof of renormalizability was provided by Dyson, Salam and Ward. Basically the idea is that although absolute calculations cannot be made with the formalism as it stands, still relative calculations can be made. The parameters of the theory such as mass and charge are redefined to absorb the terms which contribute the infinite quantities. The renormalized mass and charge are taken to have the experimentally observed values. Renormalization, as such, was a huge but puzzling step forward. It provided the tools to make impressive calculations of relativistic corrections in the spectrum of the atom but the meaning and the value of the bare quantities remained a mystery. The very fact that renormalization works is an indication that there ought to be a mathematically consistent way to solve the ^hierarchy without encountering undefinable quantities. The practical success of renormalized theory argues against a fundamental flaw in the theory. For these reasons the following scheme was developed to solve the equations without recourse to renormalization methods. The procedure which has made possible an unformalized solution of the SwingerDyson equations has three elements to it. The first is the assumption of a reasonable starting point in terms of an approximate form for the photon propagator. The second is the generalization
PAGE 19
14 of the Ward Identity which provides a neat formula for systematically truncating the hierarchy of the vertex equation at increasing levels of accuracy. The last element consists of transforming the electron and vertex integral equations into linked differential equations. The differential equations are more tractable than the integral equations to numerical and analytic approaches to the solution. A description of these three elements is the object of the following three sections. 22 Initial Approximations The question which must be raised first is whether it is best to solve first for the electron propagator, S(p), or first for the photon propagator, D (k 2 ). This can be decided by a consideration of which of the two is easiest to approximate. Since Lorentz 30 first offered the idea, as early as 1909, it has been a popular view to consider the mass of the electron as mechanical in origin. This was based on the observation that the electron, when accelerated by interaction with the electromagnetic field, behaves as though it were gaining mass. It is appealing to imagine that the bare mass of the electron may be zero and the self interaction with its own electromagnetic field is what "dresses" the electron in its apparent mass. If the bare mass is zero or very small 1 " the self energy of the electron will be of the.order of magnitude of its rest energy. In units of inverse time, the rest energy of the electron is 1.2x 10 14 megacycles. On the other hand, renormalized perturbation type calculations of the Lamb shift show the vacuum polarization contributes only about 27 megacycles. The disparity between these + 0bviously the assumption the bare mass of the electron is zero is predicated on the additional requirement that there be no other forms of interaction which contribute to the mass of the electron.
PAGE 20
15 two numbers argues that the self energy of the electron may be more important than the vacuum polarization of the photon. Therefore, it is reasonable to start with an approximate form of the photon propagator and solve first equation (a) for the electron propagator. The electron propagator is a function of the scalar p\ (Y y P U )Â» and as such its most general form is expressible in terms of two functions, one the coefficient of unity and the other the coefficient of p 1 . Because of the simplicity of Eq. (28) relative to Eq. (27), it is of interest to focus on the form of S(p) , S _1 (p) A(p 2 ) B(p 2 )p\ (215) The vertex function is a matrix function of the Dirac gamma matrices and the four vectors, p and I, the electron and photon momentum respectively. Its most general form is rVR,p) =^f +y\ + P ^4 +J f G 2 , H , H, + ia A Â°Y 2 e Y Y 4> a P 3pk + k A J + k X \ K + k A K L + ia a3 P a kgk A M (216) 2 2 where F, G , G, , G 2 , H Q , H ] , H 2 and I are functions of p , k and u. The u is the angle variable defined as P k a a pi ? i Â• Ip z I* kT (2u:
PAGE 21
16 They are coefficients of that part of r which is transverse to k . 2 2 Similarly J, K, L and M are functions of p , k and u. They are coefficients of the longitudinal part of r . The general form of the photon propagator is known from its relativ2 istic covariant properties. Since D (k ) is a second rank tensor which 2 depends only on the fourvector R, the photon momentum, D (k ) can be taken to have two components. One component is proportional to k k and the other is proportional to g . Thus D (k 2 ) = d,(k 2 ) g /k 2 + d (k 2 ) k k /k 4 . (218) yv 1 uv 2 u v In addition to knowing the general forms of the three functions, the electron propagator, the vertex and the photon propagator, the limiting form of the functions is known on the mass shell. In the 2 2 2 limit as the momenta approach the mass shell where p Â•* m , k Â•* 0, the functions approach the following limits, S _1 (p) Z"Vm) (219) r X (p+k\p) Z" 1 Y X (220 D (k 2 ) +Z, ^ . (221) yv 3 k 2
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17 The constants of proportionality to be determined by the theory are Z ? and l y First order perturbation theory finds them to be zero or infinite. They are called renormalization constants because the standard procedure is to use them to redefine the series expansions so that the divergences are absorbed. The initial step of solving for the electron propagator requires a reasonable first estimate of the dressed function D^ v and r . The approximation that was used for the photon propagator was D (k 2 ) Z,[% (1b) %. (222) yv 3 k 2 k 4 13 14 This choice was motivated by the results of earlier investigations. Studies of the asymptotic forms of the propagators which were not inconsistent with finite renormalization constants found the photon propagator to be in the Landau gauge for at least the lowest order in perturbation theory. The Landau gauge is a special case (b=0) of the general form given in Eq. (222). The approximation that was used for the vertex function was obtained from the Ward Identity which related the vertex to the inverse electron propagator, k r^p+lc.p) S" 1 (p+R) S _1 (p). (223) In the limit as the photon momentum, R, grows small this can be expressed in a differential form, y
PAGE 23
18 As illustrated by Eq. (214), when the bare form of the functions is used to generate a first approximation to the dressed function, it is found that "serious" divergences occur. It is desirable then to avoid the substitution of bare quantities for dressed quantities. Therefore if the general form of the electron propagator were used in the differential Ward Identity.it would be possible to generate a general form for the vertex at vanishing R, I A (p,p) =^[A(p 2 ) +0B(P 2 )]. (225) A To solve the electron propagator equation a knowledge of the vertex function r A (p+k,p) is needed. Equation (225) represents a very good approximation to the vertex function where k  is small. In considering the electron propagator Eqs. (27) and (29), it can be seen that this approximation is at its best when, as the argument of D^ v vanishes, the integrand is at a maximum. This coincidence of the region of best approximation with the region of most importance argues that rVk.p) ^[A(P 2 ) +0B(P 2 )] (226) is a reasonable first approximation for the vertex. By using the approximations in equations (222) and (226) for the photon propagator and vertex, a solution to the electron propagator equation can be found. A description of the method of solution and its conclusions are found in the next chapter. Once a solution to the electron equation had yielded a functional form for the electron propagator functions A(p 2 ) and B(p 2 ), a solution to the vertex equation was sought. The next section describes the method by which the vertex equation
PAGE 24
19 was approximated so that it no longer depended on the higher levels of the hierarchy. 23 Approximating the Vertex Equation with Green's Generalized Ward Identity A construction of the infinite hierarchy of equations can be selected so that a highly repetitive pattern emerges which can be used to systematically separate the equations at any chosen level of complexity. Instead of the infinite series which appears in Eq. (c) the vertex equation can be written as r y =l!! + A(p,q) o where ie 2 f A v (p,q> Â—A Y 3 D aB (k 2 )S(pR)r y (pR,pqR) (2nr J S(pqR)r a (pqk,pq)d 4 k (21) ie 2 f + _!o 3d (k 2 )S(pR)E ya (pTi,q,pq)d 4 k. (227) forn 4 J a P A new function has appeared, E_ ya , a function of three external momenta, which corresponds to four independent points in spacetime; hence E ya is a fourpoint function. Just as the twopoint functions ^(p) and D (k 2 ) were related to the threepoint function, r (p,p+R), and as the threepoint vertex, r X , was related to the fourpoint function E yct (pR,q,ql<), so the fourpoint function, E ya , can be related to a fivepoint function and so forth. The greater the number of momenta involved the lower the contribution from such a cross section should be.
PAGE 25
20 Each new npoint function is created by pulling a dressed photon and dressed electron propagator out of an n1 point diagram, creating in this way a new vertex or point. In Table 21 the equations for the twopoint electron propagator, the threepoint vertex, and the fourpoint E yv are given to show the pattern that is emerging. The repeated structure in the relations that link an n point diagram to an n+1 point diagram suggests there should be a generalization of the wellknown Ward Identity which states q A y (pq,p) = Efpq) + E(p). < 2 28) A generalization of this for the fourand fivepoint diagrams would be q E yv (p+q,p,R) = A V (p+q,pR,K) A V (p,pk\R) (229) q F yva (p+q,p,k,Â£) = E^p+q.k^pq .*.Â£)E v (p,*4p.t.I). (230) I o This generalization of the Ward Identity was first proven by H. S. Green in 1953. These identities exactly define the longitudinal components of the n+1 point diagram in terms of the difference of two npoint diagrams. These identities can be used to truncate the hierarchy of equations by approximating any given diagram by its longitudinal component. In this manner the vertex function could be solved for if E y was approximated through an application of Eq. (229). In conclusion, it has been proposed the electron propagator can be found by approximating the threepoint vertex by Ward's Identity.
PAGE 26
21 TABLE 21 . RELATIONSHIP BETWEEN NPOINT DIAGRAMS Two point diagram S _1 (p) = S'Vp) E(p) where S~ = $ m I =^\ J^(p,pq)S(pq)D uv t\h% Three point diagram Â£ y (p,pq) =lj + A y (p,pq) where r = y " 2 f A y ^4k a (p^qÂ»pq) s rp^)D av (k 2 )Y v d 4 k Four point diagram ya (pK,q,pq) = D ya + E ya (pK,q,pq) where D^ = r A (p^,pq^)S(pq^)r v (pqk,pq) .2 E ya = _^0 O ya3 (pR^R,q,pq)S(p^)D (Â£ 2 ) Y V (2u) 4 " Five point diagram O^pKa.K.q.H) =Oo ae + F yaB (P^^^q>pq) where(J) (2ir) e 2 ^ua34> MO (pkÂ£s,Â£,k,q,pq)S(ps)D yv (s ) Y V d S tt) J
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22 y\ idler* KT* y^ + 10Â•\ IdQ_ lOD
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23 Although the Ward Identity provides an exact relationship for the longitudinal component of the vertex, it only yields the limit of the transverse part for small photon momentum. To obtain the transverse part for large values of the photon momentum a solution to the vertex equation must be found. This can be done by using the firstorder generalized Ward Identity to determine an approximation for the fourpoint diagram. This process has two wery important aspects to it. One is that at all levels in the solution to the SchwingerDyson equations the Ward Identity is exactly preserved. The second is never are bare functions substituted for dressed functionsa procedure which has always been associated with divergences. 24 Converting the Integral Equations into Differential Equations Another essential step in the whole procedure of reducing these equations to a tractable form without giving in to the drastic approximations, which have characterized earlier attempts at a solution, was the conversion of the integral equations into differential equations. This method was developed by H. S. Green in connection with the BetheSaltpeter equation. It was first used for a study of the Schwinger20 Dyson equations for the electron by Bose and Biswas. Whenever the photon propagator appears under the integral it can be used to eliminate the integration. The photon propagator was taken to have the form, D (k 2 ) =Z.[% + (1bAS. (231) yv 3 k 2 k 4
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24 ?cause ik 3k u In (k 2 'JJV 2 4 k k vu 4 ' r k the photon propagator can be put into the alternate form, 9 D,,Â„(k 6 ) = Z. yv v u k _J ;232' (233; Notice also that the D' Alemberian operator, _9 3_ k ' 8k u 3k^ ' (234; /ill operate on W to yield a delta function, _Â• = 4TT 2 i6(k 2 : K r (235: This delta function can be used to trivially perform the integration over integrals of the form, iV) = ie 2 / Dijv ((qk) 2 )Y y F Av (q 2 ) d 4 q. [236) Substituting Eq. (233) into Eq. (236) results in I A (k 2 ) = ie 2 0. *(lb>. (k q ) 3(k Q ) v V V U u ln((k.q) ) Â£ (b+1) ^ 2 (kq)t PrAV/ 2\ .4 . Y F (q )d q [237) Apply the D' Alember tian ~V
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25 = 2e 2 (b+1 )rr 2 Y v F Av (k 2 ) + ie 2 ((1b) ^Â—h F X V)d 4 q. (238) J 3K v (kqT There is one remaining part under the integral. This can be removed by either of two ways. The first is operate on the equation with Y a _JL = y This yields the following thirdorder differential equation 3k a * k D k lV) =2e 2 /(b + l)v/V) + 4e 2 Tr 2 (lb) gf F X V). (2" 39 ) The second method is to identify the remaining integral as some function G X (k 2 ) such that G X (k 2 ) ifdb) JL % Â— U F^(q 2 )d 4 q. (240) Then the problem of solving Eq. (238) becomes the problem of solving the pair of equations, i D k I A (k 2 ) 2e 2 (b + l)Tr 2 Y /V) + e 2 G X (k 2 ) ?G X (k 2 ) = 4, 2 (lb) JF Av (k 2 ). (24D v By either of these routes a solution to an integral equation of the form (236) is equivalent to a solution to the differential Eq. (239) or the pair of differential Eqs. (241) when the appropriate boundary conditions are satisfied.
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26 To briefly summarize this chapter, a scheme has been laid out by which the first two unformalized SchwingerDyson equations could be solved. The first step of the scheme involved solving for the electron propagator. To solve the electron equation required using Eq. (222) and (226) to initially approximate the vertex function and the photon function. The second stage of the scheme involved solving the vertex equation to determine the form of the transverse part of the vertex for other than very small photon momenta. To make this possible the higher order terms were also approximated using a generalization of the Ward Identity. Finally the integral equations relating these functions were to be converted into differential equations to make an analytic solution easier. In the next chapter, Chapter III, a survey of the work that was done on the electron equation is given. In Chapter IV, an introduction to the details of solving the vertex equation is given.
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CHAPTER III THE ELECTRON PROPAGATOR EQUATION Recall the form of the electron propagator previously given in Eq. (27) where S , the Fourier transform of the bare propagatator, is given by s = JÂ— . (3D o $ nv The m is the bare mass of the electron, that is, the mass the electron would have if the electromagnetic interaction could be turned off. The _E(p) equation was given as . 2 ie 2 f I(P) " Â— ^4 ^(p,q)S(q)D yv (pq)Y V d 4 q. (32) (2tt) J On the basis of rationalizations detailed in section 22 two approximations are invoked to sever the connection of the electron equation from the vertex equation and the photon propagator equation. These were. and D (pq) (pq) z (pq) 4 h (3 3 Â» r i (p*i,p)^s1 (p). < 3 4 ) A When these substitutions are made the electron propagator Eq. (31) becomes 27
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28 . ie 2 fas'fp) [ ^> (1b)y V v 4 v (pq) Z (pq) d qy Â• (35) Because the electron propagator is a scalar function of only the electron momentum, its most general form is given by S _1 (p) = A(p 2 ) + 0B(p 2 ). ( 3 ' 6) So that 3p. ^ S" T (p)= 2p y A'(p 2 ) + y y B(p 2 ) + 2p u B'(p 2 ) (37) where A . (p Z,.Â«l4l and B'(P 2 )=^V dp" 1 d P (38) Substituting Eqs. (32), (37) and (38) into the electron propagator Eq. (36) yields, ie A( P 2 ) + ^B(p 2 ) = i m Q + ^4 ((2p y A'(p 2 ) (2tt) J K. + Y y B(p 2 ) +2p^B'(p 2 ))(A(q 2 ) HB(q 2 )) fc (Pq) (P " q) } d qy Â• (39) The equation was converted into a differential equation by application of the D' Alembertian operator, as described in section 24.
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29 For the purpose of performing all of the needed matrix multiplications, a table of products of gamma matrices was prepared. This table appears in Appendix A. Equation (310) then separates into twolinearly independent differential equations, the coefficient equations for the unit matrix and p\ The unit matrix equation is 2 Â„., e 2 f(4b) AB , (2+b)p 2 ( B ' A_AJ_Ep _ J> (310) A"p^ + 2A' =^(^4^Â— 2 /^ Â„/
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30 A description of the solution and a simple analytic determination 21 of the asymptotic behavior of these functions was presented in the paper on the electron propagator equation. In that paper a different approach was taken. The integral equations were not converted into differential equations by the action of the D' Alembertian operator. Instead the variables of integration were converted to hyperspherical coordinates; a Wick rotation was performed; then the (pq) factor was expanded in terms of Gegenbaur polynomials, ^ n MThese polynomials have an orthogonality condition which was used to simplify the integrations. This useful procedure is illustrated in detail in Appendix D. The solutions to the differential equations had to be restricted to be particular solutions of the integral equations by the boundary 2 conditions. The boundary conditions for the four momentum, p , approaching the mass shell is known. There the electron propagator is proportional to the bare propagator. The asymptotic boundary conditions, where the magnitude of p 2 is undefinitely large, are not explicitly known. 14 This has been an object of study of a large nimber of papers. Interesting things can be determined about the asymptotic form of the solution when the differential equations are substituted into the integral equations. This was carried out in the third section of that same paper. There it was found that the functions A and B, of the electron propagator, must approach constants for large p . In order that they could approach constants the gauge parameter b had to be set to zero. With b equal to zero, the photon propagator was set in the Landau gauge. At the same time it was demonstrated that a finite solution required the bare mass to be zero.
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31 An expression for the function A was fitted to the tabulated numerical solution over a finite range of momenta. The accuracy of the fit was around 0.1 percent. To an even greater accuracy the function B was observed to equal the constant one. Thus the functions A and B appeared to be well represented by A(p 2).. n _p2, e (1 P 2)/p2 (315) B(p 2 ) = 1.0 ( 3 " 16 ) where e = 1.74517 x 10" 3 . The tabulated values of A then predicted an asymptotic form of A = p 2 r Â£ < 3 i7) and an asymptotic B of Bl. ( 3 " 18 ) When these asymptotic expressions were substituted back into the integral equation the power law for A was explicitly determined. It was found that e = (3a/4Tr) + (3ci/4tt) 2 + 3(3a/4Tr) 3 + ... (319) 13 where a is the fine structure' constant. Baker and Johnson obtained almost the same expression for the power law of A. They concluded e = (3a/4TT) + j(3a/47r) 2 + ... ( 3 "20) Last of all it was possible to see by comparing the limit of the electron propagator as the mass shell is approached, to the propagator expressed in terms of the renormalized propagator that the renormalization constant: ZÂ„, was equal to unity.
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32 All of these resulte were in agreement with the results given by Johnson, Baker and Willey. They had also concluded that the bare mass was zero and they had determined a wery similar value for the power law of the asymptotic expression for A. This paper represented an extension to the understanding of the electron propagator because it went beyond trying to determine an asymptotic form of the propagator which was consistent with finite renormalization constants. This paper actually determined an approximate expression for the electron propagator which was good for all momenta. The electron propagator was determined to be s l(p )=M ,.p2^lP 2 )/P 2 (331) where e = 1.74517 x 10" 3 . In this manner it was shown that the electron propagator could be determined using approximations of a far less drastic nature than had been tried before. No infinite quantities were encountered. Encouraged by the success of this first stage of the project, an attempt to solve the vertex equation was ventured.
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CHAPTER IV THE VERTEX EQUATION 41 Introduction The next step in the process of seeking a solution to the SchwingerDyson equations is solving the vertex equation, restated here, T A (p,q) = Y X + A A (p,q)where i 2 A *(p,q) Â„^ r Y Â«D (k 2 )S(pR)r X (pR,qR)S(qR)r 3 (qk,q)d 4 k (2tt) J . 2 + ^ fy\ (k 2 )S(pR)E Xa (pk,qR,q)d 4 k. (41) (2tt) As with the electron propagator equation, the solution to this equation is preceded by three elements of preparation. The first, the photon propagator is taken to have the same form as was utilized in the electron propagator equation. However, it is now fixed in the Landau gauge so that D (k 2 ) = Z~ yv v 3 Lk fg k k " L 2 " k 4 J (42) The second element is to recognize that the fourpoint diagram, E eA , which is defined in terms of the fivepoint diagram, which in turn is defined in terms of all higher order diagrams, an infinite progression, 33
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34 must be expressed in some closed and approximate form. Just as the vertex function could be related to the electron function through Ward's Identity, so the fourpoint diagram can be related to the vertex through a Generalized Ward's Identity. There are two possible longitudinal components of the fourpoint diagram, one is longitudinal with respect to k v and the other which is longitudinal with respect to q . The relationships between the longitudinal components and the vertex function as given by the generalized Ward Identity are k^pR.q.pq) = A A (p,pq)A X (pk\pq) ( 4 " 3 ) q,E Av (pU,~pq) A V (P^P) A V (pR,pq). (44) A It is possible to substitute r X for A X and r v for A v in these relations because the difference between r and A is a constant. Using this fact and putting Eqs. (44) and (45) in the differential form it can be seen that I Av (p,q,pq) =^rYp,pq) (45) E A Tpfc,0,p) =^r v (pk,p). (46) 3 Px Thus in this procedure the complete E v will be approximated by these two parts , E Xv (pR,q, P q) gfr X (p,pq) + 4r V (pR,p) . (47) For the purposes of simplification it is observed that the expression r A (pk,pq)S(pqR)l V (pqR,pq) gÂ£r A (p,pq) (48)
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35 can be reduced to A [r A (pk,pqk)S(pqk)] S" 1 (pqR). (49) To obtain this simplification use is made of the fact that ^[S(pq)] SVpq) KpqJ^S^pq). ( 4 " 10 ) and the vertex function was again approximated by r v (pq^Pq)^r'(pq). (4n) 3p By substituting the expressions in Eqs. (48) and (49) into the vertex Eq. (41), one obtains a simplified vertex equation, " 2 f r A (p,q) Â„ / + Â— fi, Y Â«D (k 2 )S(plc) ^I V (pk,P)A (2tt) J A ie 2 T + _1!P_ y\ 6 (k 2 ) S(plc) ^[lVk,"pqfc)S(pqK)l (2tt) J v S^pq^d^. (412) The last step in preparing the vertex equation was to operate with the D'Alembertian, n 2 = JLJL (413) K v 8p The D'Alembertian has the desirable feature of operating on the Fourier transform of photon propagator, here in the Landau gauge, to produce a Dirac delta function,
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36 D 2 D uv (pr) 2/i g yv 6(pr)Z 3 Z 2 ^ 1 . I4H] 3 a(p U. r U ) 9(p v_ r v ) { _~ r) d Therefore, D 2 r A (p,r+P> ^r /V[2 2 i 9 pv 6(pr)]S("r) ~[r A (?,?+q)S(i ; +q)" 1 ]_S(i ; +q)d 4 r y + _LiL /> ? 1 "LÂ— s(?) (2tt) 4 J a(p M r y ) 3(p V r v ) ( P r) 2 ~ _L [r A (F,?+q)S(?+q) _1 ] S(F+q)d 4 r ar v 2 e 8/ (Y^P) 4~ [^(prp+q^p+q)1 ] S(p+q)j 2 le (2^) 4 j 8(p y r^) 3(p v r v ) (pr) ]S(r) 2 J Â— [r X (r,r+q)S(r+q) _1 ] S(r+q)d 4 r. (415 2 2 In the above the definition Z 3 e Q = e was used. At the cost of having to solve higher order differential equations the last integral can be eliminated by the action of P = Y U Â— Â• 9P U This yields the final form, y 3 r A (p,r + p) = 4 y v % F Av (p,p + q) + ^F Av (p ,p+q) ] (416;
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37 where and o 2 2 f = fD, e = e /4tt F Av = S(p) /[r A (p,p+q) S(p+q) _1 ] S(p+q). (417) 3p v 42 The Main Computer Program It is only necessary to solve the vertex equation for the transverse components of the vertex since by Ward's Identity the exact longitudinal components are known in terms of the solution to the electron equation. The main computer program is a realization of the equation where ft V rans = ^) & [4.ns ( ^~ p) ^ ^ ^ Hereafter the subscript "trans" will be dropped and it will be understood that any A superscript is taken to be transverse. Thus, for any general function Q , Q^Q A k a Q a 4(4 " 19) In the vertex equation there appear two independent four momenta, R, the photon momentum and p, the electron momentum. Scalar functions 2 therefore will be functions of the variables p and ., = Â£d< (420) pk
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38 where p = p  2 and k =  k  2 2 2 and the functions will be parametrized by k . The range of p is from negative to positive infinity. The variable u has the same range owing to the indefinite metric. The general expression for the vertex function in terms of such scalar functions is given by \ H H r x (p+k,p) = \ F( P 2 u) + P A [*,K] fa (p 2 >u) + [ Y \j*] j )p 2 >u) p K p k p + [Y\K] 4 (P 2 .u) + y\(p 2 u) + P^ 4 (P 2 ,u) k P + p*l4(p 2 u) +E to * Y 5 Vv k a i(p 2 ,u). (42! ) P P In addition to the general expression for the vertex, the general expression of the Fourier transform of the electron propagator, as given in Eq. (216), is needed. So that ,,s _ A(p 2 ) pB(p 2 ) (4 _22) MPJ o o p o, 2, and S{W = A(( P+ k) 2 ) (frK)B((p*k) 2 ) ) . (4 . 23 ) Atp+k) 2 ) (p+K) 2 B 2 (( P +kn For simplicity the following notation will be observed throughout, A ] = A(p 2 ) A 2 = A((p+k) 2 ) B 1 = B(p 2 ) B 2 = B((p+k) 2 ) D 1 = A 2 p 2 B 2 D 2 A 2 (p+k) 2 B 2 . (424)
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39 A.0B, Thus S(p) =L 1T 1 (425) S(p+k) = S g L . (426] U 2 Equation (418) presents the differential vertex equation in a straightforward and simple form. Unfortunately when substitution of the scalar functions of Eqs. (421), (422) and (423) is made and the products of the gamma matrices are taken, thousands of terms need to be resolved. Once expanded in this way it is clear the matrix equation is equivalent to eight equations which are coefficients of the linearly . , . ... A X, Ai, ABatb 5 , . aX. Aa independent matrices: y . P w P K, e P ^Y Y la P a Â» ia a ^p kÂ„p and p . The description of the process of identification of these eight equations will consume the next several sections. In section 43 the lefthand side of the eight equations will be explicitly given, and in section 44 the righthand side of the eight equations will be given. These eight equations plus a set of boundary conditions will be used to determine the eight transverse vertex functions F, ^G^ H H,H 9 and I. O 1 L 43 LeftHand Sides of the Eight Equations The formation of the lefthand sides of the eight equations which comprise the matrix equation for the vertex is a relatively straightforward operation. The operation of f on any general function, f, of p and u will yield *f(p 2 , U ) . 2Kf
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40 2 The prime denotes a partial derivative with respect to the scalar p and an asterisk denotes a partial derivative with respect to u. When y 3 was applied to T X the coefficients of the eight linearly independent matrices were obtained. These are given in Table 41. TABLE 41 LeftHand Side of the Eight Vertex Equations A Equation 1 Coefficient of p Â• ** 2 *** (if t 23Â„V^*nf**"Cu 2 ) V " 14 2 P P P P p P *' 9 F * ' 2 ' ' ' _l 8 u F + 2(1/) * + 20 F"4u F + 8 p^F P P Â• 9 ^n II 9 *** 9 , 9^9 *** 64 \H + (3086 u 2 ) + 26\(lu 2 )H o ^(li/)' H Q p p p P *> + 64 ^r H '+ (44 u 2 16) f4 ^{lu 2 )H**' 32 u l{ P P P 8 (lV)H 16 pSH Q * ** Hi H, 9 Hi o *** + 18 4 + 46 u 4 + (20 /6)V2 ^(1 V)^ p P P P i H, ,i *' A 9 **' " *" 28 4 " 36 "T H l + V 1_u )H 1 + 40 H 1 8 uH ] P P P 2 \\l")Â±. (428) + 16 P H 1 ) P i
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41 E quation 2 Coefficient of p K f* n ** F*' *" (1u 2 ) ^*** p p p p 8V* J VC + V;^'C*W>C p P P p P + 64 H Q + 16 p^H Q H l 4u ** Â„ H * P P P ** *** H H + 18 ^ H* + [10 u 2 4(lu 2 )] Â§" 2u(lu 2 )2 ir 28 \ H*' p 4 ^ P p p **' u + 48 h" 8 uH*" + 16 p 2 Hp" + 4(lu 2 )2 yc P Equation 3 Coefficient of y * ** p k p 2 p 2 p 2 (429) LI . 8u H 4 ( 1 . u 2)_o + 8 ^ P P 2^1 , o. in . Â„n ..2< **i 12 uH, + 4(l/)i+ 20 H^ + 4(lif ^ + 64 p 1 "^ P 4 < Â• Â• 8u * + 16 p 4 H 1 Â™ H 1 P * ** *** + (18 u 2 6) 14 u(ltT)f+ 2(1U
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42 Equation 4 Coefficient of e aW y Y^k P a K B O II * ? ?. ** ft ' " \h q 10^Ho + \(1u 2 )H o + 8_H + 8H q P p p p Â•TH r 10\H, +^(lÂ«m +^H, + 8 H 1 P P P P ** *** H H 18 \ H* [10 u 2 4(lu 2 )] ^ + 2u(lu 2 ) 2 1 P P P * **Â• H i H + 28 u  48 H^ + 8 uH*" 16 p 2 ^" 40u 2 )2 ^. (431) p ' '" ' P Equation 5 Coefficient of ia p g k p * *** *> l G, ** G, G, *Â„ i^ + 7 Vl (lV)V4 44 G] F P P P P * ** *** Gq Gp 9 Up o bp + 9  + 23 u 4 [310 (/) ] 4Â— u(li/) V 9 4 p 4 p H P 1 **i Gp ^.1 Up H ^n p 111 14 Â§ 18^ G 2 + 2(lu ) ^+20 G 2 4 uG 2 +8 p d G 2 P P P * ** *** + 9 4+ 23 u K[310 (u 2 ) ] V(Tu 2 )^ L TP P P P 1 *i **i T T ? T " *" 14 K18 u 4+2 0u ) ^p+20 I 4 ul P P P + 8 p 2 l'"]. (432)
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43 Equation 6 Coefficient of 10 P a * ** *** *' G 9 GÂ„ o G G 9 u 4U7u 2 ) 4u(lu 2 ) V 14 u p 4 p 4 P P G**' ,i *n O ' ' ' + 2(1 V) Â—V + 24 G Q 4 u G Q + 8 p G Q P * ** Â„' G i G n o G, G 1 + 44 + 5 u4(1u 2 )^444G 1 p 4 P P P ** 9u I . [23 (u 2 ) 8]K + 10(lV)u^T p 4 P P (lu 2 ) 2 ^+ 14 u K(818u 2 ) ^ P P P 2u(lu 2 ) ^g20 ul 4(lu )I P Equation 7 Coefficient of ia k a G GÂ„ 9 G * *' Â£ [3 _ + 5uf (lu 2 )V8G 4p 2 G o P P^ P P * ** G9 G 9 ? G 9 i . 3 _ _ 5 2 + (1 ./) _^_ + 8 Q 2 P P P 9 11 T t 2 \ 1 + 4 p^G 9 3 4 5 u ~ + (1u ) 92 P P P (433) 2 i' +10 u I*' 2(lu 2 ) I**' 28 p 2 l" 8p 4 l'"]. (434)
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44 Equation 8 Coefficient of p * ** G *** G * 9 u V [2 7 (u 2 )]^u(lu 2 )^14 u^p 4 p 4 P P ** G + 2(lu 2 ) \+ 24 GJ 4 u G Q + 8 p Go * ** Â„*' G, G, o G, Gi G 1 12 4 15 u 4 + 3(lu Z ) V + 12 4 10 u 2p 4 p 4 V P P **i + 2(lu 2 ) V + 36 G!j +8 p 2 G}" + 9 u  P P * ** r ** 9 G 9 ? G ? , 2x2 2 [8 23 u 2 ] 4 10 u(lu 2 ) 4r + ^" u ) ~H P P P Â•A* 1 G 9 ? G ? 2\ G 2 14 u 4 +(818 iT) 4+ 2 u(lu ) Â— *p 2 P P + 20 u GÂ£ + 4(lu 2 )G*"+ 8 p 2 u GJ' ' . (435) 44 The RightHand Sides of the Eight Equations The righthand side of the vertex equation, 46 Y / V +V XV ], (436) is compiled through multiple layers of matrix multiplication. It is best represented, not by an exhaustive itemization of each and every term but by definition of the various layers. The first layer is to define the matrix function F f Xv = s(p+fc) ^ (p+K,p) + sr P+ fc)rVfc,p) ^s(p)s" 1 (p) Â°Pm \ v V
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45 A ,f A 9 A^ o B'B ? B ? v A 2 B 2 + . va B 2 + . v A 2 B 2 B 2 A 2^ = S(p+R)Z Av where ,Xv 3r , ^Xr v, 3P, where + r [2p V PPD + 2p 0MPD y ABD + 0Y BBD] A~A~ o BÂ«B2 ppd + 4^ p 4(437) (438) (439) MPD 9 DrtM^ (440) ABD = A 2 B 2 (441) BBD = ^ . U 2 (442) The Z Xv is a tensor with twentyeight different linearly independent combinations of the available matrices. These twentyeight form a group, fourteen elements of which are odd in gamma matrices, fourteen elements are even. "odd Av A v x,y. Av, Z 1 + 0p p Z 2 + 0p A k Z 3 + g /vv KZ 4 A v A, v A v. A v. + Kp V z 5 + Kp A k v z 6 + p A Y v z 7 + Y A P Z
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46 'Av^V/Ao^^Mi Av8d) 5 , , AcxvcJ) 5 . 7 '$"12 + e Y V3 k o k Z 14(443) ,Av KÂ« ^\ + > X *Xz + ^V 6 Z e 3 1o% 9% Â°\p/p% 6 * A^ We7 ia ta p a p v Z e8 where r h Z, = G BBD + A + 2 p ABD ul BBD 1 Â° ? l P K Z =2F^ ^2 ^ h pk H, g, g; g, g, ppd 2 J + 2 4 u 1 2 1 p P P P + 4u jj MPD 4 4MPD P P kG l k H o k Z=^Â±^G 9 BBD 2 S ABD + Â£ I BBD 3 3 p 2 2 p Z, = G + 2 H 9 ABD + pk I BBD 4 p 2 2 (444) (445) (446) (447) (448) Z 5 =\G 2 u\G; + G 2 BBD + 2 G2 I G w PPD P P H n k + 2  ABD 4 H Q MPD 4 H 2 MPD Â£ I BBD P (449)
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47 V? 7 = 4 ABD + G BBD + 4 + G, BBD + uG 2 BBD 7 pk Â° p z 8 = 2 G o " T G o " 2 G o PPD + G o BBD + 4 1 H l MP Â° . 2 4 ABD + 4 Â£ uH 9 MPD + ul BBD pk L k u "2 G, 9 = k p ln = k _Â£ _ 2 H 2 ABD p k I BBD k G BBD 2  ABD + Â£ I BBD P Z 10 = " p U 2 DDU ' J Â— P Z n =4H 2 MPDV + 2fl'uV*2flPPD ZÂ„ = G BBD 2 [ ABD + u ! BBD Z n = 2 H 2 ABD + Â•* I + Pk I BBD 7 ___L_ + 2 I r u ^2 4ppd + h BBD Z ei kp 3 P.k kp 3 P k Pk G l + 2 G MPD + 2 G 1 MPD \ ABD + 2 uG 2 MPD + 2 u Pk Hi H o 2 ^ BBD + 2 uG 2 MPD + 2 u ^ + 2 f BBD pk (450) (451) (452) (453) (454) (455) (456) (457) (458) (459)
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48 Ze 2 ^ G 2 ABD 2 H Q BBD 2 Ho BBD Z = I ABD + 2 H 9 BBD e 3 p I (4eo; (46i ; Z e 4 = G ABD + 2 4 + 2 Â£ H, BBD + 2 Â£ u Ho BBD pk ' k "1 (462) Z = L G ABD + 2 Â£ H, BBD + 2 Â£ uH ? BBD es pk o k l k Â£ "e6 , H H 2 G 9 MPD 4 4 + 4  P 2 P 4 P 2 H o k 4 4 PPD + 2 I MPD 2 u (463) (464) e ? p 3 k P 3 k HHZ e8 =2G MPD + 23 1 4^ + 2u^ + 4^PPD P 3 k H l pk eg pk 2 u I MPD * H, "eio P 2 Go ABD + 2 4 + 2 H BBD + 2 H 9 BBD (465) (466) (467) (468) Z eÂ„
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49 " e 13 H 2 4 + 2 H ? BBD + Â£ I ABD P ' H. o Z = 2 Â£ 2 H BBD Â£ I ABD ei4 p 2 2 P Since Av 2 )z Xv 1 .Av B l S(p')Z A *t z (pf+K)Z Av Feven = pV*, + P Vw 2 + ^V B W 3 + 'A A v. a A . aE + c^ 5 + pVia a \p 6 H 6 + P >a ia W W 7 t io X VpX + io oX kÂ„k v W + 1o ov p\M aA. _v, , , _Â• ^Aa_ ,.v, P a p w 8 T lu V "9 T lu F V10 A va 4 ipVVll + ia UA k a p"W 12 ia"> a k"W 13 Av "odd .Av, A_v A. v. Av, V ] + Rfp A p v V 2 + ^p A k v V 3 + g AV KV 4 X_v, .A,;,.V A v. Av + KPP V V 5 + p A Kk v V 6 + P W 7 + yW 8 + / k v Vg + ^%\^/ ho Xa3 VV3 k aP Vv il + ^Wl2 + ^Va V 13 + ^Ve k a k V where 13 N "12(471) (472) (473) (474) (475) W 1 AD Z + BD[(p 2 +pk u)Z 2 (1 +^)Z 5 Z 7 Z g ] (476) 9 2 2 W ? = AD Z + BD[(p +pk u)Z 3 (k +pk u)Z g k Z y Z g Â£ "2 W= AD ZÂ„Â„ + BD[Z iq + k 2 Z 1? ] (476] (477) (478)
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50 W 4 = AD Z + DB[(p 2 +pku)Z 12 (1 +  u)Z 13 W, = AD I + BD[(p 2 +pku)Z 1 (1 +1 u)Z 4 ] W 6 = AD Z e6 + BD[k 2 Z 2 Z 5 Z 1Q Z n ] W, = AD Z., + BD[Z Z 6 Z 1(J Z H ] 7 "" ^e7 "" e W 8 = AD Z eg + BD[Z 8 + (1 +{ u)Z n + Z 12 ] W g = AD Z + BD[Z g + (p 2 +pku)Z ]4 + Z ]3 ] W 1Q = AD Z elQ + BD[k 2 Z 7 + (p 2 +pku)Z 10 Z ]3 ] w n = AD z en + BD[ " z 7 + (1 + ^ u)z io " z i2 ] W 12 = AD Z ei2 + BD[k 2 Z 8 + (p 2 +pk U )Z n + Z 13 ] 9 2 W l3 = AD Z + BD[Z g + (k +pku)Z 14 + k Z ]2 ] W 14 = AD Z e 14 + BD[k2z i " Z 4 ] and where V] =ADZ 1 + BD[Z e5 +  U Z eu+ Z eu ] V 2 =ADZ 2 + BD[Z ei + ^^)Z e6 ^ e8 + Z e 11 ] V 3 = AD Z 3 + BO[ Z e2 + pku Z e7 + Z el 3 +k2Z e 7 +k2Z eil ] V 4 = AD Z, + BD[p 2 Z ei4 k 2 Z e5 pku Z ei4 ] V 5 =ADZ 5 + BD[p 2 Z e6 Z elo Z ei2 k 2 Z ei pkuZ e6 ] V 6 = AD Z 6 + BD[(p 2 +pku)Z e7 Z e g Z^ Z^] (479) (480) (481) (482) (483) (484) (485) (486) (487) (488) (489) (490) (491) (492) (493) (494) (495)
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51 ? = AD Z 7 BD[Z + { u Z eiQ (p 2 +P ku)Z ell + Z^] (496) V 8 = AD Z 8 + BD[Z (p 2 P ku)Z + (1+ u)Z fil2 ] (497) V g = AD Z g + BD[(k 2 +pku)Z eg (p 2 +pku)Z ei3 + k^] (498) V 1Q AD Z 1Q + BD[Z e3 + Z eiQ + A _l (499) V ll = AD Z ll + BD[Z e 12 " Z e 3 + k2z e 8 ] (4100) Z + k 2 Z 12 V 12 = AD Z 12 + BD[(1 + l U )Z e3 + Z e4 ] (4101) V 13 = AD Z ]3 + BD[k 2 Z e4 + (p 2 pku)Z e3 ] (4102) V H AD Z U + BD[Z eg + Z ei3 Z e3 ] (4103) Another way to write the righthand side of the vertex equation is to notice 4 * v Xv+ if 3 (4 104) The next layer of definition is to choose T Xa = [ 9 4 a 1F XV (4105) L 2 3 av 2 av J where T = p^V + P Vl + e XavB y 5 kp 6 T even v v ej v e 2 a p e 3 + io vX T e 4 9 iV T e5 P X P V T C6 io a6 k 6 P X P U k T e?
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52 ctA v ACt .Vn io"Vt " A p v T e , 2 + <Â°~V T e 13 . aBi Xv T + 10 B k a p 6 g T eu ^dd *% + ^ VT 2 + Â»^Â» 3 + gXVKT 04 X v T . . X_v, + KpVt 05 + KpVl ^ PVT 07 + Y^P T Q X, v T j. va3 op Â°14 where the TÂ„,=lÂ«,^ W 6I H 7I W 8! W n ei 2 "1 2 k e2 = 2 H 2 ' 2 F "6 T =  Wj, + i P 2 W ,+\ Pku W 7 +  W 1(J + j W 12 + \ W H c; w 3 jw i2 lw i3+ ln 14 T e4 4 W 4I W 5 + I W 9 + if W 127P\^ W 13 T =4w c w a 1^ 12 4p 2 W 8+ 1^W 13 1w 9 05 2 W 5 ' 2 w 4 " 2 1T12 2 T = W e 6 T e 7
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53 1 ,, , 1 T e = ! W 10 + \ W 2 + I P 2W 6 + \ Pku W 7 + 2 W 14 + 2 W 12 T ei2 = l W 12"I W 3I W 14 + I W 13 1 ,, 1 u _ 1 T Q ,. =W U+ JW 3 iÂ« 12 2 W 13 14 T o 2 = V 2 T O3 =! V 3i V 5V 10I V ll T 04 ! V 4l V 9l p2v ilV l T o 5 = l V 5i V 3 + V 10 + i V H 3 " 2 pku V 14 (4117) (4118) (4119) (4120) (4121) (4122) (4123) (4124) (4125) (4126) (4127) T o 6 V 6 T 0? 3Â« 7 *\ V, \ p 2 V 2 4f V 3 \ f V 5 1 V 6 + \ V 8 (4128) 3Â„ l u IE" iv. TÂ„ s =iVi V l2 k'll^2 "14 08 T 9 =I V 9? V 4 + 1" 2V 11 V 13+ IpkuV u T 010 =!V 10 1V 3+ V 5+ 1V 10+ 1V 11 T .nV " 1 T 012 " 2 V 12 + ? V l S W 8*7â€¢ V H + i V H (4129) (4130) (4131) (4132) (4133)
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54 T 013 = 2V 13! V 4 + ! p2v n +;UuV 14 + 2 V 9 (4 " 134) T oi4 V Â™ The righthand side of the vertex equation, . e _L_ T *a , (4136) r a is collapsed into a group of eight linearly independent matrix terms by the summation over a. It is clear that in order to obtain the firstorder partial derivatives of T Aa type functions the firstorder partial derivatives of the W Xv and Z Xv had to be known. Initial work on this main program tried to avoid this confrontation by seeking the derivatives of l Xa by a standard numerical process. This was found unsatisfactory for two reasons. One was the numerical calculation added significantly to the time parameter of the program. The other was the precision which this time bought was inadequate. This method was shortly abandoned in favor of compiling the algebraic forms of the partial derivatives of T Xa , W Av andJ Xv . These are recorded in the program included in Appendix C. In conclusion the righthand side of the vertex equation is comprised of eight linearly independent matrix terms. The coefficients of each of these terms are given in Table 42.
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55 TABLE 42 RightHand Side of the Vertex Equations Equation 1 Coefficient of p j> ' ii * 2 ' d ' e [2T 4T +6T +2 p T + 2 Â£ u T bL ^ 'oi T? 02 02 H o 2 k o3 + pk T o 3 Z V p 2 V' Equation 2 Coefficient of p K E[T Â°3 + 2T o 4 7 T oV 5T 05 + 2p2T Â°5 i 9 \j * U * Â„ + 2 P ku T o 6 + (, u ' ? T o 6 P VEquation 3 Coefficient of y ^ + T Â„ 7 + 4 T o 8 + 2 P 2 T o 8 + 2 I u T o 9 pk og J Equation 4 Coefficient of e aW y Ya^Pr o i V * * ell + 4 T + 2 p T + TÂ„ +2T L o 10 'on v on p o 12 o 13 " T _ 2 pku t' +(1u 2 ) T* ]. p 2 'o 13 H 'oh P o 14 J Equation 5 Coefficient of ia k a P R P e[6 T ee + 2 p 2 T ; 6 + 2 pku T^ +  (1u 2 ) T^ + 2T ^o T + T +2T 4T . e 10 p 2 eio P en e 14 p 2 e, 4 J
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56 Equation 6 Coefficient of ia P a ^ 4 y: 4 5T e 8 2 P 2T e 8 T eil 2 M j _ (iu 2 ) 4T * ]. k 'e 13 U ' pk l e 13 J Equation 7 Coefficient of ia k e[ T* + 2 pku T* + (1u 2 )  T* + T L P e 4 F e 9 P e 9 e 10 + 4 T +2 pV T + T J. ei2 e 12 ei3 ei4 J Equation 8 Coefficient of p u T e5 ]
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CHAPTER V CHECKING THE ALGEBRA In performing all the myriad matrix operations necessary to express the vertex equation we found that working out the algebra was extensive, repetitive and subject to error whenever the practitioner's strictest attention lapsed. After an unoriginal effort was made at wading through the whole expression term by term, it was necessary to confirm the results. Faced with the prospect of having to repeat the monumental process, we drew the conclusion that this kind of work is better done by machine. Efforts were made to solicit the use of a computer language capable of symbolic algebra. The software chosen was a programming language from the University of Utah called Reduce. The program offered a great variety of general algebraic calculating facilities. Amongst those of interest were symbolic differentiation, automatic and user controlled simplification of expressions, calculations with gamma matrices and tensor operations. With all of these capabilities it offered a \ery promising approach to the unwieldy task. It was found that Reduce provided a direct and easily acquired check on the lefthand side of the equation. The partial differentiation was performed, including all nine possible mixed derivatives with respect to p and u up to third order. A quick and accurate check of the original 57
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58 lefthand side was achieved. Furthermore, this confirmation was achieved with only a reasonable expenditure of time invested in becoming familiar with the language. However Reduce was less easy to make use of on the righthand side where, in addition to first and second order mixed partial derivatives, there were three layers of matrix operations to undergo. The size of the arrays quickly outgrew the allotted workspace in the machine. All of the calculations had to be performed in steps and then the results were summed afterwards. A further investment of time would have been necessary to learn how to design the output to be displayed in a form more amenable to easy checking. Despite this user related ineptitude, a complete expansion of the righthand side was obtained. One of the reasons Reduce was found to be less useful than expected on the righthand side was the operations, though limited in variety, involved a proliferation of terms. It was almost a waste to bring all of the ingenious operational flexibility of Reduce to bear on what was only a problem of tensor and matrix multiplication and largescale sorting of terms. The righthand side could be written down in a way that was better designed for checking by giving up the luxury of exhibiting the righthand side in terms of the basic functions A, B, F, G Q , G 2 , H Q , H , fi, and I. The final form settled upon for the righthand side, as it appeared in the previous chapter, was expressed in a hierarchy of definitions. The righthand side was given in terms of the 28 components of T Xa tensor. The 28 components of T Xa were given in terms of the 28
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59 components of the F Av tensor. The F Av tensor was defined in terms of the 28 components of the Z Av tensor. Finally the 28 components of the Z Av tensor were defined in terms of the basic functions, A, B, F, G Q , G 1 , G ?) H , H p H 2 and I. Each layer of redefinition represents the execution of another tensor operation. Table 51 summarizes the five steps taken to define the righthand side. The first step was to define a tensor, i^ =_ 2ppd r A P v abd rV + bbdtV v + 2MPD r A 0p v . (51) The abbreviations PPD, ABD, BBD and MPD, represent combinations of the electron functions, A and B, which were defined in Eqs. (439), (440), (441) and (442). Step 2 forms a new tensor, W Av = {J + *)Z Av . (52) Step 3 forms yet another tensor out of the former two tensors, F Av _ ^ z Av _ h w Av ( 5 _3) h D 1 L D 1 W where D ] = (A 2 p 2 B 2 ). Step 4 forms the last tensor, T Aa = [g a v + \ Y a Y v ]F AV (54) Finally, in Step 5, the a index is contracted by a differentiation with respect to p a . The rightside equals _JL T Xa . (55) 8p
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60 TABLE 51 Formation of the RightHand Side of the Vertex Equation Step l. T Z Xv = 2PPD T A p v ABD r\ v + BBD r X Y V + 2MPD T X z!p v Step 2. W Xv (0 + i)l Xv c+ t c Xv _ A l 7 Av B 1 w Av Step 3. F = KZ nW u l u l Step 4. T = [g v + 2 Y Y V J F Step 5. Righthand side = e Â— T a dp + The abbreviations PPD, BBD, ABD, and MPD were defined in Eqs. (439) through (442). The abbreviations Aj and D 1 were defined in Eq. (424)
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61 After the righthand side is separated into layers so that the full length and breadth of the righthand side is disguised, a large number of operations still have to be performed. Even a simple multiplication like r A pJ or (tf+jOz Av involves a large number of steps. Consider the procedure necessary to perform the multiplication of a single element of r\ say E Xa6 SVaP 6 W ' Wlth *' 2 e e %x$ a P 3 P Â° pk = {[2gy a + 2g^ + 2g^] kaP3 p^ = [Ta B \p p A + ipku c Aa p a+ ip 2 a aX k a ] ^ (56) Table Al was used to obtain the product y Y.Y^ in tne second line. Equation A12 was used to go from the third to fourth line. It is easy to forget a sign or reverse the order of two indices when a large number of such operations are performed. However it is possible to express the multiplication rules for matrix operations in a very simple way that allows the practitioner to do the same calculation in his head without ever consulting Table Al. This simplified method can be used to supply an algorithm to enable a computer to do the same kinds of manipulations in a common language like Fortran which does not possess symbolic capabilities. The basis of this method was designed by H. S. Green.
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62 To explain the simple multiplication method it is necessary to alter slightly the form of the definitions of vector functions like r and tensor functions like Z Av . The expressions used in Chapter IV were evolved through a historical process that did not necessarily produce the most symmetric arrangement. In this chapter it will be shown that when some small changes are made, things become very much easier. In Chapter IV, P K + e XoB y 5 Y P D k X Â• (57) Now a slightly altered form will be used. r A = p\ + y\ + P V 3 + p A M 4 + P x ia^k a p 3 Â£ 5 + ia A V 6 + Â™ X Xh + ^' r\W8 (5 ' 8) where Â£ 1
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63 The new form of r A is equivalent but notice that the order of the indices in the sixth and seventh components has been changed. Throughout this chapter it will be assumed that any general vector, say V , is expanded in terms of these same components. V A = p A Vl + y A v 2 + pV 3 + P A *v 4 + pVk^ + \v 6 + ia X \v 7 . XaB* 5 . n u (510) Similarly it will be required that any general tensor, T v , will be expressed in terms of the following components only. t xv = P y tl + P y,*t 2 + pV^t 3 + P Via a6 k a p 6 t 4 v A. v. Act , v. Act. . + P V Y t 5 + p la p o t 6 + p ,o k a t 7 pV^V^tg pVtg + PÂ»1o"P a t, , Â• vA. , Xa\xj> 5Â„ _ j. , Aav 5 t t + na t ]3 + e ^y Y(J) P a t 14 + Â£ Y Y^ a t 15 + e XVa3 Y 5 k a P 6 t 16 V pVt ]7 + p\^t 18 + p X k^t lg + pVioVe^o + A v t 21 + ^ia% a t 22 + kWt^ + kV a ^\k a P B t 24 + g Av t 25 + g A > 26 + g Xv ^t 27 + g Xv ia a6 k a P 3 t 28 . (511)
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64 Now that the groups of matrices have been carefully selected for any general vector V A or tensor T Av , the following notation will prove extremely useful. The components of a vector will be denoted by the following brackets. p A = (p\ 0, 0, 0) p\ = (pY 0, 0, 0) ?H = (p\ k, o, o) p X ia a6 k a P 3 = (pY k, 0, 0) Y A = (0, 0, Y \ 0) ia A V = (*, 0, Y \ 0) 10 Aa k = (0, i, y , 0) a c XaB \\k a P 3 = (Iff, fc, Y \ 0). (512) The components of a tensor, T Av , will be denoted by the following brackets. p A p v = (pV, 0, 0, 0) pV* = (pW, oÂ» o> o) p A p v k = (p A p v (c, 0, 0, 0) P A p v ia aB k a P 6 = (pVn, L 0, 0)
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65 P V (p v > o, Y \ o) p V io% (p V ^, 0, Y \ 0) p v ia A \ = ( P \ k, y\ 0) pV^Y^k^p (p v . i. y\ 0) pV (P X > 0Â» 0' y v ) P^ia^Mp^. 0, 0, Y V ) p A ia V \ = (p\ i, 0, Y V ) p VÂ«W Y 5 Y kapp = ( p *Â„, i, 0, Y V ) vA A Vx ia VA = (0, 0, Y \ Y V ) ^ aV ^\k a (0J,Ay v ) Â£ AVa6 Y 5 k a P 6 (*, l(. y\ Y V ) . vX A v> ia VA = (0, 0, Y \ Y V ) : XaV *Y%P a M^O.A/) ;AaV VV a = (0, 4, Y \ Y V ) : AVae Y 5 k a P 6 = (K, *. Y \ Y V )
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66 pV = ( P \ k v , o, o) p A k v pf = (py, k v , o, o) phH = ( P \ k v k, o, o) p A k v ia a3 k Pft = (pV, kH t 0, 0) a p Y V = (o, k v , Y \ o) k v ia A \ = (0, k\ Y \ 0) k v ia A \ = (0, k v /, A 0) k VÂ«BVy a P B = (Â„, k V ^ Y \ 0) g Av = (0, 0, g Av , 0) g Av t* = (0, 0, g Av , 0) Av i = (0, /, g Av , 0) g Av ia a \p 6 (pf, J, g Av , 0). (513) The rule for the mapping of each component into the bracket notation is straightforward. The first position in each bracket holds all electron momenta terms; p , p v and p 1 . The second position holds all photon momenta terms, k v and i. (The k A term does not appear here because only the transverse part of the vertex equation has been considered. However it is possible to extend the bracket notation to include longitudinal components without any additional complications.) The third position holds Y and g Av . The fourth and last position holds y .
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67 Now to convert the component into its bracket merely tabulate the occurrence of Y 3 P g > y\, y\ Y V , p\ P V > ^ and 9 VV and P ut them int0 their appropriate "home" positions. For example, consider p ia k a It contains p\ y v and k Y a so its bracket is (p\ L 0, Y V )Notice that although there is a unique bracket notation for each component, it is not necessarily possible to guess from the bracket notation what the original choice was for the component. (0, i, 0, 0) = io aB P a k 3 or (p\ i, 0, 0) = ia ap k a p 6 ? The originally selected meaning of (p", I, 0, 0) must be preserved so it is the latter relationship which is the correct one. Now it will be shown that multiplying the components of the vector V A or tensor T Av by a tern like ft k\y v P v Â»k v> etc. is simply done by observing a few rules. To multiply by a unit matrix vector like p v or k merely add the vector to its correct position. v (*. L y\ 0)p v (p V (*, I y\ 0) (514) To multiply by a matrix like y V , i or t from the right, move the matrix across the bracket from right to left. Each time the matrix crosses another matrix take the scalar product of the two. Remove the scalar products to the outside of the brackets. (pY p, 0, 0)jrf P 2 (p\ 0, 0, 0). (515)
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68 One term will occur for each scalar product and one term will occur when the multiplying gamma matrix reaches its "home" position, ( P V, k\ 0, 0)* (pV k V K, 0, 0) + pku(p X , k v , 0, 0). (516) Finally, reverse the sign of alternate scalar products, (pV It. 0, y v )t = ( P Y k v l(, 0, 0) k 2 ( P V 0, 0, y V ) + P ku(p\ L o, Y v ). ( 5 " 17) In order to multiply a tensor or vector from the left the rules remain the same except that the multiplying term is moved across the bracket from left to right. Recall the example given earlier of the multiplication of XaB4> 5 k I b u from the r ight. This example can now be written as Y Y 4> crB pk J r (0, /, y A , 0) ^ i [(pV L 0, 0) pku(*. o, /, 0) + P 2 (0J,A0)]^. (518) Translating the bracket notation back to the original components it is found that 2. Aa, t I + P io k a ] ^ APerJk = Lpi Â° k Â« p b 2,Aa. ,1 _ (519) This is the same result which was achieved in Eq. (56), yet it was achieved without using Table Al. With little practice the products can be arrived at as quickly as they can be written down. This process
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69 of bracket manipulation is not only many times easier for the human mind, but it also lends itself to the writing of an algorithm for executing similar products in Fortran. It is also possible to predict easily the products of a contraction of a vector V a or tensor T Xa with another vector like p a or gamma matrix like y . To contract with a momentum, merely perform the implied con'a traction. To contract with a gamma matrix it is necessary to first move the gamma matrix across the bracket, taking alternate signs of each possible scalar product. While doing so, perform the implied contractions on each term in the product. A summary of the rules for bracket operations is given in Table 52. Returning to Table 51 it is easy to see that a knowledge of the following operations is all that is needed to generate the righthand side, V X 0, V A p v , vV, yV, i T Xv , i T Xv and yj Xv , where V X stands for any general vector function and T Xv stands for any general tensor function. Using the expressions for V and T v given in Eqs. (510) and (511) and using the method of bracket operations, these seven products are easily written down. (1) vV = P A (P 2 V 2 + pku v 3 + v 5 ) + (p X ^(v 1 + pku v 4 + v 6 ) + p A k(p 2 v 4 + v 7 ) + p x ia a6 k a p 6 (v 8 v 3 ) + Y X ( P 2 v 6 pku v 7 ) + ia Xa p a ( v 5 pku v 8 ) + ia Xa Mp 2 v 8 ) + e Xae VV a P3(v 7 ) (520)
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70 TABLE 52 Bracket Operations A. Multiplying by a momentum: Add the momentum to its "home" position. B. Contracting with a momentum: Merely contract the momentum with its proper complement and move the scalar product to the outside of the bracket. C. Multiplying by a gamma matrix: (i ) Move the matrix across the bracket from right to left in order to multiply from the right. Reverse directions to multiply from the left. (ii) Take alternate signs of each scalar product that can be formed. (iii) If there is no matrix in "home" position, include a term with the multiplying matrix in the "home" position. D. Contracting with a gamma matrix: (i) Perform steps (i), (ii). (ii) Perform the implied contraction as the gamma matrix is moved across the bracket.
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71 (2) V A p v pV(v,) + pV0(v 2 ) + P X P^(v 3 ) + P Via aB k a P 3 (v 4 ) P V P X (v 5 ) + p V ia X \(v 6 ) P V ia A \(v 8 ) + pV^y\k a P 6 (v 8 ) (521) iAa, (3) V A Y V pV(v ? ) + P A P^(v 4 ) pV(v 6 ) + P v 1a Aa k a (v 8 ) A. va A. va, + P V(v 1 ) + p'ia^p^) + p A ia^k a (v 3 ) + p A e^\k a P B (v 4 ) + ia^(v 5 ) + Â£ n 5 Va (v 6 ) + p A k V (v 3 ) + pW(v 4 ) + p A k v )((v 7 ) + p A k v ia ae k a P B (v 8 ) + Y x k v (v 7 ) + k v ia A \(v 8 ) + g Av (v 5 ) + g Av 0(v 6 ) + g A ^(v 7 ) + 9 AV ia a6 k a P 6 (v 8 ) (522) (4) y v V A = p A p v (v 2 ) + pV(((v 4 ) + P V(v 6 ) + P V ia\(v 8 ) + pV(v,) + P^a^pjvg) + p A ia va k a (v 3 ) + p A e Va6 \%k a P 6 (v 4 ) * ia vA (v 5 ) + e A ^y\p a (v 6 ) + B XaV *Y 5 T^ a (v 7 ) + e A Vp 3 k a (v 8 ) + pV(v 3 ) p A k v ^(v 4 ) + y A k v (v 7 ) + k v ia Aa p a (v 8 ) + g Av (v 5 ) + 9 AV 0(V 6 ) + g Av X(v 7 ) + g Av ia^k a P e (v 8 ) (523) (5) il Xv = p A p v (pku y k 2 t 3 ) + p A P v 0(k Z t 4 ) + p A p^(t 1+ pku t 4 ) + p A p v ia a6 k a P 6 (t 2 )
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72 v. Act / ,2. + pV(pku t 6 + \ft 7 ) + P V ia Aa p a (kH 8 ) + p V ia X \(t 5 ^ Pku t 8 ) + P V Â£ W y\k aP3 (t 6 ) + pV(pku t 1(J + k 2 t n ) + p A ia va P a (k 2 t 12 ) + p X iak a (t g + pku t 12 ) + P A ^VY/ a P 6 (t 10 ) + ia^( P kut 19 + k 2 t 15 ) +Â£ Aav V Y4) P a (k 2 t 16 ) + Â£ AaV V%k a (t 13 + pku t 16 ) + e X ^y 5 P 6 k a (t 14 ) + p X k v (t g+ pku t 18 + k 2 t ]9 ) + p X k v p((t 10 k 2 t 20 ) + p\ v Ht ]7 + pku t 20 t n ) + P Vi oB k a p B (t 12 t 18 ) + Y A k v (t 13 +pku t 22 + k 2 t 23 ) + k v ia Aa p a (t 14 k 2 t 24 ) + kV v k (t 15+ t 21+P kut 24 ) + kV a6 \\k a P 3 (5 16 t 22 ) + g Av (pku t 26 + k 2 t 27 ) + g Av 0(k 2 t 28 ) + g Av /(t 25 + pku t 28 ) + g Xv ia a3 k a P 3 (t 26 ) (6) i*T Xv = p X tf (p 2 t 2 + t 5 + t g + pku t 3 ) + p X p v ^(t 1 pku t 4 t 6 t 1Q ) + p p v l((p t 4 t 7 t^) + P Vl0Â°\pB (t 3 + t 8 + t 12 ) + pV(p 2 t 6 + pku t ? t 13 ) (524)
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73 + P V io X \(t 5 Pku t 8 + t 14 ) + p v ia A \(p 2 t 8 + t 15 ) + P V e Xa3 V^k a p g (t 7 t 16 ) + p A v (p 2 t 10 + t 13 + pku t^) + p A ia v \(t 9 pku t 12 t H ) + p X ia va k a (p 2 t 12 + t 15 ) + P V^ Y %k a p B (t 11+ 1 16 ) + ia VX (p 2 t 14 + pku t 15 ) ^ rtV a (t 13 pku t 16 ) + e^ Y \k a (p 2 t 16 ) + P V(p 2 t 18 + pku t 19 + t 21 ) + p x k v ^)t 1? pku t 20 t 22 ) + p x k v /(p 2 t 20 t 23 ) + p A k v ia a6 k a Pe( t 19 +t 24 ) + Y X k v (p 2 t 99 + pku tÂ« J Y^ IP i22 + k V ia Aa p a (t 21 pku t 24 )
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74 + k v ia X \(p 2 t 24 ) + ^yVa^) + g Av (p 2 t 26 + pku t 27 ) + g Xv 0(t 25 pku t 28 ) + g Av *(p 2 t 28 ) , Xv. oBi, n 1+ \ (525) + 9 io k P R (t ?7 ) V 27 (7 ) Y T Av = p A (p 2 t 2 + pku t 3 + t 5 + 4t g + pku t 18 + k t ig + t 26 ) + p A ^(t 1 pku t 4 t 5 3t 1Q P t 2Q ) + P H(p 2 t 4 t ? 3t n + t 17 + Pku t 2Q + t 28 ) + p X ia a6 k a P 3 (t 3 + t 8 + 4t 12 t 18 ) + Y X (P 2 t 6 + Pku t 7 3t 13 + Pku t 22 + k t 23 + t 25 ) + ia A \(t 5 pku t 8 + 4t u k 2 t 24 + t 26 ) + io A \(p 2 t 8 + 4t 15 + t 21 Pku t 24 t 2? ) + e AaB \%k a P 3 (t 7 3t 16 + t 22 + t 28 ) (526) These seven equations complete the work necessary to execute a check on the correctness of the righthand side as it was described in Chapter IV.
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CHAPTER VI THE MASS SHELL SOLUTION 61 An Approximate Solution In Chapter IV the transverse part of the vertex equation was decomposed into a set of eight linked differential equations out of the original integral expression for the matrix function. The eight differential equations established relations for the eight unknown scalar functions, F, G , G ] , G 2 , H , Hi , H 2 and I. These scalar functions exactly describe the transverse vertex for a given photon momentumsquared, k . > ~ H r A (p + k,p) mSL F (p 2 , u ) + p A [^,K] Â— 2 (p 2 ,u) p kp^ H H + [y\*] if (p 2 ,u) + [ y \k] 4 (p 2 Â» u ) + AV' u) + P V ^ (P 2 ,u) + p X K I (P 2 ,u) + E H 5 Y.k a P v ^ (p 2 ,u). (61 P The circumflex signifies the transverse component. The scalar functions 2 are shown as functions of the electron momentumsquared, p , and the relative angle function, u, where u = %Â—? Â— r Â• Tne scalar functions (IpVI) 2 n are parametrized by the value of k . The eight equations were checked by a process described in Chapter V. The eight linked equations which are third order in derivatives including all nine possible mixed 75
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76 derivatives with respect to p and u, were complicated to write down, difficult to check and promised to be significantly more difficult to solve. Therefore it was important to start with a good guess of the correct solution. From the experience of solving the electron equation, where it was learned that the mass shell solution dominated a wide region around the mass shell, it was hoped that the vertex equation would similarly be a slowly changing function. If this were so, then a solution to the eight differential equations which worked near the mass shell might be extended into the asymptotic region by simple variations of the eight scalar functions. In 1981 H. S. Green communicated an approximate solution to the second order tensor equation, v 2 r A = d F vA + g ^lpVA^ (6 _ 2 ) L ' v v The third order tensor equation, Eq. (416), is obtained from this equation by the operation of i. The tensor functions Green found to be approximate solutions to Eq. (62) were used to identify the approximate form of the eight scalar functions. The approximations employed were commensurate with the known behavior of the vertex and the electron propagator near the mass shell. In Eq. (62) the tensor pvX= s (iV 35" [r A (P 1 p 2 )s(p 2 )]s1 (p 2 ) (63) was approximated by A y , where A =
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77 This amounts to neglecting sO^Cgjr rH?f z ) r x (p 1 3 2 )s(p 2 )(p 2 )(4s1 (p 2 )Y v )]. (65) V v This term approaches zero as r A approaches its mass shell limit of y . Near the mass shell the appropriate form of the second order vertex equation is given by V 2 r X =. e ^ Yv AV + r 1 A X ?]. (66) This is a far simpler equation than (62). This equation can be decomposed into a set of four linearly independent matrix equations by the following device. Take the product of Eq (66) with 1, y , v and y , then take the trace of each product. To facilitate this the following definitions were made: C A = I tr [r A ] (67a) (67b) < v 1 tr [r\ v ] (670 C X = i tr [r\ ] (67d) yvp 4 L 'yvp J D A = 1 tr [A A ] (67e) D l I tr [A \ ] (6_7f) D X = i tr [A X Y ] ( 6 " 7 3) yv 4 'yv D * = 1 tr [A X Y ] (67h) yvp 4 yvp
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78 where and V = 2 [y A'V Y yvp ~ 2 {y A' Y yv (68) Now it can be seen that the set of four equations below is equivalent to Eq. (66). V 2 C X = 3eD A V 2 C A = 2 e D A 2 e 9 V 2 aV u y y v ) A 2e9 V" 2 8 P D A +2e8 V~ 2 3 P D A V C yv e yv " " "y" vp ' 'v yp V 2 C A ... 2ea.r 2 3Â°DL2e V^pyo yvp vpa 2 Â£ 3 V~ 2 3 a D A . p yva (69a) (69b) (69c) (69d) The D A D A , D A and D A tensors are evaluated by using the ' y' yv yvp y y definition of A A , Eq. (64) in Eqs. (67e) through (67h) V X 1 ... rrX D A =\ tr [r A (^ +A 2 )(0 1 +A 1 )]/D 12 = [(p 1 .p 2+ A 1 A 2 )C A + (A lP ^A 2 pi;)C A P^C A v ]/D 12 D A ={tr [r A (^ 2 +A 2 )Y u (^ 1 +A 1 )]/D 12 (610a) 'v (610b) [(A 2 P ly +A lP2y )C A + (A^gPTPg)^ + (P 1y P2 + P 2y PiK (A 2 P^A 2 P^)C A v + ( P > 2 P C A vp )]/D 12 D yv = I tr [rX( ^2 +A 2V^ +A l )]/D 12 = t(PlyP2v%hv) cA ^ A 2Plv^P2v ) ^ (A 2hy^P2y)< +( hV A l A 2) C iv (PlvP2 + P2vP^y P + ^lyP2 + P2yP^ C vp + ( A 2Pl +A lP2 P ) C U /D 12 (6 " 10c)
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79 C = 1 tr [r VVW W} /D 12 A . / _ \^A . /Â„ Â„ \r^ [( Pll> p 2vp 2p P lv )C p + 1p )C p + C pp + (A 2Plp +A lP2p )C pv MA 1 A 2 P 1 P 2 )cJ, vp (P lu P^P 2y PpC (P lv P^P 2v P?)C X py Â„ + C ]/D 12(6 " 10d) These definitions are somewhat intimidating in appearance but the full effect of assuming the vertex function will be dominated by y in the region of the mass shell has not yet been fully explored. If r is well expressed by y for some range of values of p and u, then this means C A is well expressed by 6 A and the other tensors C , C and u y MV C A have neqliqible effect relative to C A . Furthermore, the electron yvp V propagator has the simple form lj$m. Under these assumptions the D A , D A , D A and D A tensors take on the greatly simplified appearance of ' y yv yvp D * = 2mp_ (6lla) D 12 D A = [(m 2 Pl .p 2 )6 A + (P ly +P 2u )P X ]/D 12 (6 " llb) D yv =m "(P 1v P 2 v)<(PlyP2y^>i2 {6 " llc) D yvp = t(P 1v P2vP2yPlv )6 p + ( PlvP2pP2vPl P )6 i + (PlpP 2 yP2pPly> 5 >i2 (6 " 11d) 2 2 2 2 where D, 2 = (P _m )(P2" m )Â• t A , 6 is defined in Appendix A.
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80 Green's work using these definitions of the D type tensors to solve the four equations, (69a) through (69d), for the four tensors C , C y , C A and C A , is reproduced in Appendix B. In the next three sections yv yvp' his expressions for c\ C A , C A v and C A vp are used to determine the nearthemassshell forms of the eight scalar functions F, G , G, , G 2 , H Q , H 1 H 2 and I. 62 The F and I Functions In Appendix B, Eq (B32) gives the expression for the C tensor C =4emp v/" 1 S [^i n (i3) +1] (612) where x 6 = [p + (B+l)l 2 and u 6 =m 2 l(l6 2 )k 2 . (613) Furthermore, from the definition of c\ Eq. (67a), the scalar function F is determined. C A =)tr (i X ) = p X f (614) Combining Eqs. (612) and (614) the scalar function is identified in terms of an integration over beta, f~\ dÂ„ uÂ„ x n F"!Â«p/ ^^ln(l^) + l]. (615) Obtaining the I function is slightly more involved but in the end it will be seen that the expression for I is simply related to the expression for F. The defining equation for C^ is given in Appendix B, Eq. (B59) and it states,
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VH F2 yvp 7_ l x g "g/ 81 (616) Using the fact that when 4 v x then 1 U fi X R i(lx?)l"0u!) (617) V 2 *=Â— L(618) V U B so that Eq. (B16) can be reexpressed as V 2 (pVPo3 P C X ) = ekW/* dB. ( 6 " 19 ) It follows that pVp^a p C X ekV /* dB. ( 6 " 2 Â°) From the definition of C A vp the scalar function I is defined. ip^^Svp^^vw + ^V p k pPy > 5 >vPyVv ]} i(6 " 21) From the above it can be shown that it follows that From Eq. (617) it can be shown that 2 9 X/$d 3 = / ZpM^ln (1 % + L >dB. (623) 7 ^ x Z U B X B 3
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82 By substituting Eqs. (622) and (623) into Eq. (620) the function I is identified, I = _ e pk / hi In (1jJB) +f]d3. (624) M xÂ„ u 3 3 4 It is now apparent that a simple relationship has emerged. That is, I 4 F. (625) 3m 63 The G , G, , and G Functions o I c The C X tensor is defined in the appendix by the following statement, u C* = CV d X Z* (B48) y y vi where C "f J tl^^]ln d>3 4 ^l x 3 3 3 f U1V ^ (1 pf ) + (1^) In (lp)> (B40) Pi P2 and where <H"/ < m HÂ»^ [( v ,)ln(, ^ ) + ,]dB lV^r + iÂ»/ l ^Â« e )iiÂ»(i^) 1n (1 "tt) d3 (6 " 26) 3 ^ x 3 U B
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83 fl q d3= f^ (lu R )f[(^D In (1^) + l]dB (627) ^vi 1 ^ 1 ^ 3 6 Then CK= (li) In (1pij) + (1^) In (1 p). Pi p 2 >Kh A * C A = C 6 A 6 C v v v (631) (632; ^f/yef CK jfh 6 ^
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84 + P A p y [l Â£ /l X 3 d3 + p\[\e (2/ + IK/i^B + I^e) Je/lJdB]. (633) From the definition of c\ Eq. (67b), the identification of G Q , G ] , and G ? can be made. c*= I tr [i\] r G = 6 A G + p A p 1 + p X k  . (634) y o F H y P 2 * y pk From a comparison of Eqs. (633) and (634) the final expressions for the three G functions are formed. Go=f/^BCKf/l 3 d3 (635) G ] = 4 s / !> {6 " 36) G 2 = (2p 2 u + pk)(/l*d(3 +/ljd0) f/l*pkdB. ( 6 37 ) 64 The H , H ] , H 2 Functions The defining equation for C A is shown in the appendix to be 3 n yv C X IS [6 A k 8 a k 3 A )8 (6 X k 3 a k 9 A )9 1 flgdB yv 16 L y a y'v'va y y J 3 (6 A k 6 A k ) [ I.dB (B55) v y v v y 'Â„/ 6 yv , em / A where UdB = / H 2 &) Ml ^) In (1 ^)]dB (638) b /l * U B B B
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85 / I 6 d6 =/ 1[ ^ ,n0 "i )]d6 (6 " 39) 1 fz dz L 2 (z) = J ln(lz) f . (640) In order to perform the implied operations first notice that 3x 3 I(X B> " "Spf ^=^a + < 6+1 ' k a )lX (6 41) where and I x =l_ (642) 9x n 3 a 9 I = 26 a I X + k k a (g+l) 2 I XX + p a k 2(3+1 )i xx r\ n n v + 2p n k a (B+l)I XX + 4p n p a I XX . (643) Completing the operations implied in Eq. (B55) yields a new expression. + (6jk v 6ik y )[2/l X d8 + 2k 2 /l 5 XX 6 2 d8 + (2pku+2k 2 )/l xx 3d3 + 2pku/l xx d3 + /igdp] + (5 X p v 6^p )[4k 2 /l XX 6d3 + 4pku Jl XX d3]>. (644) The relation between the tensor function C A and the three scalar H functions is given by the definition C A = I tr [r\ ]Â• (67c) yv ^ ' yv
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86 This can be expanded to reveal the relation between C and the three r yv H functions. C A = (p p k p A p k ) Â—J yv x ^ *v y ^ *y v' 2 A A 2H 1 v y y v' k 2H + (6 A p 6 A p ) Â— Â£ . (646) vy y*v p By comparison of Eqs. (644) and (646) the three H functions are now expressible in the following way, H o = " kp 2 _/"l xx de (647) H, = ^ [pk 2 /l XX BdB2(p 2 ku+ ^)/l xx de] (648) 2 , emk r /Dku . k \ /" T xx. Ql + "F [( V + T ) J ! 5 d6] + ^jjr/lÂ£ X B 2 dB + y^ (pku + ^)/ljj x Bd3. (649) 65 Summary of the Mass Shell Solution All of the preceding description of the eight scalar functions, F, 6 , G, , Go, H , H, , H 2 and I has established the transverse part of the vertex function in any region which is typified by the satisfaction of two constraints. One constraint is that the electron propagator be well represented by $ m
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87 where m is the experimental mass of the electron. This means that the function A(p?) or A(p^) must be essentially constant and equal to the experimental mass. The second constraint is that r must be dominated by the contribution from y X so that the transverse vertex is expressible in a series expansion, r X X + jo. L X + (JL) 2 L A + ( JL)3 L * + ... (650) where a is the fine structure constant. If it is possible to assume that the first term in the expansion dominates and if the electron propagator is on the mass shell then it is expected that the functions F through I will satisfy the eight differential equations which have been written into the Main Program. These eight function solutions have been collected together for ease of reference in Table 61. By using y X as the first order contribution to r a solution was found to the vertex equation up to second order. In effect the solution which was determined represents an identification of the L 1 term in Eq. (650). If this new improved version of r = y +l } were put into the vertex equation the vertex could be determined up to third order. There is a practical limit to how far this process should be carried toward self consistency. The coupling constant, ^, is a very small number, 2.32 x 10" 3 . Where it is true that r A is dominated by y the third order corrections would make little difference to the solution. An even more important consideration; the determination of the vertex function is only one step in a larger iterative procedure which seeks to reevaluate the electron and photon propagators to self consistency.
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88 TABLE 61, THE MASS SHELL FUNCTIONS F = emp /^dB I =  pkjl } dB G = 1 +/l 2 dBCK/l 3 d3 G, = e Jf T x P 2 /i x 3 de o 2 .  pk (Mm + dc/i^b + /i x 4 d ] f P k/ >\ H, f [P^/lfsdS 2(p 2 ku + ^)/l X *dB] H 2 ^ [/l X dB j/l 6 d6] ^ Iff ^)[/l^6 t /l>B] 3 /> + onk f j xx o 2 . 32 J A 5 1 U o X R U R t 2 X R 4 = (1 ^r " *r } ln (1 " tt } * x 3 3 6 I 3 =( m 2 u B )J[(^l)ln(l^) + l] I 4 GÂ»S> ^ in (1 ^) ? 2 2 Pi 2 P? CK = (1 K) In (l 4) + (1 \) In (1 Â§) p^ m p 2 m
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89 TABLE 61 (Continued) .^uJt^e^V"' 1 ^'5 ] x^ k >:,: >i >, Ij i^l^ in (1 J) ^ ^ ] U 6 X o U Q X R I 5 =L 2 a) + (l^)lnO^) L 2 (z) =/ in (lz)f 6 x B u 6 5 u g x 3 U B x 2 B U B X 6 2 t xx 1 u a X R U R 1 i .^ c B ln(1 * + j_]_^ 3 2 B 2 ~3 ln (1 "Tj "7 ^ 2 ] X B 6 X 3 (1 u 6 ) X B where . . rp lejii tf 2 In R 2 W 2 uÂ„ = m ^ I B )k . a e Â— fine structure constant.
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90 This project involved an initial approximation for the photon propagator, from that a calculation of the electron propagator, from that a calculation of the vertex. Future plans for the project looked toward using the information gained by this work to calculate the photon propagator. From this new photon propagator a new electron propagator could be calculated. From these two a new vertex. Each cycle of calculation should contribute a finer resolution of the exact solutions. There would be no point in refining the vertex function much beyond the level to which the electron propagator is known. The functional form of the electron propagator is accurate to about 0.1% of the value of the function. Where y X is dominant, corrections to order (Â— ) in r would be expected to amount to less than a 0.1% difference. A direct consequence of solving the vertex equation only up to the A L, term will be that the eighth equation, the coefficient of .Tf 1 Xvy 5 k will not be so i ve d. it happens that the righthand side ' '(j> v r u of the eighth equation is second order in ^. Therefore to this level of solution the eighth equation should equal zero on the right>hand side. For this reason equation eight will not be used as a criterion in assessing the correctness of the solution. In the next chapter a test of the viability of the mass shell solution is made, and a description of the programming methods necessary to enact it is given.
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CHAPTER VII VERIFICATION OF THE MASS SHELL SOLUTION 71 The Mass Shell Program Chapter VI established the functional forms of the eight scalar vertex functions. There is no single absolutely superior way to compile these functions which were summarized in Table 61. The Mass Shell ProgramÂ—the Fortran ProgramÂ—which computes the values of the functions at given points, assumed many strategies. The decision of how to compute the functions was influenced by concern for ease of assembly, the demand for machine time, and the inherent error in each method. Three basic categories of method were tested, and we became familiar with their merits. These three categories are described in section 73. Once the mass shell solution was computed by one of these methods, it had to be interfaced with the Main Program by way of a matrix of anas? numbers. The data matrix contains the value of each of the eight scalar functions and the nine possible partial derivatives of each function at 2 one or more points in the infinite plane of the variables p and u. Also included as data are the simpler electron propagator functions A and B (which were given in Eqs. (315) and (316)) and their derivatives. The Main Program uses the data matrix to evaluate the leftand righthand sides (L.H.S. and R.H.S.) of each of the eight equations at each point provided. The Main Program evaluates the relative error in each equation at each point. 91
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92 Â„,,.._ . _ .. . (L.H.S. R.H.S.) of Equation i ,, n Relative Error of Equation i = L.H.S. of Equation i [7 " V for i = 1,2,... ,8. The relative error is reported as a percentage error in the output. An optimal solution will have a minimal error over the entire plane defined by p 2 and u. The question of what constitutes an acceptable minimal error invites an analysis of what are the possible sources of error which contribute to the Main Program and the Mass Shell Program. The relative error in the eight differential equations is the effect of a number of contributory causes. There is inherent error in the data matrix just because the Mass Shell Solution is an approximate solution to the eight differential equations. There is inherent error in the electron propagator functions A and B because they represent only approximate solutions to the electron equation. There are truncation errors introduced by the numerical procedures used to perform integrations and differentiations. Finally there is roundoff errorÂ— the inevitable outcome of any calculation which is carried out to a fixed finite number of figures. All of these sources of error had to be either maintained below a preset, tolerable level or, where they could not be controlled, they at least had to be understood well enough so that we could recognize when they were contributing to a significant loss of information. The next section will consider what were the sources of error in the Main Program and how these were controlled. The following section will discuss the three principal methods used to evaluate the vertex functions, F, G , G,, G 2 , H , H, , H 2 and I, and how each method affected the overall level of uncertainty.
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93 72 Contributions to Error in the Main Program For the purpose of analyzing the kind of error that is being generated within the Main Program alone, we will assume for the time being that the data matrix of the eight scalar functions, their derivatives, the two scalar electron functions and their derivatives, could be supplied to the Main Program without error. If this could be done what would be the remaining sources of error and how would they propagate through the Main Program? One of the earliest versions of the Main Program defined the partial derivatives of the twentyeight T Aa components by taking firstorder differences. (This tensor appears in the R.H.S. and it was defined in Eqs. (495) through (4116).) This saved writing the analytic expressions for the partial derivatives of the twentyeight components of T which, in turn, had to be expressed in terms of the partial derivatives of twentyeight components of F Xv which, in turn, had to be expressed in terms of the partial derivatives of the twentyeight components of Z and W Av which, in turn, were at last expressed in terms of the partial derivatives of the eight vertex functions and two electron functions provided by the data matrix. (See Figure 51 to be reminded of the hierarchy of tensors which define the R.H.S.) The numerical procedure for evaluating the partial derivatives of the twentyeight components of T Aa was quickly abandoned because the T Aa components are very often extremely large and slowly changing, and roundoff error eliminated most useful details. It is a characteristic f the R.H.S. that, not only are the T Aa components large, but that the R.H.S. is very much smaller
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94 than its components. The R.H.S. is very sensitive to roundoff error because it involves the difference of many large and almost equal terms. In order to effectively calculate the R.H.S. analytic expressions of the ,. , L r ou rtat r XV yXV _Â„j ,,,AV partial derivatives of the components of the T , F , L ana w tensors had to be supplied and the data matrix had to provide the functions and their derivatives to more than six figures. Once the numerical means of calculating the derivatives of the components of T Aa was discarded, the only remaining source of error in the Main Program is the accumulative effects of roundoff error. In hexadecimal based arithmetic the roundoff error for a single operation will be proportional to the 16" t+1 , where t is the number of digits in the mantissa when the number is expressed as a fraction times the base raised the power of the exponent. For a calculation of standard precision, the number of figures in the mantissa is only 8. This is not enough to provide an accurate evaluation of the R.H.S. It was necessary to increase this precision to 16 significant figures. This meant 15 the roundoff error for a single operation was proportional to 16 , a yery small number. However as the hundreds of thousands of single operations of multiplication and addition take place this error will grow systematically through the program. As mentioned before, the problem grows particularly severe on the R.H.S. of the vertex equation when the differences of large and nearly equal quantities are taken. It would be a huge task to draw a process graph to follow the propagation of the approximate error throughout the program. Even if this was done the projected error would be an upperbound with a large possible
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95 deviation since the error would be assumed maximal at each individual step. As an alternative measure to estimate the intrinsic roundoff error, both the Main Program and the Mass Shell Program were converted to quadruple precision. The stability of the program results under the change from double precision to quadruple precision was taken to indicate that the data matrix was being supplied with numbers of sufficient accuracy that roundoff error was not a matter of principal concern. In this way the Main Program was established to be working satisfactorily. The really significant problem of error management lay in the design of the Mass Shell Program. 73 The Mass Shell Program The Mass Shell Program takes the functional form of the F, G Q , G 1 , G ? , H , H, , H ? and I functions and computes the value of the functions and all the derivatives at selected points. The vertex functions are parametrized by k 2 and are dependent on the variables p and u, where V "S (7 . 2) u = * ' ( P â€¢uâ€¢ ' 2 Due to the indefinite metric the domain of p is from Â°Â° to + Â°Â°. P 2 P/ g p p v D y (73) Therefore both (p 2 )* and (k 2 )* can take on imaginary values. This leaves a dilemma in the interpretation of the meanings of the symbols p and k used in the definition of u and in the definitions of the functions in
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96 Table 61. Initially this difficulty was circumvented, rather than 2 resolved, by merely agreeing to consider only positive values of p and k 2 . In the following chapter designs are included to allow for the negative momenta and the full range of u. Before agreeing to consider only the positive momentum squared, it was possible to tackle the real business of turning Table 61 into a reliable program which could provide a matrix of numbers representing 2 2 the eight functions at a given selection of points in (p ,y,k ) space. Also needed are the nine possible mixed derivatives of each vertex function. Let r(p 2 ,y) represent any one of the eight vertex functions. Then *r(p ,u) = y Â— dp AvftSO* W J KjTM ? (7 " 4) where P.^ and r*.& 9p In order to obtain the lefthand side of the vertex equation, i r , it i ii iii * ** *** *' ** is necessary to evaluate r,r ,r ,r,r ,r ,r ,r and * r for each vertex function. The object of the Mass Shell Program was to evaluate by some method the many types of integrals over beta which define the vertex function, and then by some other method obtain all of the above mentioned partial
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97 derivatives. At the same time the Mass Shell Program was constrained to evaluate these functions and derivatives in a way which was conservative of machine time since many evaluations at different point were going to be made. It was also constrained to be very precise in its evaluations since it had been found that the R.H.S. of the vertex equation involved the subtraction of nearly equal quantities. This required that the data matrix be reliable to more than six significant figures. The evaluation of the beta integrals was a challenge to both of these constraints since the integrands become singular at those values of the integration variable where ^ = or ^ = 1. (75) Any integration method used has to be preceded by an identification of any such singular points so that a treatment could be prescribed. A number of different approaches were made at forming the Mass Shell Program. They are divided into three main types. The first type chosen was to evaluate the functions with extreme precision and then take first, second and third order differences to form the derivatives. The second type wrote out the derivatives in terms of the analytic expressions derived from Table 61. Then the functions and the derivatives were all evaluated through integrations over a great number of beta integrals. The third type of approach was to perform analytic evaluations of the beta integrals to define the functions, then apply the method of taking differences to obtain the derivatives. Each method has its own special advantages and disadvantages which shall be further discussed. Familiarity
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98 with the peculiarities of each approach led to a prescription for the most efficient acquisition of the data matrix. Method 1 One way to determine all the scalar functions and their derivatives was to determine the functions to a minimum of 12 significant figures at a number of points and use these values to evaluate the derivatives by the method of taking differences. The main advantage of this method is that it performs the minimim number of different integrations over beta. Two principal schemes were used to perform the integrations over beta, Gaussian quadrature and a modified trapezoidal rule. For the Gaussian quadrature subroutine, ninetysix points were used. This meant that if the integrand under the beta integral could be well approximated by a polynomial of the ninety sixth degree or less, there would be no truncation error involved in the use of the subroutine. Gaussian quadrature is typically fast and efficient. The subroutine used generally gave excellent results in a small fraction of time consumed by the trapezoidal rule subroutine. However, where the range of integration included a point where the integrand was singular, Gaussian quadrature is not expected to be reliable. A monitor was needed to signal the occurrence of places where the integration needed to be performed in a different manner. This was done by writing out the analytic expressions for the first partial derivatives of each function in terms of the beta integrals and then evaluating the
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99 first partial derivations by both the integrations over beta and by the first order differences of the functions. Where the two evaluations of the first order partial derivatives did not agree to at least eight significant figures, a warning message was automatically issued. This monitoring device was very useful as an alert. However it didn't solve the problem of the singular points. It only brought them to one's attention. To deal with these special points in the integration over beta, an additional subroutine was added. This subroutine provided an alternative route to evaluate the beta integrals. It was a more time consuming path and therefore its use was flagged by the positive identification of a singular point. The problem integrals were then reexpressed so that the integrands no longer diverge. As an illustration, consider the following integral which is used in the definition of the F function, r 1 uÂ„ 'i x s X B B In the case that there is a 6 Q such that for 1 < 3 Q < 1 there exists po 1 Xo then ln (1 &) will diverge. The integral can be reexpressed. Let , X 6 , % A dl o dl Z = Â— E ; Z = Â— " and ^= ^ u 3 Â° U 6o 36 o 9e Â»6=3
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100 Then In (lZ)d M In (1Z) U 3 \  2 2 9Z Z(3=+l) Z(Bl) 9Z. In (lZ)dZ In (1Z) uÂ„ 8Z 3 o 93 2 2 9Z x r X R _2. 6 o 33 J d3 9Z (Zl) In (1Z) + (1Z) Z(3+D JZ(31) In this way, that part of the integral which the numerical technique evaluates with the greatest difficulty has been performed analytically. Another way to handle the special points was to substitute a kind of trapezoidal rule for the Gaussian quadrature. The trapezoidal rule is in general less sensitive to the occurrence of isolated special points. An efficient modification of the trapezoidal rule, called Romberg integration, was applied. The Romberg method is an example of a deferred approach to the limit. It applies the trapezoidal rule repeatedly. Each iteration alters the step size in a specific way such that the largest contribution to the truncation error at each step is cancelled by the following iteration. 25
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10' Although the Romberg subroutine was slow and expensive compared to the Gaussian subroutine, it did provide a more certain evaluation of the integrals because the level of precision desired was preset by the tolerance. This routine was committed to grind away, subdividing the area under the integrand into small and smaller trapezoids, until the required self consistency was attained. The calculation of the derivatives of the functions by the numerical method loses much of the precision which is attained by the Gaussian or Romberg integration subroutines. The nine partial derivatives of each of the eight vertex functions were calculated from the evaluation of the functions at seventeen points. The truncation error in the first order derivatives, r 1 , (let r represent any one of the eight functions and let the I indicate the first differentiation with respect to either of the h 4 V variables) was proportional to 4k r v . The truncation error in the second 4 order derivatives, r 11 , was proportional to ^ T and the error in the third order derivative, r 111 , was proportional to jr . The third order derivative suffered from the highest degree of truncation error. This truncation error could be reduced by decreasing the step size, h. However, it was found that roundoff error began to dominate when the step size was smaller than 0.01% of the variable magnitude. This established a practical limitonthe accuracy of the differentiation process which reflected the balancing of the two opposing trends of truncation error versus roundoff error.
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102 Method 2 Method 2 sought to eliminate the need for evaluating the nine partial derivatives of the eight functions by the method of taking differences. To do this it was necessary to use the definitions of the functions in Table 61 to obtain analytic expressions for all the derivatives in terms of the integrals over beta. This eliminates the serious limitations Method 1 had encountered on the precision of the third order derivatives, but it does so at the cost of introducing a host of new beta integrals to be evaluated. On the other hand the beta integrals do not have to be evaluated to the same extreme degree of precision that they had to be in Method 1 because the integrals were not later to be used to evaluate first, second and third order differences. The analytic expressions for all the derivatives are not provided here because they are lengthy. The expressions for the derivatives can be found, as they appear in the Mass Shell Program, in Appendix C. Method 2 did not include special treatment for the many new beta X B integrals for the cases where there exists a B o such that Â— = 1 or O Uo Xr, P Â— Â— = 0. Therefore a test is made for each point to be calculated to determine if 3 lies in the range from 1 to 1. In the event such a 3 q isfound, the rejection of the point is automatic when Gaussian quadrature is used. In those cases it is easier to return to Method 1 or switch to the slower Romberg integration than to try to reexpress all the new beta integrals so that S could cross the singularity.
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103 Method 3 There is a third approach possible to the problem of accurately evaluating the functions in Table 61. When it is very important to know the functions with great accuracy, it is better to perform the integrations over beta analytically. The analytic expressions for the beta integrals 2 2 are different for ^ > 4 and ^< 4. Therefore it is necessary to work m m out the analytic expressions twice. This approach was first taken by Ruben Mendez Placito. 26 He worked out the analytic expressions for the .2 eight scalar functions for K < 4 Then he a PP lied a numerical procedure m to evaluate the derivatives. By this method the value of the eight functions is precisely given, but the derivatives are subject to the same squaring off of truncation versus roundoff error which was mentioned in the discussion of Method 1. The leading terms on the L.H.S. of the vertex equations are the terms with the highest order derivatives. Therefore the error generated in creating the third order derivatives goes directly into the relative error of each equation. It is clear that the ultimate improvement upon all three methods would be to eliminate all the numerical processes of integration and differentiation. The improvement that Method 3 represents for the functions should be made also for all the derivatives. It is also clear that such a maneuver, done in duplicate 2 2 f or J< > 4 anc  J< < 4, would represent a considerable investment in m m effort.
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104 74 Summary of Results of Mass Shell Program The three methods just described were used to test the Mass Shell Solution. The expectation was that the Mass Shell Solution would provide a reasonable solution to the vertex equation in the vicinity of the mass shell. Using units which set the experimental mass of the electron 2 to unity, the mass shell is given by the sphere described by p =1, k 2 = and Â°Â° < u < Â°Â°. This region was sampled. Because each of the three methods is associatd with a different kind of approximation, the differences in the results they yielded provided information about the importance of these approximations. Around the mass shell, Method 1 provided a 1% error in the seven of the eight differential equations considered. The 1% error in the differential equations represents the accumulation of all the errors throughout both programs and therefore the systematic error in the functions can be taken to be less than IX. In addition it was found that Method 1 could not approach the mass shell too closely. The error grows dramatically if p 2 is less than 1.01 and k 2 is less than 10" . At the point p 2 equals 1.0001, k 2 equals 1 x 10" 12 , for all values of u, there is no resemblance of a solution remaining. It was not immediately clear whether this failure was due to the Mass Shell Solution itself or due to a loss of precision where it was most needed. The explanation for the problem was evident when Method 2 was 2 applied. At the same point that Method 1 totally broke down, p equals 1.0001 and k 2 equals 1 x 10" 12 , Method 2 yielded less than 2% error for the first four equations and less than 5% error for the last three
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105 2 10 equations. The error climbs only slowly as p approaches 1.0 + 1 x 10 and k approaches 1 x 10 . As the mass shell is approached the third order derivatives dominate the L.H.S. of the differential equation. Method 2 provides a better resolution of the third order derivatives and demonstrates that this is exactly what is needed to go arbitrarily 7 2 close to p = 1 and k = 0. By this same reasoning Method 3 was not expected and was not found to perform well in the very immediate vicinity of the mass shell. However at the moderate range out from the mass shell it performed reliably. It was interesting to discover that near the mass shell using Method 2 in conjunction with the Romberg subroutine, with a very high level of stability enforced, yielded the same results as the Gaussian subroutine. Apparently the integrals over beta were not contributing significant error in this region. The really important factor is control over the level of precision in the higher order derivatives. The most surprising result that came out of testing the Mass Shell Solution was the way the solution still continued to work at points of intermediate and even large distances from the mass shell. Holding k 2 = 10" 3 and increasing p 2 from 1.0 to 1.0 x 10 still found all three methods performing well. Method 3 generally had errors less than 1% while the other two methods hovered around 1%. Method 2 and Method 3 did not perform well for those values of u that caused the beta to be singular. However, Method 1 did perform properly for those values of u. It was not possible to test the Mass Shell Solution for values of 7 12 p greater than 1.0 x 10 because the size of numbers calculated exceeded the capacity of the machine and overflows resulted.
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106 ? 2 When p was held constant and k was allowed to grow, it was found that all three methods concurred that there is a natural upper bound on the allowable range of k 2 . The Mass Shell Solution consistently fails o 2 2 when k is greater than ten times p , for any value of p . In conclusion, these trials had shown that the Mass Shell Solution is a solution to the vertex equation near the mass shell as it chould be. Perhaps even more importantly it was shown to have an unexpectedly large range of applicability covering the area, 1 < p 2 < 10 12 < k 2 < 10 p 2 Â°Â° < u < Â°Â°. As a result of the testing we have a prescription for the method to be used to evaluate the Mass Shell Solution, For the Mass Shell Region Method 2, Gaussian Quadrature For the Mass Shell Region with 3 integral singularities Method 2, Romberg Integration For the Region beyond Method 1 , Gaussian Quadrature the Mass Shell. We would also predict that if Method 3 were extended so that, not only the functions were expressed analytically, but also all the derivatives, then it would out perform both of the earlier Methods in all regions. The Mass Shell Solution has presented some further questions. 2 How would the Mass Shell solution perform for values of p greater than 1 x 10 ? Is there a way to test it beyond this upper bound?
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107 2 2 Why does the solution begin to fail when k is greater than 10 p ? The next chapter addresses itself to the exploration of these difficulties.
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CHAPTER VIII EXTENDING THE MASS SHELL SOLUTION 81 A Scaling Symmetry The work described in Chapter VII found the Mass Shell Solution (M.S.S.) to be a viable solution to the vertex equations in the mass shell region and also extending over an unexpectedly large range in the 2 magnitude of p . There remained two principal areas of concern. The first was, would the M.S.S. continue to be a solution where the magni9 12 tude of p was greater than 10 , or was the gradual incline in percent 2 error (up to 3%), as p approaches the machine imposed upper bound, an indication of the failure of the analytic approximation. The second concern was what was the cause of the upper bound on the magnitude of 2 k and would an understanding of the cause lead to a prescription for its resolution? In the course of addressing these matters, the M.S.S. was modified to make the examination of the vertex equations more flexible. The vertex function is a dimensionless function of the electron momentum p, the photon momentum R and the experimental mass m. For convenience, the mass is normally taken to be unity. It is possible to let the fourvector k" define an axis of the coordinate system. The significant quantities then are the magnitude of k relative to the mass, the magnitude of p relative to k and the angle between p and k, denoted u. The indefinite metric introduces the inconvenient feature that fourmomenta squared need not 108
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109 be positive definite. In all of the previously described work the problem 2 2 was circumvented by considering only those values of p and k which were greater than zero. No difficulty was then encountered in evaluating u nor in evaluating the functions described in Table 61. However, a real 2 2 advantage of allowing p and k to take on negative values simultaneously is that no singularities are encountered in the evaluation of the beta integrals. With no singularities present, the Gaussian method of integration could be used with impunity. Any failure of the M.S.S. to fit the vertex equations could not be attributed to an awkward estimation of the beta integrals and a more definitive test of the M.S.S. could be made. The strategy employed to extend the M.S.S. into the range where the square of both momenta were negative was to regroup the products of all ? 2 the momenta so that, nowhere, do the terms p = (p ) 2 and k = (k ) 2 appear. Two new terms were introduced called pk and pok to replace p times k and p divided by k. X p x k Â•+ pk where pk = Â± (p k ) 2 (81) 7 2 p/k + pok where pok = (p /k ) 2 (82) The Â± sign in the definition of the variable pk is taken to be positive when the electron and photon momenta squared are both positive. Conversely, the Â± sign is taken to be negative when both squared momenta are negative. This reflects the fact that the product or quotient of pk and pok should 2 2 recover the sign of the negative momenta when both p and k are negative. Odd combinations of momenta occur in the F, H , Hj and H 2 functions as they appear in Table 61. This is easily repaired by multiplying the
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no ox 2 1 functions by an additional (k ) 2 . The addition of (k ) 2 is compensated for by dividing the F, H , H, and fL functions by the same factor where they appear in the vertex equations. Thus the notation used to express the transverse component of the vertex function is changed in a small way to now be ' r x (p,p + k) =Â£F(p 2 ,u) + y\(p 2 ,u) +Â£/ G^p^u) A X, , H + ^G 2 (p 2 ,u) + p A [0,k] fc (p 2 , u ) H H + [/.*] pj; (P 2 .") + h A ,k] (p 2 ,u) '*Â™V V ,/ a i(p ! ,Â«). (83) The expressions of the functions has also taken on a slightly different form. Table 61 is now to be replaced by Table 81 which includes this and other extensions, yet to be described. No attempt has been made to generalize the Main and Mass Shell Programs to handle the alternate possibility of momenta of opposite sign, 2 2 that is, p = k . It is not expected that this should incur any special obstacles other than the rather considerable initial inconvenience of tracking through the programs a large number of sign changes. 'The bar over the eight vertex functions is to remind the reader that the expressions in Table 81 should be used, not the earlier version of the expressions found in Table 61.
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Ill The next extension which was made was one that allows the study of points taken for p 2 /m 2 > 10 12 . Because the vertex function is dimensionless, it is invariant under scale transformations which treat equally the electron momentum, the photon momentum and the mass. Thus if a solution could be obtained for the coordinates (p 2 , k 2 , m 2 , u) = (101, IxlO" 3 , 100, u), (84) this would be same as the solution at the point, (p 2 , k 2 , m 2 , u) = (1.01, IxlO 5 , 1.0, u). (85) 9 9 9 It is the relative magnitudes of p , k and m which identify a point for some particular value of u. The angle variable, u, is independent of the scale changes. Although it is not possible to evaluate the vertex at p 2 > 10 12 because of the high powers of p accumulated in the course of the Main Program, it is possible to let m become arbitrarily small relative to p 2 and k 2 . The mass appears only up to squared powers and doesn't threaten the capacity of the machine. The solution at the point (p 2 , k 2 , m 2 , u) = (1.0, IxlO" 10 , IxlO' 80 , u) (86) is equivalent to the point (p 2 ,k 2 ,m 2 ,u)^ (IxlO 80 , IxlO 70 , 1.0, u). (87) By this strategy it is possible to test the M.S.S. at points which are ? 2 80 equivalent to those which have p /m attain the limiting value of 10 or less.
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11 This scaling symmetry was used to test the validity of the M.S.S. in the asymptotic p region. It was found that the M.S.S. as represented by the functions in Table 61 loses its resemblance to a solutipn rapidly as 2 2 12 the magnitude of p /m grows larger than 10 . Before going on to describe the work that was done to correct the 2 ... difficulty at asymptoticly large p , it may be worth mentioning in passing that the scaling symmetry provided a handy diagnostic tool for detecting bugs. After any major alteration of either the Main Program or the M.S.S. Program it was found useful to test them out at a set of equivalent points. Even small and hard to find errors were unequally treated at such equivalent points and their occurrence could be readily detected. This tool was routinely applied after any major changes in the programs. 2 82 The Large p Region 2 The deterioration of the accuracy of the M.S.S. for large p begs for a reexamination of the premises upon which its derivation was based. In arriving at the solutions described in Table 61 it was assumed that Y was the dominant term in r . It was assumed that all other terms were smaller by at least one order of magnitude of the fine structure constant. 7 2 2 For large values of p (relative to k and u ) , this approximation is not in contradiction with the value of the vertex function obtained from the M.S.S. In fact, as p grows very large, all the functions except G Q decrease at a rate of 1/p or faster. The function, G Q , approaches a constant equal to unity. Therefore the approximation used to derive the 2 M.S.S. not only holds but is actually better in the asymptotic p region
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113 than it is in the mass shell region. By this criterion alone the M.S.S. 2 should be expected to work in the asymptotic p region. However, in deriving the M.S.S. another approximation was made. It was assumed A(p^) A(p) m, (88) where m is the experimental mass. This approximation is completely 2 wrong in the large p region since as p 2 +~, A(p^) + and A(p 2 ) 0. (89) In deriving the M.S.S. as it appears in Table 61, the approximation in Eq. (88) was used to provide a simplified expression of the D A , D A , D A and D A tensors (first defined in Eq. (67)). If the approxy' yv yvp imation had not been employed the tensors, D , D , D and D , would have taken the form, D A = [A ]P A + A 2 p A ]/D 12 < = C^VPl h )6 l + ( hy + P 2 y)P^ /D 12 D yWA 2 P lv V 2 X (VlyA lP2y^>!2 C = [(p ly p 2v " P 2 yPlX + (Plv P 2pP2vPlp )6 y + (p. p 9 p, p, j6 A j/D, 9 (Bll) v Hp 2y K 2pHu v 12 where D ]2 (p^A^) (p 2 A 2 ) and A ] = A(p ] ), A 2 = A(p 2 ) instead of that given by Eq. (611 ).
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114 These new expressions for the D X , D A , D A v and D A tensors lead to a new and necessarily more complicated set of differential equations which have to be solved to define the C A , C A , C X and C A tensors, which in turn, would lead to an identification of the F. G , Gj , G 2 , H Q , H^ , H 2 and I functions. As an example of how this less restricted form of the D tensor affects the problem, consider the new equation for the C tensor which leads to the identification of the F function, V 2 C X = 3e A ]P A A 2 pÂ£D 12 D 12 j (810.; It is impossible to solve this equation by the same methods employed in Chapter VI and Appendix B. The transformation functions given in Eqs. (B21) through (B30) no longer apply. An alternative to resolving these more complicated expressions for the eight vertex functions is suggested by the observation that the mass 7 2 2 function, A(p ), is slowly changing. The derivatives of A(pj ) and A(p 2 ) ? 7 are \iery small and A(p,) and A(p 2 ) will behave much like constants over a finite range of p . Furthermore A(pj ) is approximately equal to A(p 2 ) 7 7 where the ratio, p /k , is large. A useful adulteration of the M.S.S. would be to replace m 2 by A^ 2 and m by (A 1 +A 2 )/2, wherever they appear in the definition of the functions. This crude recasting would not be 2 2 expected to be favorable in any region where k >> p (because A(p?) i A(p 2 )). However, it has already been found that the M.S.S. is 2 2 not valid where k > p , so no loss will be suffered on that account. Furthermore, since as the mass shell is approached, the mass function
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115 approaches the value of the mass, the original M.S.S. will be recovered and no harm will be done to the solution in the region of the mass shell. Some tinkering was done with this fomula. The best results were obtained when the replacement of m by AjA 2 and m by (A+A 2 )/2 was made almost everywhere. The exception lay in the CK function which appears 2 in Table 81 and which defines the vertex function G Q . The m in the natural logarithm is not altered because when it was altered it introduced a small error which grew in inverse proportion to the scale of the mass. The changes that were made are summarized in Table 81. Table 81 represents the Extended Mass Shell Solution (E. M.S.S.). The E. M.S.S. performs dramatically better than the M.S.S. Figure 81 has been prepared to illustrate how extensive the solution is. ? 2 For any relative value of k to m the graph plots the dependence of 2 2 2 2 2 the solution on the variables p /k and u k /p . The error is less 2 2 2 5 than three percent in the region bounded on the right by u k /p =1x10 ? ? 3 and on the bottom by p /k = 1 x 10 . However, when the approach to the mass shell is extremely close there is a rise in the error in the sixth and seventh vertex equations which was attributed in Chapter 7 to the difficulty in obtaining extremely accurate third order derivatives in the function G . Otherwise the functions represent a very regular and predictable solution. The solution is marred by only the limitation that u 2 k 2 /p 2 < 10 5 and p 2 /k 2 > 10" 3 .
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116 TABLE 81 THE EXTENDED MASS SHELL FUNCTIONS F =  e(A 1 +A 2 )^/l 1 d3 I = { a/l, dB G o = 1 + f/ I 2 dB T CK !/ i 3 I, df 2 r T x Io df v f p 7 G 2 =  (2p 2 u + *)[/l> +/lJdB] ~  ^/l X 3 dB V^fe (A 1+ A 2 )k 2 p 2 /l^ X d3 2 H l = 4 (A 1+ A 2 )[Â£k 2 /l^ X 3dB 2(p 2 k 2 u + Y)/ T 5 X d6] H 2 =^(A 1+ A 2 )k 2 [/l X dS + ^/l 6 dB] + ^(A 1+ A 2 )k 2 (f + ^)[/l^ X d3 + /l XX 3d3] JL I4 64 k 4 (A 1 +A 2 )/lg X 3 d3 9 9 1 Â£ = (pV) 2 pk i 2 b 2. (P 'k . i r& 'l^lxf'l 1 ^ 1 vi'^&M 1 ? i 3 = ' fi iV"B'^!Ci' 1 ,n "" ^ ii
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TABLE 8 1 (Continued) 117 V^lW T a ln i 1 u CK (1^lnfl ^1) + (1 m ~2 P2 In 1 "7 m 2y ?! 1 " , 4) t i ,n ( 1 2 "> I' 5 1 Vo*c '"f I' V VHpf' 1 In 1 L 2 (z) in (1z)f 5 y 3 L > M 13)4 + ~>; ln(l i) + ^ r xx 1 where L x 2 6 v V x, l i i \ 3 2 4 ln x. 1 * v 3 1 x 6 = [p +^(B+l)k] 2 ^ = A 1 A 2" n3 2 )k 2 a = fine structure constant
PAGE 123
x 118 ~1 N^ '2 x or o a: a: UJ N ? D '2 o o' i ml x
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lift 2 83 The Large k Region It is not difficult to see that the solution represented by either Table 61 or Table 81 is not going to be a valid solution to the vertex 2 equations when the magnitude of k grows very large relative to the magni2 tude of p . A necessary condition for the self consistency of the M.S.S. is that it should be dominated by the leading term G = 1. However, since the functions F, I, H , H, , Ho and G 2 are proportional to the first and 2 second powers of k, it is indicated that for fixed p and sufficiently large k, these functions will grow to dominate over the contribution 2 from G = 1. Actual determinations of the M.S.S. in the large k region 2 3 2 (that region where k > 10 p ) do show these functions are much larger than G = 1. In addition, when the M.S.S. was extended by the replace2 ment of m by (A,+Ap)/2 it was assumed that p, was not greatly different 2 2 2 2 2 2 from p 2 . This corresponds to assuming u k  < p  and k  < p . For these reasons, it is easy to see that neither the M.S.S. nor the 2 2 2 E. M.S.S. should perform properly in any region where u k  Â» p  or k 2  Â»  P 2 . The correction to the problem is easily proposed though the actual achievement of the correction may be very much more difficult. To correct the problem would necessitate returning to the beginning and assuming some 2 approximate form for the large k behavior of the functions F, I, H , H, , Hp and G~. The differential equations from this approximation will be too complicated to solve directly. As was done in Chapter VI and Appendix B, the closest approximation to the real equations which is solvable would have to be used as an initial guess. Finally the new functions would
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120 have to satisfy the boundary condition that they should smoothly join the functions in the E.M.S.S. Basicly the problem is that the solution that we have, has built into it certain expectations about the relative magnitudes of the variables. The solution could be said to have a welldefined region of convergence. A trivial example is useful to illustrate this difficulty. Consider a function, f, of two variables p and q, given by f(p,q) = 1 2p3q' (81T) Occasionally a function of the sort is needed in the form of a series expansion, f(p,q) ^ 1 2p + [2pJ Â•Â•' (812) Some finite number of terms in the expansion will represent the function to a particular level of accuracy as long as Â£ < *. Thus, f(2,l) could p o be calculated by Eq. (814) but f(l,2) could not be. If the value of f (1 ,2) was desired, it is necessary to return to the basic function, Eq. (813) and rewrite the expansion so that, f(p,q)=4Jl + (â€¢) &?.. *?+{% 3q (813) In this illustration it is of course exceedingly simple to write out the second expansion. If the second expansion was harder to come by, it might be faster to look for a symmetry in the function in Eq. (813) which could provide a relationship which could predict the value of f(l,2) in terms of the first expansion. It is clear that the righthand side of
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121 3 Eq. (813) is unaffected under a transformation which replaces p by ^ q 2 and q by ~ p and reverses the sign of the whole expression. Thus, f(p,q) = f(q,p), (814) 1 and therefore, f(l,2) = f (3,2/3) . The value of f(l,2) may be obtained by evaluating the negative of f(3,2/3). The value of f(3,2/3) can be obtained from the first expansion. It can be seen then that if the second expansion is for some reason difficult to procure, the first expansion plus a symmetry relation will serve the same purpose. It would be most useful if some kind of symmetry could be found to evaluate the vertex functions, outside of the region of three per cent error, in terms of the functions at a point inside the region of three percent error. With this thought in mind consider the relative magnitudes of the incoming and outgoing electron momenta of the vertex function r' i (p,q), where q = p + L The incoming electron momentum squared is p , and as long as p /k > 10~ and u k /p < 10 , r y (p,q) can be determined 2 2 2 from Table 81. Yet the outgoing momentum, squared, q =p + 2p*k" + k , is not restricted like the incoming momentum, 2 <%<<*>. (815) vr If there were a symmetry relation between r^(q,p) and r^(p,q), then the vertex could be evaluated for cases where the incoming momentum squared 3 2 was less than 10 k . There is an intuitive feeling that there should be a relationship between the amplitude for a process that goes in with momentum p and
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122 out with momentum q and the amplitude for a process that goes in with momentum q and out with momentum p. 27 In a report by A. A. Broyles, the symmetries of the vertex function were exhaustibly analyzed. He used the following abbreviated notation for the complete vertex, r y (p,q) = F*(p,q) + Y V GV"PÂ»q) + ^H^ (p.q) LT + iY 5 Y V I^,(P,q). (816) For comparison, the complete vertex from Eq. (217) is repeated here with the notation consistent with the use of Table 81 to define the eight transverse scalar functions, and consistent with Ward's Identity having been applied to identify the longitudinal scalar functions. ,y rj T (p,q) =jk F(P 2 .k 2 Â»u,m) + y y G (p 2 ,k 2 ,u,m) + A ^(p^.u.m) +^ G 2 (p 2 ,k 2 ,u,m) 2ip^ a V B 4r ( P 2 * k2 ' u ' m) 2 ^\ ^(P 2 .k 2 Â»u,m) P K K + ^ [A(q 2 ) . A(p 2 )] _i^ + k^. (8 . 17 ; T Because the reader has become accustomed to notation representing the transverse vertex by r^, the designation T[_j will be used to signify the complete vertex equation.
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123 For future reference we will equate Eqs. (816) and (817) and make the following identifications. FTp,q) = jÂ£ F(p 2 ,k 2 ,u,m) + iÂ£ [A(q 2 ) A(p 2 )] y G y (p,q) = g^G (p 2 ,k 2 ,u,m) + Â—Â£ G, (p 2 ,k 2 ,u,m) P P\ ,,2,2 , k V k "P + p1T G 2 ( P ' k ' U ' m) 2~ + T2 77 77 H y vn (p,q) = 2ip\k n ^ (p 2 ,k 2 ,u,m) 2ig y ) P n ^ (p 2 ,k 2 ,u,m) P K ^2 ,2,2 2ig u k 4 (p ,k ,u,m) 3 Â«v n 1,2 ^ k I^ v (p,q) ie yaB v P 6 k a ^ (p 2 k 2 u,m). (818) Having established the new notation in relation to the more familiar one we now examine the symmetries of Eq. (816). Under charge conjugation of the Y y matrix transforms in the following way, (C Y )Y y *(C Y )" 1 Y y , (819) u r .2o where C = iy Y Â» and since YÂ°Y y yÂ° = y (where T represents the transpose operation), c y c=Y yT (8 " 20) The vertex function transforms under charge conjugation in the same fashion as Y y does , C'V^ C = r*J . (821)
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124 This places the following restrictions on the tensor functions F y , G y , v H M and I M , Â•v *v F y (p,q)* = F y (p,q) (822) GMp,q)* = G y ,(p,q) (823) H^(p.q)* = H^ vn (p,q) (824) lÂ» (p.q)* = iMp.q). (825) The y y matrices are selfadjoint and the vertex function will transform like y y so that I* T (p,q) = Y Â°r^ T (q,p)* T Y Â° = rj^p.q) (826) where the bar over r y stands for the adjoint operation. When the adjoint operation is applied to Eqs. (822) through (825) the following relations are obtained, F y (p,q) =F y (q,p) (827) G y v (p,q) = G y v (q,=p) (828) H^(p,q) = H y vn (q,p) (829) iMp.q) =I y (q,p). (830) Thus rf T (q,p) = F y (p\q) + Y V G y v (p,q) + a vn H^ (p.q.) i Y V lVJ (P.q). ( 8 31 )
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125 Eq. (818) can be used to identify the tensors on the right of Eq. (831) in terms of the scalar functions. We find rf T (q,p) ={Â£ F(p 2 ,k 2 ,u,m) +^ [A(q 2 ) A(p 2 )] + Y \(p 2 ,k 2 ,u,m) + p^ 1 (P 2 >k 2 u,m) P >Â§ G 2 (p^, U ,Â™,f ,f + 2ia a6 p k R P y T2 (P 2 ' k2 ' u ' m ) + 2ia ya p a ^ (p 2 ,k 2 ,u,m) + 2ia y \4 (p 2 ,k 2 ,u,m) " eyaBV A v P 3 k a X( p 2 >k 2 jU , m) . (8 _ 32) We return to our principal interest in the transverse components of the vertex only. Furthermore, because Figure 81 holds for any ratio of 2 2 k  to m the eight transverse scalar functions are identified by the 2 2 variables which form the axis of graph in Figure 81, namely, p /k and 2 ? 2 2 2 u k/p plus a statement of the magnitude of k /m . Therefore, we will write the transverse vertex as rVi (q.P) =jÂ£ F(p 2 /k 2 ,u 2 k 2 /p 2 ) +Y y G n (p 2 /k 2 ,u 2 k 2 /p 2 ) pk "( U ~2 W /* Â» u * IV I T V "> "pk + p^ 4 (pV.uV/p 2 ) + p y k J (p 2 /k 2 ,u 2 k 2 /p 2 ) P u p^WiA (p 2 /k 2 ,u 2 k 2 / P 2 ) pV
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126 [Y y ,0] 4 (p 2 /k 2 ,u 2 k 2 /p 2 ) [y y ,k]  (P 2 .k 2 ,u,m) e^ Y 5 y v P 3 k a i R (p 2 /k 2 ,u 2 k 2 /p 2 ) (833) 2 2 for q = p + k" and for some fixed k /m . A comparison with Eq. (83) reveals that the difference between the expressions r y (p,q) and r y (q,p) is only that the signs of the H Hi , TL and T functions have been reversed. 2 2 2 2 2 r y (q,p) is a function of the variables q /k and u k /q for a 2 2 particular k /m . u Q = ^\^ (834) q (q 2 ) 2 (k 2 ) 2 It has been related to a particular combination of the eight scalar func2 2 2 2 2 tions which are defined by the coordinates p /k , u k /p for the same 2 2 k /m . Since q = p + k, it follows that 2 2/ 22\J S^J2^ 2 (JCR) + 1 (835) and 2.2 . ? ,2 V = [(u 2 p 2 /k 2 ) 2 + 1] 2 (%V ( Q 36 ) q Vq / 2 2 2 2 2 We would like to show that any value of q /k and u k /q outside 2 2 2 2 2 of the three percent error region maps onto a point (p /k ,u k /p ) in the interior of thethree percent error region. To do this we will examine 2 2 three cases. In the first case we can select q /k to have some
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127 2 2 2 5 arbitrary value and show that u k /q increasing from 10 to infinity 2 2 2 2 2 maps to a line of (p /k , u k /p ) points in the one percent region. Consider Figure 82 which is a replica of Figure 81. The vertical axis describes the ratio of the incoming momentum squared to the photon momentum squared. The horizontal axis is the quotient of the square of the angle variable for the incoming momentum and vertical axis variable. The shaded area covers the three percent error region. For our example, let q 2 /k 2 = b 2 and let u 2 k 2 /q 2 = 10 5 a 2 . When a is allowed to vary from one to infinity a line of points is drawn on the graph, denoted by the arrow. 2 2 2 2 2 Next we solve for p /k and u k /p , p 2 /k 2 = b 2 + 2(a 2 10 5 ) i b 2 + 1 and ,2.2,2 _ T(a 2 IP 5 )" b 2 l] 2 ry + 2(a^ lcn 2 b ir For any fixed value of b, varying a from 1 to Â°Â° draws a line of 2 2 2 2 2 points with increasing p /k and u k /p approaching 1/4. This line is the new arrow that appears in Figure 83. By a similar analysis it can be shown that for a fixed value of 222? 22 32 u k /p = b and a value of q /k = 10 /a where a ranges from 1 to Â°Â° q ~~ a line is drawn from the lower boundary of the one percent region. The line appears in Figure 84, pointing vertically down. The infinite line 2 2 2 2 2 of (q /k , u k /q ) points is mapped onto a short finite line of 9 9 9 9 9 (p /k , u k /p ) points which approach the value (1,1).
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128
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129
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130
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131 2 By repeating this process of drawing horizontal lines for b fixed and vertical oines for a fixed and negative, it is possible to see how the (q 2 /k 2 , u 2 k 2 /q 2 ) region maps onto the (p 2 /k 2 , u k /p ) region. In Figure 85 an illustration is given of this relationship. In conclusion we find that Table 81 gives a description of the 2 2 3 solution to the transverse vertex equation for the region p /k > 10 and u 2 k 2 /p 2 < 10 5 . It is possible to evaluate the solution outside of the region by the relation given in Eq. (833) for r y (q,p) where the coordinates of incoming momentum q are related to those of an incoming momentum p by Eqs. (835) and (836).
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132 \
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CHAPTER IX THE CONCLUSION A solution has been found to the SchwingerDyson equations based on an approximation scheme which is characterized by the following: (1) the photon propagator is approximated by its form near the mass shell, (2) the infinite hierarchy of the vertex equation 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 diagrams. It was found that,in order to obtain a finite solution to the electron propagator, the photon propagator had to be fixed in the Landau gauge, The bare mass of the electron was found to be zero as predicted by 1 3 Johnson, Wiley and Baker. Unlike Johnson, Baker and Wil ley., who concentrated on finding the asymptotic form of the electron propagator, a complete solution for all momenta was found. It took on a very simple form, S(p) n ! o. (92) A(p^) 4 B(p^) The functions A and B were defined in Eqs. (315) and (316). An analytic solution to a simplified approximation of the vertex equation was found by H.S. Green. His solution forms the basis of the 133
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134 Mass Shell Solution. This expression formed an extraordinarily successful solution to the complete vertex equation (approximate only to the extent of points (1) and (2) above). This analytic solution was represented by a numerical process and tested for its range of applicability. It was found to be a good representation of the solution, not only in the region for which it was designed but also in the region of an electron momentum squared as large as 10 . This solution was then extended by making some deductions about how the solution should behave in the asymptotic region. The Extended Mass Shell Solution was found to be a good solution for the range of the variables, p 2 /k 2 > 10" 3 u 2 k 2 /p 2 < 10 5 , (93) where p is the incoming momentum squared of the vertex, r y (p,p+R), k is the photon momentum squared and u = p Tc/(p k ) . The Extended Mass Shell Solution provides an identification of the transverse vertex function. r y (p,p+K) ={Â£f(p 2 A 2 , u 2 k 2 /p 2 , k 2 /m 2 ) + Y y G Q (p 2 /k 2 , u 2 k 2 /p 2 , k 2 /m 2 ) + ^G 1 (p 2 /k 2 , u 2 k 2 /p 2 , k 2 /m 2 ) P _ + P y ^
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135 + [Y y ,0]^ (pV, u 2 k 2 /p 2 , k 2 /m 2 ) + [y y .k] (p 2 /k 2 , u 2 k 2 /p 2 , k 2 /m 2 ) k* + ^ y\p v k a ^ (pV. u 2 k 2 / P 2 , k 2 /m 2 ) (94) where the eight scalar functions F, G, Gj , G,,, H Q , Hj , H 2 and I are defined in Table 82 and pk is defined in Eq. (81). The longitudinal part of the vertex is given by ^(p.p+R) = Â£ [A((p+fc) 2 ) A(p 2 )] ty + ^l . (95) long k k k The above description of the transverse part of the vertex fails outside the region bounded by Eq. (93). However a symmetry principle can be used to prescribe the value of the transverse vertex at a point, (q 2 /k 2 , u 2 k /q , k An )Â» outside of the region, in terms of the eight 7 7 7 7 7 2 2 scalar functions evaluated at the point, (p /k , u k /p , k /m ). In this case T y (q,p) ={Â£F(p 2 /k 2 , u 2 k 2 /p 2 , k 2 /m 2 ) + Y \(pW. u 2 k 2 /p 2 , k 2 /m 2 ) + Â£^G 1 (p 2 /k 2 , u 2 k 2 /p 2 , k 2 /m 2 ) P + p^^ (p 2 /k 2 , u 2 k 2 /p 2 , k 2 m 2 ) P y [M] A (p 2 /k 2 , u 2 k 2 /p 2 , k 2 /m 2 )
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136 [Y y ,Ff]^ (p 2 /k 2 . A 2 / P 2 , kV) e ^ ^Vv k a h (p2/k2> u2k2/p2 ' k2/m2) (9_5) where the point (q 2 /k 2 , u 2 k 2 /q 2 ) is related to the point (p 2 /k 2 , u 2 k 2 /p 2 ) by Eqs. (835) and (836). This solution represents a substantial improvement over all earlier 15 attempts to find a nonperturbative solution to the vertex equation. Work by other researchers has been characterized by a dependence on renormalization techniques to keep the solution finite, or else only asymptotic forms of the eight transverse functions were sought, usually involving more drastic approximations than have been applied here. Furthermore, due to the great complexity of the equations, typically only the asymptotic forms of the T, G , G, , and GL were solved for.. Thus, the Extended Mass Shell Solution to the vertex equation is sharply contrasted with earlier solutions by its completeness. It cannot be said that the combined solutions, described here, for the electron propagator and the vertex, constitute by any means a complete resolution of the uncertainties present in the theory of Quantum Electrodynamics. However, the success of this method, to this level of approximation, is encouraging evidence that the unreasonable infinite quantities that occur in perturbation calculations of the selfenergy of the electron and the vertex part are not essential. This work has demonstrated that a consistent, noninfinite solution for the electron propagator and the vertex can be found given an approximate form for the photon propagator.
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137 Future work would look toward solving all three of the SchwingerDyson equations simultaneously for the three basic functions, the vertex, and electron and photon propagators. The present work gives rise to favorable expectations that a simultaneous solution to all three equations would continue to be characterized by the absence of any divergences.
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APPENDIXES
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APPENDIX A DIRAC GAMMA MATRICES: DEFINITIONS AND PRODUCT RULES P. A.M. Dirac constructed a relativisticly covariant equation for the motion of the electron. In order to achieve this, the form of the equation needed to be invariant under a Lorentz transformation. An invariant scalar quantity is expressed as a product between a contravariant and covariant vector. x^x = g xV (Al) u 3 UV The metric tensor is defined by, g =0 for u f v y,v = 0,1,2,3 9 jj = 1 for j = 1,3 (A2) 9oo = ' The contravariant metric tensor is defined by the relation, g uv g . (A3) The contraction of g yot g is known as 6 y . 6 U = g ya g . (A4) It follows that 6 M = for u f v v 6 V 1 for u = v . v (A5) 139
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140 The Dirac equation for the relativistic particle with spin needed to be linear in both time and space derivatives alike and needed to correspond to the second order KleinGordon equation which provided for the correct energy momentum relation for a relativistic particle. To accomplish this, a familiar representation of the Dirac equation (although not the original one), employs a set of matrices called Dirac gamma matrices. o Y " Y J 1 oL0 1a j " _a j J (A6) where cf 1 represents the Paul i spin matrices. The Dirac gamma matrices are related to the metric tensor by an anti commutation relation, V v . v u _ uv Y Y + y Y = 2 9 Â• (A7) All possible products of gamma matrices constructs a group of 16 linearly independent 4x4 matrices. These are U Y y '
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141 The definition of y is not standardized throughout the literature. g This designation agrees with that used by J. M. Janch and F. Rohrlich but differs from that used by J. Bjorken and S. Drell by an i. In working out the expansion of Eq. (416), products of the sixteen matrices are taken repeatedly. A multiplication table for all possible products was constructed and appears in Table Al. In constructing the table the following definition of the Levi Civita antisymmetric tensor was used, e . . = unless a, 6, y and 6 are all different '1234 1234 e '+ 1 for even permutations of the indices 1 for odd permutations of the indices "1 for even permutations of the indices (A10) .+ 1 for odd permutations of the indices The following identities have proven useful: 4e Aay \ 5 Y({) = {y a ,[y\ y y ]} (Al 1) where { } enclose the anti commutator and [ ] enclose the commutator. e X * YV e Aor a^ 26 A c^ + 26 v X Av + 2g v a yX (A12) e^ e ^ a^ 4a Y \ (A13)
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142 o iÂ—
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APPENDIX B DERIVATION OF GREEN'S MASS SHELL SOLUTION Bl The Vertex Equation Near the Mass Shell The equation for the vertex function is vV( Pl ,p 2 ) = 4 y v F VA + 3 V 7 _1 F VX ) (Bl where and f vX = s( Pl ) *Â£[r A ( Pl ,p 2 )s(p 2 )] s _1 (p 2 ). (B3) This eauation can be modified slightly by the approximation f vX + s( Pl )r x ( Pl p 2 )s(p 2 )Y v . (B4) This will be a good approximation in the region of the mass shell where r approaches y and S (p) is the slowly changing function, pf + A(p ). The following definition is introduced, a x = s( Pl )r x ( Pl ,p 2 )s(p 2 ). (B5) Now a solution is sought to the modified vertex equation, V 2 r X = e ()y v A X y V + d v fh\ v ). (B6) 143
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144 This equation is expected on the basis of the approximation in Eq. (B4) to be a faithful representation of (Bl) equation near the mass shell. The following tensors are defined C X = \ tr [r A ] (B7) c i = I tr [r \ ] (B_8) C A =itr [r\ ] (B9) yv 4 L 'yv J C 4tr [r\ ] (B10) yvp 4 L ' yvp J D X =Â±tr[A A ] (B1D D X =Â±tr[A\] (B12) D A = i tr [A A Y ] ( B " 13 ) yv ^ L 'yvp J D * l tr[A \ ] (B14) yvp 4 L 'yvp1 where Y yv = 1 ^ Y y ,Y v^ Y yvp = 2 {y y' Y vp } " Take the product of Eq. (B6) with each matrix, n, y Â» Y yv and y and perform the trace. This converts Eq. (A6) into a system of 'yvp v four equations: V 2 C A = 3eD X (B15a) V 2 C A = 2eD A 2ed v" 2 9 V D A (B15b) y y y v
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145 V 2 C A = e D A 2 E 3 V~ 2 3 P D A yv yv y vp + 2e3 V~ 2 3 P D A (B15c) v yp V 2 C A = 2 e 3 V _2 3 a D A yvp y vpa 2e3 V _2 3 a D A v pya 2e3 V" 2 3 a D A (B15d) p \ivo. The electron propagator is given by 0, + A, ^o + A ? S(pO =~k 4 and S(pJ = f Â£ (B16) p 2 A 2 P 2 A 2 where 2, A, p 1 = p + R, p 2 = p, A(p 1 ) = A,, A(p 2 ) r, 2 and the A function is the electron propagator function which was 2 2 2 2 determined in Chapter III. Allow the abbreviation D,p = (Pi" A i ) (p 2 A 2 ) Â• When this is done the D tensors are expressed as D X =\ tr [r A (p( 2 +A 2 )(^+A 1 )]/D 12 = [(pfPg +A 1 A 2 )C A + (A^ + A 2 pV)C A " PlPKv ]/D 12 ( B 17 a ) D y = 1 tr Cr X (p* 2 +A 2 ) Yy (0 1 +A )]/D 12 = [(A 2 P 1y + A lP2y )C A + (A 1 A 2 p r p 2 )C A + (P ly pJ ^ P 2y P^)cJ + (A 2 pfA lP pc A v+ p> p C A vp ]/D12 (B17b)
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146 = C(PlyP2vP 2 yPlv )cA + (A 2PlvA lP2v )C i " ( Vly " V2X + (P n P 2 + A,A 2 )CJ; V " (PlvP2 + P2vPXp + (PlyP2 + P 2 yP?> cX vp + (A 2 pf + A ]P P)C y A vp ]/D 12 (B17c) D y X vp = l tr[r VVW l +A l )]/D 12 = [( PlyP2v " P2yPlv )C p + { Plv P 2p " ?2 Pi K +( PlpP2yP 2p Ply) C i +(A 2Ply +A lP2y) C ip + (A Pi +A n p Q )C A + (A p n +A,p, )c\, v 2Hv l r 2v py 2Hp r2p' yv (A,A 2 Pl P 2 )C^ p (p,^f*1Vl)C + (PlvP2 +P 2vPlÂ» C ppa + (PlpP2 + P 2 pPl'C < B 17 d > The C tensors can be expressed in terms of the eight scalar functions of the transverse vertex, where r. is given by
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147 the associated tensors are C A =P X ^ (B19a) C A = 6 A G n + p A p I + p A k 4 (B19b) C A = (p A p k p A p k ) *% + (6 A k 6 A k ) 2 4 yv v ^ ^v y y V .22 v v y y v' .2 + (6 A P 6 A p ) r^(B19c) v v\i y K v' kp v ' C A = [6 A (k p k p ) + 6 A (k p k p ) yvp L y v p r v v r p v v y r p p r y + 6 A (k p k p )] J" Â• (B19d) p v v*y y K v /J kp The system of equations contained in (B15) is still a heavy challenge to analyze. An additional simplification is desirable. It will be assumed that the G term in r is dominant and has its near the mass shell value of 1, and that the functions A, and A~ will be closely approximated by their on the mass shall values of m , the experimental electron mass. These are exactly the assumptions which were made in approximating F v by A y in Eq. (B6). No new restrictions to the solution have been made. The effect of G being dominant and equal to 1 is C becomes 6 and the contributions of C , C and C are small relay y yv yvp tive to C and can be neglected in Eq. (B17). The new simplified version of (B17) is D A = 2 m p A /D ]2 (B20a) D A = [(m 2 p r p 2 )6 A + (P ly +P 2y )p A ]/D ]2 (B20b)
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148 C =m[(P 1v" P 2v)<(PluP 2 X ]/D 12 (B20c] D L = [(p lu P 2/ P 2uPlv )6 p + (PlvP2pP 2 vPl )6 A P M + (p, p p p, )6 l/D,0. (B20d) The new expressions for the D tensors will be used to solve the four equations in (B15). To facilitate the solution observe the following general relationships. Let x = (x+a) 2 = x 2 + 2xa + a . a If then Let ,2 *a r*(j) = Â— ^ x m a Â»A) =1(1^) m (1% ma m (B21 (B22) x D = [p+ (3+1)4 R]' and u = m 2 (l3 2 )k 2 . '2 (B23) If A " (p 2 m 2 )(p 2 m 2 ) then d3 1 (x 3 u 6 )' 1 P dB ln n \ (B24) (B25) If ln V^4> 2 2 Pm Pom :b26:
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149 then K=l dMb&) '1 d3(l/) In (1/). (B27) if v\f_' i^S )ln(1 ^ * then */: where so that L 2 (z) d$(mu a ) In (1z) W" O^lnO^) dz (B28) (B29) (B30) ^o L 2 (z)dz = z L 2 (z) + (z1) In (1z) z. Using these general transformation rules, the system of Eqs. (1315) can be solved. B2 Solution to Equation (B15a) Substituting the definition of D in (B20a) into Eq. (B15a) yields V 2 C A 6emp* (p2m 2 )(p2m 2 ; = 3&\d' In 2 2 p r m ~ P 2 m 2 2 h P2 (B31)
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By using the definition of $ B in (B27) and letting $' (z) = y<>(z) , C A can be determined. 150 C A = 3m d\ 1P X /?^ *' ( ^ /I u 6 3 X '1 d ru '1 x; ln ^u: )+1 (B32) This equation is used in Chapter VI to evaluate the scalar function F(p 2 ,y) through the relationship provided by (B19a). B3 Solving Equation B15b S yields Substituting the definition of D A in (B20b) into Eq. (B15b) X Â„ r X. V 2 C A = 2e i^pfX * (Piu^p y where V 2 E A =3 V (p 2 m 2 )(p 2 m 2 ) VP^X + (P 1v + P 2v )P X (p 2 m 2 )(p^) ] 1 * > x * Let C A = C 6 A 3 C , y u y (B33) (B34) (B35) Since (m 2 p r P 2 )6 A + (P 1y + P 2 y^P A / 2 2 W 2 2n (p^m )(p 2 m ) 1 9 2 3p A 2 2 ( Ply +p 2y } , / P r m f 2 2v (PP 2 ) In 2 2 p 2 m
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151 2 , 2 2. 1 c +^^ln (p 2 m 2 )(p 2 m 2 ) p 2 p 2 (p 2 m 2 )J (B36) Eq. (B33) can be reexpressed in terms of two equations. ( Ply + P2y } 2 * 1 V^C = 2e ^ u 2 PiPÂ« In L H H 2 / 2 2\ p,m 4Â— y + 9 E Â„2 2 I y p 9 m and V d C = 2e 2 / 2 2 1 m p 1# p ? , /Pm ! L + ^ In ' ,2 2 W 2 2x ' 2 2 "' 2 2 (p^m )(p 1 m ) p 1 p 2 (p 2 m / (B37) with = 2e 1 1 1 1 k 2 /2 2 (p 2 m 2 ) " 2 (p 2 m 2 ) (p 2 m 2 )(p 2 m 2 ) (pfp) In / 2 2\l p r m ~2~T I P2 _m V^E m p, p 1 K 2 + 1 In (p 2 m 2 )(p 2 m 2 ) " (p 2 p 2 ) '" lp 2 m 2 n 2 m 2 ' p^m 1 u 12 2 Pm ~~ 2 2 . , P 2" m ' (B38) 2m' In 2 2 p^m (p 2 m 2 )(p 2 m 2 ) (p 2 p 2 ) "" \^p" (B39) From (B38) and using the definitions of <2>, $ A and B it can be seen that 2 2 Pi Pc C =e*(Â£) Â£$(Â£) + elC$ A + 2 E^r
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152 a ' U ft k 2 V 0Â£\) In (1^) 3 2xÂ„ V d6 (l\) In L. m p 9 m (B40) From (B37) an expression for the transverse and longitudinal * ~* * ra 2 components of C , C and k C k /k , can be derived r u y y a V^C = 2e y 2 2 p p,m ^Â— In' ! , 2~~27 '"I 2 2 HP 1 P 2 ) \ P2 _m + v] = 2e 1 l 2 Wl V U 3 y 2e 3 y 1^ dBlndJj+E 2e 3 fi \]f (m 2 u3) In (1 3) H.V1 x b u 3. (B41) since E = 2m d $ A 2* E f l ^(x 6 + m 2 u 3 )ln(l^) d3 (B42) Using the $ transformation in (B41) it is found that y 2 y c = 1 e 3 '1 1 f 9 r x r uÂ„ x * ((m 2 u 6 )[L 2 (^) + (l^)ln(l^) d6 4 e P y ,/ ~ (ni 2 uj f(^l) In (1^) + 1 l /l u 3 6 L x 3 U B d3 . (B43)
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153 The expression for the longitudinal part of C in Eq. / 2 2\ . ' In I 4. a 3 F i 4 2 2 , 3d V 2 k y C* =4e ! B37 ) yields (B44) Rearranging this expression, V 2 [kVe(kp + ) E] =4e In / n 2 m 2 ^~~2 \ P2" m , \ (kp + J ^)V 2 E 4 Â£ L 2 2 i , p i "2" II2T2/ " 4 ,Â„2 Â„2 W Â„2 m 2. 2 2 2 T m (p 1 p 2 ) PgnT/ H (Pim t )(P2"i c ) J 9 Using the expression (B39) for V E it can be seen that k a C* = e(k'pV)E 2 (pfV) In (14 m 2 (p 2 m 2 ) In (14) (p 2 p 2 ) m , (kD k 2 r *Â»s> In (l^)dB. (B45) (B46) By combining the longitudinal part with the transverse part C is defined vK/' i (m2 u Â°> "(u R l) x Â— Â§ In (1/) + 1 dB \S~2 + 1> I Y^^ ln (1 F) d6 Â• (B " 47) V V 6 C J1 X R P U R
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154 Therefore (B40) and (B47) provide the needed information to define C since U C A = cV 3 A C* . (B48) y y v Equations (B40), (B47) and (B48) are used in Chapter VI to evaluate the scalar functions G , G 1 and G 2 through the relationship provided by (B19b). B4 Solving Equation (B15c) To find the definition of the C A it is necessary to solve Eq. (B15c), V 2 C A =eD A 2e3 V~ 2 3 P D A +2 Â£ 3V" 2 3 P D A . Â• (B15c) yv yv y vp v yp Alternatively, since V 2 3 V C A = e 3 V D A (B49) yv yv so that eV'W = 3 V C A (B50) yv yv Eq. (B15c) could be equivalently expressed as V 2 C A + 23 3 P C A 23 3 P C A Â— eD X ,. (B51) yv y vp v yp yv Inserting the definition of D pv given in (B20c) into this yields V 2 3 V C A = OT(6 A k 3 a k 3 A ) n ? ] * n . (B52) ^ v ya y (p 2 m 2 )(p 2 m 2 )
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155 From the $Â« relationship it can be seen that 8 V C X = m(& X k 3 a k d x )$* yv y a y A 1 Jrt x = ldn(6 X k 9 a k 9 X ) / ^ ln ( 1 "ir)(B " 53) 8 M U 71 x u 3 This can be used to form the terms ^^ P ^ p and 3 y 9 P C X p in Eq. (B50). Equation (B50) then becomes V 2 C A = dn[(6 A k 9 a k 3 A )3 yv u a y > i r in + (6 A k 3 a k 9 A )3 ] / ^ In (lTj^d an(
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156 B5 Solving Equation (B15d) The last equation, (B15d), can be put into a simpler form by performing a differentiation with respect to p p . This yields 7 2 a p r A __ 2^0* V t 3 M C yvp yvp (B56) Usinq the definition of D in Eq. (B20d) this becomes D yvp v2 a p r X _ ? f a A JPl P 2P2 hj V d K C =<:e:( d Â— * Â— 9" 2" Â— 2"^ yvp (p r m )(p 2 m ) + 3 c r ( PlvP2pP2yPlp ) + (PlpP2yP2pPly )5 v ]A L (p 2 m 2 )(p 2 m 2 ) J J v_ v When Eq. (B56) is multiplied by pj p 2 a new equation is formed, (B57) K l r 2 yvp .22 A_P_ 2 2 W 2 _2 3> (k 2 P V 2 2 W 2 2 L \(Pinr)(P2nr) (Pim^)(P2m t ) 4ek 2 p A (p 2 m 2 )(p 2 m 2 ) In the above statement use was made of the fact that pVp^V 2 3 p C X = V 2 pV p^3 p C X , F 1 F 2 uvo H K 2 yvp' (B58) a consequence of the antisymmetry of C . Using the $ A transform again yvp gives *m*
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APPENDIX C FORTRAN PROGRAMS Presented on the following pages are the Fortran programs which were used to code the Main Program and the Extended Mass Shell Program. 157
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158 IMPLICIT REAL*8 (AH.OZ) C THIS IS THE MAIN PROGRAM C THE EIGHT DIFFERENCIAL EQUATIONS ARE DEFINED FOR THE VERTEX EQUATION C THIS PROGRAM READS THE VALUE OF P2,U,K2,SKL,SIGN ( ELECTRON MOMENTUM C SQUARED, ANGLE VARIABLE, PHOTON MOMENTUM SQUARED, THE SCALE AND THE C SIGN OF P2 AND K2, BOTH MUST HAVE THE SAME SIGN.) DIMENSION DX2(8) COMMON/ SIGNPK/ SIGN COMMON/ SCALE/ SKL WRITE(6,701) 701 FORMATC NI.IX FOR RESULTS, FUNCTIONS, STORAGE') READ(9,*) NI.LX C THE NI = 1 GIVES JUST THE RESULTS OF THE EQUATIONS C THE NI =2 GIVE RESULTS PLUS FUNCTIONS C LX = 1 IS DEFAULT C IX = 2 PUTS THE RESULTS PLUS FUNCTIONS INTO STOREPRINT READ(2,122) P2 ,U,AK2 , SKL, SIGN 122 FORMAT(D24.16,4D11.3) WRITE(6,74) P2,U,AK2,SKL 74 FORMATC THE POINT IS P2,U,K2; Â» ,3D15.7/ ' THE SCALE' ,D15.7) WRITE(4,74) P2,U,AK2,SKL SM=1.D0*SKL C WRITE(6,17) P2,U,AK2,SM IF(IX.EQ.2)WRITE(4,17) P2.U.AK2 AM2=SKL*SKL IF(IX.EQ.2)WRITE(10,17) P2,U,AK2,AM2 17 FORMATC AT THE POINT P2.U.K2 ,M2' ,D12.4 .3D12.4) X=FTBM(P2,U,AK2,SM,NI,IX) C IF(NI.EQ.2)WRITE(6,21) X 21 FORMAT (' THE FTBM EQUALS' ,D20. 7) STOP END C THIS SUBROUTINE TAKES THE VALUE OF THE EIGHT FUNCTIONS AND ALL C DERIVATIVES FROM THE DATA MATRIX (THE EXTENDED MASS SHELL SOLUTION) C FORMAT STATEMENT 113 C AND CONTRUCTS THE LEFT AND RIGHT HAND SIDES. FUNCTION FTBM(P2,U,K2,SM,NI,IX) IMPLICIT REAL*8 (AH.OZ) DIMENSION D2(8),TM(8),X(20) DATA EP/2.322819D3/ COMMON/ S IGNPK/ S IGN COMMON/ SCALE/ SKL REAL*8 K2,LPL,LSALP,LPLSN,LSLN,LGL,LPLG,LPP,LE DIMENSION EROR(10,5),E(8) COMMON/S IMP/LPL, LS ALP , LPLSN, LSLN, LGL, LPLG , LPP , LE , C RPL,RSALP,RPLSN,RSLN,RGL,RPLG,RPP,RE COMMON/ SUBTT/TE( 14) ,TES(14) ,TEP(14) , A, AP, APP,B,BP,TO( 14) ,T0S(14) , C TOP(14) COMMON/ FUNS/XF(8 , 10 , 1,1) DIMENSION XM(3,10) 1=1 J=l DO 13 IF=1,8 13 READ(2,113) (XF( IF, IT, 1 , 1) , IT=1 ,10)
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159 113 FORMAT(3D24.16/3D24.16/3D24.16/D24.16) PK=SIGN*DSQRT(P2*K2) POK=DSQRT(P2/K2) AKOP=DSQRT(K2/P2) R=P2 SK2 PKU=PK*U U2=U*U TU=1.D0U2 P4=R*R P6=P4*P2 CALL TT(R,S,U,I,J) F=XF(1,1,I,J) FP=XF(1,2,I,J) FPP=XF(1,3,I,J) FPPP=XF(1,4,I,J) FS=XF(1,5,I,J) FSS=XF(1,6,I,J) FSSS=XF(1,7,I,J) FSP=XF(1,8,I,J) FSSP=XF(1,9,I,J) FSPP=XF(1,10,I,J) G0=XF(2,1,I,J) G0P=XF(2,2,I,J) G0PP=XF(2,3,I,J) G0PPP=XF(2,4,I,J) G0S=XF(2,5,I,J) G0SS=XF(2,6,I,J) G0SSS=XF(2,7,I,J) G0SP=XF(2,8,I,J) G0SSP=XF(2,9,I,J) G0SPP=XF(2,10,I,J) G1XF(3,1,I,J) G1P=XF(3,2,I,J) G1PP=XF(3,3,I,J) G1PPP=XF(3,4,I,J) G1S=XF(3,5,I,J) G1SS=XF(3,6,I,J) G1SSS=XF(3,7,I,J) G1SP=XF(3,8,I,J) G1SSP=XF(3,9,I,J) G1SPP=XF(3,10,I,J) G2=XF(4,1,I,J) G2P=XF(4,2,I,J) G2PP=XF(4,3,I,J) G2PPP=XF(4,4,I,J) G2S=XF(4,5,I,J) G2SS=XF(4,6,I,J) G2SSS=XF(4,7,I,J) G2SP=XF(4,8,I,J) G2SSP=XF(4,9,I,J) G2SPP=XF(4,10,I,J) H0=XF(5,1,I,J) H0P=XF(5,2,I,J)
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160 H0PP=XF(5,3,I,J) H0PPP=XF(5,4,I,J) H0S=XF(5,5,I,J) HOSS=XF(5,6,I,J) HOSSS=XF(5,7,I,J) H0SP=OCF(5,8,I,J) HOSSP=XF(5,9,I,J) H0SPP=XF(5,10,I,J) H1=XF(6,1,I,J) H1P=XF(6,2,I,J) H1PP=XF(6,3,I,J) H1PPP=XF(6,4,I,J) H1S=XF(6,5,I,J) H1SS=XF(6,6,I,J) H1SSS=XF(6,7,I,J) H1SP=XF(6,8,I,J) H1SSP=XF(6,9,I,J) H1SPP=XF(6,10,I,J) H2=XF(7,1,I,J) H2P=XF(7,2,I,J) H2PP=XF(7,3,I,J) H2PPP=XF(7,4,I,J) H2S=XF(7,5,I,J) H2SS=XF(7,6,I,J) H2SSS=XF(7,7,I,J) H2SP=XF(7,8,I,J) H2SSP=XF(7,9,I,J) H2SPP=XF(7,10,I,J) OXF(8,l,I,J) CP=XF(8,2,I,J) CPP=XF(8,3,I,J) CPPP=XF(8,4,I,J) CS=XF(8,5,I,J) CSS=XF(8,6,I,J) CSSS=XF(8,7,I,J) CSP=XF(8,8,I,J) CSSP=XF(8,9,I,J) CSPP=XF(8,10,I,J) 677 FORMATC THE F' .D15.7/3D15.7/3D15.7/3D15.7/ ' THE GO', C D15.7/3D15.7/3D15.7/3D15.7/' THE Gl ' .D15.7/3D15.7/ C 3D15.7/3D15.7/' THE G2Â» .D15.7/3D15.7/3D15.7/3D15.7/ C ' THE HO',D15.7/3D15.7/3D15.7/3D15.7/' THE HI' ,015.7/ C 3D15.7/3D15.7/3D15.7/' THE H2' .D15.7/3D15.7/3D15.7/ C 3D15.7/' THE C ' .D15.7/3D15.7/3D15.7/3D15.7) IF(NI.EQ.2)WRITE(6,677) F, FP, FPP, FPPP, FS, FSS, FSSS , FSP, FSSP, FSPP, C GO , GOP ,GOPP .GOPPP ,GOS , GOSS .GOSSS ,GOSP .GOSSP ,GOSPP , C G1,G1P,G1PP,G1PPP,G1S,G1SS,G1SSS,G1SP,G1SSP,G1SPP, C G2,G2P,G2PP,G2PPP,G2S,G2SS,G2SSS,G2SP,G2SSP,G2SPP, C HO ,HOP ,HOPP .HOPPP ,HOS ,HOSS .HOSSS ,HOSP .HOSSP ,HOSPP , C HI, HIP, HIPP .H1PPP, HIS, HISS, H1SSS,H1SP,H1SSP,H1SPP, C H2,H2P,H2PP,H2PPP,H2S,H2SS,H2SSS,H2SP,H2SSP,H2SPP, C C, CP , CPP , CPPP , CS , CSS , CSSS , CSP , CSSP , CSPP IF(IX.EQ.2)WRITE(4,677) F, FP, FPP, FPPP, FS, FSS, FSSS, FSP, FSSP, FSPP, C GO, GOP ,GOPP , GOPPP ,GOS ,GOSS , GOSSS ,GOSP , GOSSP .GOSPP ,
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161 C G1,G1P,G1PP,G1PPP,G1S,G1SS,G1SSS,G1SP,G1SSP,G1SPP, C G2,G2P,G2PP,G2PPP,G2S,G2SS,G2SSS,G2SP,G2SSP,G2SPP, C HO ,HOP ,HOPP ,HOPPP,HOS ,HOSS ,HOSSS .HOSP.HOSSP ,HOSPP , C HI, HIP, HIPP, H1PPP, HIS, HISS, H1SSS,H1SP,H1SSP,H1SPP, C H2 ,H2P ,H2PP ,H2PPP ,H2S ,H2SS ,H2SSS ,H2SP,H2SSP,H2SPP, C C,CP,CPP,CPPP,CS,CSS,CSSS,CSP,CSSP,CSPP IF(IX.EQ.2)WRITE(10,711) F.GO.Gl ,G2 ,HO,Hl ,H2,C 711 FORMAT(4D20.9) C RIGHT HAND SIDE OF THE EQUATION RGL=EP*( TO(l) +TO(7) +4*TO(8) +2*P2*TOP(8) +2*POK*U*TOP(9) C +TU*TOS(9)/PK ) RPP=EP*( 2*TOP(l) U*TOS(l)/P2 +6*TO(2) +2*P2*TOP(2) C +2*POK*U*TOP(3) +TU*TOS(3)/PK +2*TOP(7) U*TOS(7)/P2 ) RPLG=EP*( TO(3) +2*TOP(4) U*TOS(4)/P2 +5*TO(5) +2*P2*TOP(5) C +2*PKU*TOP(6) +TU*TOS(6)/POK +TOS(7)/POK ) RE=EP*( TO(IO) +4*TO(ll) +2*P2*TOP(ll) +TOS(12)/POK +2*TOP(13) C U*TOS(13)/P2 +2*PKU*TOP(14) +TU*TOS(14)/POK ) RPL=EP*( 5*TE(1) +2*P2*TEP(1) +2*POK*U*TEP(2) +TU*TES(2)/PK C +2*TEP(5) U*TES(5)/P2 ) RSLN=EP*( TES(4)/POK +2*PKU*TEP(9) +TU*TES(9)/POK +TE(10) C +4*TE(12) +2*P2*TEP(12) TE(13) +TE(14) ) RSALP=EP*(2*TEP(4) U*TES(4)/P2 5*TE(8) 2*P2*TEP(8) TE(ll) C 2*POK*U*TEP(13) TES(13)*TU/PK ) RPLSN=EP*( 6*TE(6) +2*P2*TEP(6) +2*PKU*TEP(7) +TU*TES(7)/POK C +2*TEP(10) U*TES(10)/P2 +TES(ll)/POK +2*TEP(14) C U*TES(14)/P2 ) C LEFT HAND SIDE OF THE EQUATION LGL=( 3*F/P2 +5*U*FS/P2 TU*FSS/P2 8*FP 4*P2*FPP C 8*U*H0/P2 +4*TU*H0S/P2 +8*U*H0P C 12*U*H1SP +4*TU*H1SS/P2 +20*H1P C +4*TU*H1SSP +64*P2*H1PP +16*P4*H1PPP 8*U*H1S/P2 C +(18*U26)*H2S/P2 14*U*TU*H2SS/P2 +2*TU*TU*H2SSS/P2 C +(1628*U2)*H2SP +4*U*TU*H2SSP +48*P2*U*H2PP C +16*U*P4*H2PPP +8*P2*TU*H2SPP )/PK LPP=( 9*F/P4 +23*U*FS/P4 3*FSS/P4 +10*U2*FSS/P4 C U*TU*FSSS/P4 14*FP/P2 18*U*FSP/P2 +2*TU*FSSP/P2 C +20*FPP 4*U*FSPP +8*P2*FPPP c 64*U*H0/P4 +(3086*U2)*H0S/P4 +26*U*TU*H0SS/P4 C 2*TU*TU*H0SSS/P4 C +64*U*HOP/P2 +(44*U216)*H0SP/P2 4*U*TU*HOSSP/P2 C 32*U*HOPP 8*TU*H0SPP 16*P2*U*H0PPP C +18*H1/P4 +46*U*H1S/P4 +(20*U26)*H1SS/P4 c 2*U*TU*H1SSS/P4 28*H1P/P2 36*U*H1SP/P2 C +4*TU*H1SSP/P2 +40*H1PP 8*U*H1SPP +16*P2*H1PPP)/PK LPLG= 8*FS/P4 7*U*FSS/P4 +8*FSP/R +4*FSPP +TU*FSSS/P4 C 8*H0/P4 4*U*H0S/P4 +8*H0P/R 28*U*H0SP/R C +4*TU*H0SSP/R +64*H0PP +16*R*HOPPP C +8*H1SP/R 4*U*H1SS/P4 8*H1S/P4 C +18*U*H2S/P4 +(10*U24*TU)*H2SS/P4 2*U*TU*H2SSS/P4 C 28*U*H2SP/R +48*H2PP 8*U*H2SPP +16*R*H2PPP C +4*TU*H2SSP/R LE=(8*H0/P4 +10*U*H0S/P4 2*TU*H0SS/P4 8*HOP/R 8*H0PP C +8*H1S/P4 +10*U*H1SS/P4 2*TU*H1SSS/P4 8*H1SP/R C 8*H1SPP
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162 C +18*U*H2S/P4 +(10*U24*TU)*H2SS/P4 2*U*TU*H2SSS/P4 C 28*U*H2SP/R +48*H2PP 8*U*H2SPP +16*R*H2PPP C +4*TU*H2SSP/R ) LSLN=AKOP*(3*G0S/P2 +5*U*G0SS/P2 TU*G0SSS/P2 8*G0SP C 4*P2*G0SPP C 3*G2/P2 5*U*G2S/P2 +TU*G2SS/P2 +8*G2P +4*P2*G2PP C 3*C/P2 5*U*CS/P2 +TU*CSS/P2 2*CP +10*U*CSP C 2*TU*CSSP 28*P2*CPP 8*P4*CPPP ) LSALP= 9*U*G0S/P4 (27*U2)*G0SS/P4 U*TU*G0SSS/P4 C 14*U*G0SP/R +2*TU*G0SSP/R +24*G0PP 4*U*G0SPP C +8*R*G0PPP C +4*G1/P4 +5*U*G1S/P4 TU*G1SS/P4 4*G1P/R 4*G1PP LPLSN= AKOP*( 9*G1S/P4 +7*U*G1SS/P4 TU*G1SSS/P4 4*G1SP/P2 C 4*G1SPP C +9*G2/P4 +23*U*G2S/P4 (3.D010*U2)*G2SS/P4 U*TU*G2SSS/P4 C 14*G2P/P2 18*U*G2SP/P2 +2*TU*G2SSP/P2 +20*G2PP C 4*U*G2SPP +8*P2*G2PPP C +9*C/P4 +23*U*CS/P4 (3.D010*U2)*CSS/P4 TU*U*CSSS/P4 C 14*CP/P2 18*U*CSP/P2 +2*TU*CSSP/P2 +20*CPP C 4*U*CSPP +8*P2*CPPP ) LPL= 9*U*G0S/P4 (2.D07*U2)*G0SS/P4 U*TU*G0SSS/P4 C 14*U*G0SP/R +2*TU*G0SSP/R +24*G0PP 4*U*G0SPP C +8*R*G0PPP C 12*G1/P4 15*U*G1S/P4 +3*TU*G1SS/P4 +12*G1P/R C 10*U*G1SP/R +2*TU*G1SSP/R +36*G1PP +8*R*G1PPP C +9*U*G2/P4 (8.D023*U2)*G2S/P4 10*U*TU*G2SS/P4 C +TU*TU*G2SSS/P4 14*U*G2P/R +(8.D018*U2)*G2SP/R C +2*U*TU*G2SSP/R +20*U*G2PP +4*TU*G2SPP +8*R*U*G2PPP C EROR IS THE SQUARE OF THE DIFFERENCES OF THE LEFT AND RIGHT SIDES OF C THE EIGHT EQUATIONS. E(1)=2.0*DABS((LPLRPL)/(LPL+RPL)) E(2)=2.0*DABS((LSALPRSALP)/(LSALP+RSALP)) E(3)=2.0*DABS((LSLNRSLN)/(LSLNfRSLN)) E(4)=2.0*DABS((LPLSNRPLSN)/(LPLSN+RPLSN)) E(5)=2.0*DABS((LPPRPP)/(LPP+RPP)) E(6)=2.0*DABS((LGLRGL)/(LGL+RGL)) E(7)=2.0*DABS((LPLGRPLG)/(LPLG+RPLG)) E(8)=2.0*DABS((LERE)/(LE+RE)) IOUT=l IF(IOUT.NE.l) GO TO 98 IOUT=0 IF( IOUT.EQ.l)WRITE(6,777) (E( IN) , IN=1 ,8) 777 FORMAT (' THE LEFT MINUS RIGHT' ,4D20.7) C WRITE (6,666) LPL,LSALP,LSLN,LPLSN C WRITE (6,667) RPL.RSALP.RSLN.RPLSN C WRITE (6,666) LPP,LGL,LPLG,LE C WRITE (6,667) RPP,RGL,RPLG,RE 666 FORMAT(* LEFT '.4D14.5) 667 FORMATC RIGHT' .4D14.5) ALHSV1=LPP RHSV1=RPP ALHSV2=LPLG RHSV2=RPLG ALHSV3=LGL
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163 RHSV3=RGL ALHSV4=LE RHSV4=RE ALHSW1=LPLSN RHSW1=RPLSN ALHSW2=LSALP RHSW2=RSALP ALHSW3=LSLN RHSW3=RSLN ALHSW4=LPL RHSW4=RPL RATV1=DABS((RHSV1ALHSV1)/ALHSV1)*100.D0 RATV2=DABS( (RHSV2ALHSV2 ) / ALHSV2 ) * 1 00 . DO RATV3=DABS( (RHSV3ALHSV3 ) / ALHSV3) *1 00. DO RATV4=DABS ( ( RHSV4ALHSV4 ) / ALHSV4 ) * 100 . DO RATW1=DABS((RHSW1ALHSW1)/ALHSW1)*100.D0 RATW2=DABS((RHSW2ALHSW2)/ALHSW2)*100.D0 R ATW3=DABS ( ( RHSW3ALHSW3 ) / ALHSW3 ) * 1 00 . DO RATW4=DABS( ( ALHSW4RHS W4 ) / ALHSW4 ) * 1 00 . DO WRITE(6,987)ALHSV1,RHSV1,RATV1 WRITE( 6 ,20) ALHSV2 .RHSV2 ,RATV2 WRITE( 6 , 30 ) ALHSV3 , RHSV3 , RATV3 WRITE( 6 ,40 ) ALHSV4 ,RHSV4 ,RATV4 WRITE(6,50)ALHSW1 ,RHSW1 ,RATW1 WR ITE ( 6 , 60 ) ALHSW2 , RHSW2 , RATW2 WRITE( 6 ,70) ALHSW3 .RHSW3 ,RATW3 WRITE( 6 ,80) ALHSW4 .RHSW4 ,RATW4 IF( IX.EQ.2)WRITE(4 ,987) ALHSV1 ,RHSV1 .RATV1 IF(IX.EQ.2)WRITE(4,20)ALHSV2,RHSV2,RATV2 IF(IX.EQ.2)WRITE(4,30)ALHSV3,RHSV3,RATV3 IF(IX.EQ.2)WRITE(4,40)ALHSV4,RHSV4,RATV4 IF( IX.EQ.2)WRITE(4,50) ALHSW1 ,RHSW1 ,RATW1 I F( IX . EQ . 2 ) WR ITE ( 4 , 60 ) ALHS W2 , RHS W2 , R ATW2 IF(IX.EQ.2)WRITE(4,70)ALHSW3,RHSW3,RATW3 IF( IX.EQ.2)WRITE(4 ,80) ALHSW4 ,RHSW4 ,RATW4 987 FORMATUX.'EQ.l.LEFT'.EU.S.lX.'RIGHT'.En.S.lX, C I ERR0R= , ,F6.2, I %') 20 F0RMAT(1X, , EQ.2,LEFT=' .E12.5.1X, 'RIGHT=' ,E12.5,1X, C , ERR0R= , ,F6.2, , %') 30 FORMATCIX.'EQ.S.LEFT^' ,E12.5,1X, 'RIGHT=' ,E12.5,1X, C'ERR0R=' ,F6.2,'%') 40 F0RMAT(1X,*EQ.4,LEF]>' ,E12.5, IX, 'RIGHT=' .E12.5.1X, C'ERR0R=',F6.2, , %'/) 50 F0RMAT(1X, , EQ.5,LEFI> , .E12.5.1X, 'RIGHT=' .E12.5.1X, C , ERR0R= , ,F6.2, , % , ) 60 F0RMAT(1X, , EQ.6,LEFT=' ,E12.5,1X, 'RIGHT"' .E12.5.1X, C'ERR0R=' ,F6.2,'%') 70 F0RMAT(1X,'EQ.7,LEFT=* .E12.5.1X, 'RIGHT=' ,E12.5,1X, C'ERR0R=' ,F6.2,'%') 80 P0RMAT(1X,'EQ.8,LEFT=',E12.5,1X,'RIGHT= , ,E12.5,1X, C*ERR0R=' ,F6.2,'%') IF(IX.EQ.2)WRITE (4,793) A, AP, APP.B.BP 793 FORMAT (' A, AP, APP.B.BP' .3D15.7/2D15.7) 98 CONTINUE
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164 DIFP=0.0 DO 99 IN1,8 DIFF=DIFF+E( IN) 99 CONTINUE IF(IOUT.EQ.l) WRITE (6,400) I,J,DIFF 400 FORMATC THE SUM OF THE DIFFERENCES AT ' ,2I3,D16.6) EROR(I,J)=DIFF 10 CONTINUE FTBM=0.0 DO 717 INl.LIM DO 717 JN=l,LOT 717 FTBM=FTBM+EROR(IN,JN) RETURN END C THE FOLLOWING SUBROUTINES DEFINE THE TENSORS WHICH DEFINE THE C RIGHT HAND SIDE OF THE EQUATIONS. SUBROUTINE TT(P2,K2,U, I, J) IMPLICIT REAL*8 (AH.OZ) REAL*8 MPD,MPDP,MPDU,K2 COMMON/SIGNPK/SIGN COMM0N/SUBTT/TE(14),TES(14),TEP(l4),AOUT,APOUT,APP,BOUT,BPOUT, C T0(14),T0S(14),T0P(14) COMMON/ TWS/W1,W2,W3,W4,W5,W6,W7,W8,W9,W10,W11,W12,W13,W14 C ,W1P,W2P,W3P,W4P,W5P,W6P,W7P,W8P,W9P,W10P,W11P,W12P,W13P,W14P C ,W1U,W2U,W3U,W4U,W5U,W6U,W7U,W8U,W9U,W10U,W11U,W12U,W13U,W14U COMMON/TVS/V1 ,V2 ,V3 ,V4 ,V5 ,V6 ,V7 ,V8 ,V9 ,V10 ,V1 1 ,V12 ,V13 ,V14 C ,V1P,V2P,V3P,V4P,V5P,V6P,V7P,V8P,V9P,V10P,V11P,V12P,V13P,V14P C ,V1U,V2U,V3U,V4U,V5U,V6U,V7U,V8U,V9U,V10U,V11U,V12U,V13U,V14U COMMON/ FUNS/XF( 8 ,10,1,1) C THIS SUBROUTINE GENERATES THE FUNCTIONS TE AND TO FOR THE R.H.S. OF THE EQUATI T1=1.D0 T2=2.D0 T3=3.D0 T44.D0 T5=5.D0 T6=6.D0 F=XF(1,1,I,J) FP=XF(1,2,I,J) FPP=XF(1,3,I,J) FS=XF(1,5,I,J) FSS=XF(1,6,I,J) FSP=XF(1,8,I,J) G0=XF(2,1,I,J) G0P=XF(2,2,I,J) G0PP=XF(2,3,I,J) G0S=XF(2,5,I,J) G0SS=XF(2,6,I,J) G0SP=XF(2,8,I,J) G1=XF(3,1,I,J) G1P=XF(3,2,I,J) G1PP=XF(3,3,I,J) G1S=XF(3,5,I,J) G1SS=XF(3,6,I,J) G1SP=XF(3,8,I,J)
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165 G2=XF(4,1,I,J) G2P=XF(4,2,I,J) G2PP=XF(4,3,I,J) G2S=XF(4,5,I,J) G2SS=XF(4,6,I,J) G2SP=XF(4,8,I,J) H0=XF(5,1,I,J) H0P=XF(5,2,I,J) H0PP=XF(5,3,I,J) H0S=XF(5,5,I,J) H0SS=XF(5,6,I,J) H0SP=XF<5,8,I,J) H1=XF(6,1,I,J) H1P=XF(6,2,I,J) H1PP=XF(6,3,I,J) H1S=XF(6,5,I,J) H1SS=XF(6,6,I,J) H1SP=XF(6,8,I,J) H2=XF(7,1,I,J) H2P=XF(7,2,I,J) H2PP=XF(7,3,I,J) H2S=XF(7,5,I,J) H2SS=XF(7,6,I,J) H2SP=XF(7,8,I,J) C=XF(8,1,I,J) CP=XF(8,2,I,J) CPP=XF(8,3,I,J) CS=XF(8,5,I,J) CSS=XF(8,6,I,J) CSP=XF(8,8,I,J) IKI0 IF(IKI.EQ.l) WRITE (6,100) F, FP, FS.G0.G0P, C G0S,G1,G1P,G1S,G2, C G2P,G2S,H0,H0P,H0S,H1,H1P,H1S,H2,H2P,H2S,C,CP,CS > P2,U 100 FORMAT (' F,G,H,C,AS SEEN IN TT' /5D14.5/5D14.5/5D14.5/ C 5D14.5/4D14.5/ 1 P2.U IN TT' ,2D18.8) PK=SIGN*DSQRT(P2*K2) P4=P2*P2 P0K=DSQRT(P2/K2) AK0P=T1/P0K P12=P2+K2+2*PK*U DU=T2*PK DP=T1+AK0P*U U2=U*U P4=P2*P2 P6=P2*P4 TU=1U2 A1=XA(P12) A2=XA(P2) DFF=100*DABS((A2A1)/A2) WRITE(6,456) Al , A2.DFF.P12 ,P2 WRITE(4,456) Al , A2 ,DFF,P12,P2 456 FORMATC Al AND A2 , ,3D15.7,' %DIFF'/' PI SQ AND P2 SQ* .2D15.7) B1=XB(P12)
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166 B2=XB(P2) AP=XAP(P2) BP=XBP(P2) A1P=XAP(P12)*DP A1U=XAP(P12)*DU A2P=AP A0UT=A2 B0UT=B2 APOUTAP BPOUT=BP D1=A1*A1P12*B1*B1 D2=A2*A2P2*B2*B2 AD=A1/D1 BD=B1/D1 MPD=AP*B2/D2 BP*A2/D2 PPD=AP*A2/D2 P2*BP*B2/D2 BBD=B2*B2/D2 ABD=A2*B2/D2 APP=XAPP(P2) D1P=T2*A1*A1PDP*B1*B1 D1U=T2*A1*A1U DU*B1*B1 D2P=T2*A2*A2PB2*B2 ADP=A1P/D1 A1*D1P/D1**2 ADU=A1U/D1 A1*D1U/D1**2 AZ1 = A1U/D1 AZ2=A1*D1U/D1**2 BDP=B1*D1P/D1**2 BDU=B1*D1U/D1**2 MPDP=APP*B2/D2 AP*B2*D2P/D2**2 PPDP=APP*A2/D2 +AP*A2P/D2 AP*A2*D2P/D2**2 BBDP=B2*B2*D2P/D2**2 ABDP=A2P*B2/D2 A2*B2*D2P/D2**2 ZE1=(F/P2 +2*FP U*FS/P2 2*PPD*F +F*BBD)/PK C +2*MPD*G0 +2*G1*MPD G1*ABD/P2 C +2*U*G2*MPD +2*U*H0*BBD/PK 2*H1*BBD/PK ZE1P=(FP/P2 +2*FPP U*FSP/P2 2*PPD*FP +FP*BBD)/PK C +2*MPD*G0P +2*G1P*MPD G1P*ABD/P2 C +2*U*G2P*MPD +2*U*H0P*BBD/PK 2*H1P*BBD/PK + C (1.5DO*F/P4 FP/P2 +1.5DO*U*FS/P4 +PPD*F/P2 0.5D0*F*BBD/P2)/PK C +G1*ABD/P4 U*H0*BBD/(PK*P2) +H1*BBD/(PK*P2) C 2*PPDP*F/PK +F*BBDP/PK +2*MPDP*GO +2*G1*MPDP G1*ABDP/P2 C +2*U*G2*MPDP +2*U*H0*BBDP/PK 2*H1*BBDP/PK ZE1U=(FS/P2 +2*FSP U*FSS/P2 2*PPD*FS +FS*BBD)/PK C +2*MPD*G0S +2*G1S*MPD G1S*ABD/P2 C +2*U*G2S*MPD +2*U*H0S*BBD/PK 2*H1S*BBD/PK C FS/(P2*PK) +2*G2*MPD +2*H0*BBD/PK ZE2=( FS/P2 G2*ABD/POK 2*H0*BBD 2*H2*BBD ) ZE2P=FSP/P2 G2P*ABD/POK 2*H0P*BBD 2*H2P*BBD C FS/P4 +O.5D0*G2*ABD/(P2*POK) C G2*ABDP/POK 2*H0*BBDP 2*H2*BBDP ZE2U=FSS/P2 G2S*ABD/POK 2*H0S*BBD 2*H2S*BBD ZE3=( C*ABD/POK +2*H2*BBD ) ZE3P=CP*ABD/POK +2*H2P*BBD 0.5D0*C*ABD/(POK*P2) C +C*ABDP/POK +2*H2*BBDP
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167 ZE3U=CS*ABD/P0K +2*H2S*BBD ZE4=( GO*ABD +2*Hl/PK +2*P0K*H1*BBD +2*POK*U*H2*BBD ) ZE4P=G0P*ABD +2*H1P/PK +2*P0K*H1P*BBD +2*P0K*U*H2P*BBD C H1/(PK*P2) +H1*BBD/PK +U*H2*BBD/PK C GO*ABDP +2*P0K*H1*BBDP +2*POK*U*H2*BBDP ZE4U=G0S*ABD +2*H1S/PK +2*P0K*H1S*BBD +2*POK*U*H2S*BBD C +2*P0K*H2*BBD ZE5=( F/PK GO*ABD +2*H1*P0K*BBD +2*P0K*U*H2*BBD ) ZE5P=FP/PK GOP*ABD +2*H1P*P0K*BBD +2*POK*U*H2P*BBD C 0.5D0*F/(P2*PK) +H1*BBD/PK +U*H2*BBD/PK C GO*ABDP +2*H1*P0K*BBDP +2*POK*U*H2*BBDP ZE5U=FS/PK GOS*ABD +2*H1S*P0K*BBD +2*P0K*U*H2S*BBD C +2*POK*H2*BBD ZE6=(2*G2*MPD/POK 4*H0/P4 +4*H0P/P2 2*U*H0S/P4 C 4*H0*PPD/P2 +2*C*MPD/P0K ) ZE6P=2*G2P*MPD/POK 4*H0P/P4 +4*H0PP/P2 2*U*H0SP/P4 C 4*H0P*PPD/P2 +2*CP*MPD/P0K C +G2*MPD/(P0K*P2) +8*H0/P6 4*H0P/P4 +4*U*H0S/P6 C +4*H0*PPD/P4 C*MPD/(P0K*P2) C 2*G2*MPDP/POK 4*H0*PPDP/P2 +2*C*MPDP/P0K ZE6U=2*G2S*MPD/POK 4*H0S/P4 +4*H0SP/P2 2*U*H0SS/P4 C 4*H0S*PPD/P2 +2*CS*MPD/P0K 2*H0S/P4 ZE7=( 2*H0S/(P2*PK) ) ZE7P=2*H0SP/(P2*PK) 3*H0S/(P4*PK) ZE7U=2*H0SS/(P2*PK) ZE8=(2*G0*MPD +2*H1/(P2*PK) 4*H1P/PK +2*U*H1S/(P2*PK) C +4*H1*PPD/PK 2*U*C*MPD ) ZE8P=2*G0P*MPD +2*H1P/(P2*PK) 4*H1PP/PK +2*U*H1SP/(P2*PK) C +4*H1P*PPD/PK 2*U*CP*MPD C 3*H1/(P4*PK) +2*H1P/(P2*PK) 3*U*H1S/(P4*PK) C 2*H1*PPD/(P2*PK) c 2*G0*MPDP +4*H1*PPDP/PK 2*U*C*MPDP ZE8U=2*G0S*MPD +2*H1S/(P2*PK) 4*H1SP/PK +2*U*H1SS/(P2*PK) C +4*H1S*PPD/PK 2*U*CS*MPD +2*H1S/(P2*PK) 2*C*MPD ZE9=( 2*H2S/PK ) ZE9P=2*H2SP/PK H2S/(P2*PK) ZE9U=2*H2SS/PK ZE10=( G2*ABD/P0K +2*H0/P2 +2*H0*BBD +2*H2*BBD ) ZE10P=G2P*ABD/POK +2*H0P/P2 +2*H0P*BBD +2*H2P*BBD C 0.5D0*G2*ABD/(P2*POK) 2*H0/P4 K;2*ABDP/P0K +2*H0*BBDP C +2*H2*BBDP ZE10U=G2S*ABD/POK +2*H0S/P2 +2*H0S*BBD +2*H2S*BBD ZE11=( F*BBD/PK G1*ABD/P2 +2*U*H0*BBD/PK 2*H1*BBD/PK) ZE11PÂ«*FP*BBD/PK G1P*ABD/P2 +2*U*H0P*BBD/PK 2*H1P*BBD/PK C 0.5D0*F*BBD/(P2*PK) +G1*ABD/P4 C U*H0*BBD/(P2*PK) +H1*BBD/(P2*PK) C +F*BBDP/PK G1*ABDP/P2 +2*U*H0*BBDP/PK 2*H1*BBDP/PK ZE11U=FS*BBD/PK G1S*ABD/P2 +2*U*H0S*BBD/PK 2*H1S*BBD/PK C +2*H0*BBD/PK ZE12=( 4*H2P 2*U*H2S/P2 4*H2*PPD +2*H2*BBD C 2*PK*C*MPD +C*ABD/POK ) ZE12P=4*H2PP 2*U*H2SP/P2 4*H2P*PPD +2*H2P*BBD C 2*PK*CP*MPD +CP*ABD/POK C +2*U*H2S/P4 C*MPD/POK 0.5D0*C*ABD/(P2*POK)
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168 C 4*H2*PPDP +2*H2*BBDP 2*PK*C*MPDP +C*ABDP/POK ZE12U=Â»4*H2SP 2*U*H2SS/P2 4*H2S*PPD +2*H2S*BBD 2*PK*CS*MPD C +CS*ABD/POK 2*H2S/P2 ZE13=( 2*H1S/P2 +2*H2*BBD + C*ABD/POK ) ZE13P=2*H1SP/P2 +2*H2P*BBD +CP*ABD/POK +2*H1S/P4 0.5D0*C*ABD C /(P2*P0K) +2*H2*BBDP +C*ABDP/POK ZE13U=2*H1SS/P2 +2*H2S*BBD +CS*ABD/POK ZE14="( 2*H0/P2 2*H2*BBD C*ABD/POK ) ZE14P=2*H0P/P2 2*H2P*BBD CP*ABD/POK 2*H0/P4 C +0.5D0*C*ABD/(P2*POK) C 2*H2*BBDP C*ABDP/POK ZE14U=2*H0S/P2 2*H2S*BBD CS*ABD/POK ZlÂ» (GO*BBD +G1/P2 +2*H1*ABD/PK U*C*BBD ) Z1P=G0P*BBD +G1P/P2 +2*H1P*ABD/PK U*CP*BBD C G1/P4 H1*ABD/(P2*PK) GO*BBDP +2*H1*ABDP/PK U*C*BBDP Z1U=G0S*BBD +G1S/P2 +2*H1S*ABD/PK U*CS*BBD C*BBD Z2= ( 2*f*MPD/PK 2*G1/P4 +2*G1P/P2 U*G1S/P4 C 2*G1*PPD/P2 +4*U*H0*MPD/PK 4*H1*MPD/PK ) Z2P=2*FP*MPD/PK 2*G1P/P4 +2*G1PP/P2 U*G1SP/P4 C 2*G1P*PPD/P2 +4*U*H0P*MPD/PK 4*H1P*MPD/PK C F*MPD/(P2*PK) +4*G1/P6 2*G1P/P4 +2*U*G1S/P6 C +2*G1*PPD/P4 2*U*H0*MPD/(P2*PK) +2*H1*MPD/(P2*PK) C +2*F*MPDP/PK 2*G1*PPDP/P2 +4*U*H0*MPDP/PK 4*H1*MPDP/PK Z2U=2*FS*MPD/PK 2*G1S/P4 +2*G1SP/P2 U*G1SS/P4 C 2*G1S*PPD/P2 +4*U*H0S*MPD/PK 4*H1S*MPD/PK C G1S/P4 +4*H0*MPD/PK Z3= ( G1S/(P2*P0K) G2*BBD/P0K 2*H0*ABD/P2 +C*BBD/POK ) Z3P=G1SP/(P2*P0K) G2P*BBD/P0K 2*H0P*ABD/P2 +CP*BBD/POK C1.5D0*G1S/(P4*P0K) +0.5DO*G2*BBD/(P2*POK) C +2*H0*ABD/P4 0.5D0*C*BBD/(P2*POK) C G2*BBDP/P0K 2*H0*ABDP/P2 +C*BBDP/POK Z3U=G1SS/(P2*P0K) G2S*BBD/P0K 2*H0S*ABD/P2 +CS*BBD/POK Z4=( G2/P0K +2*H2*ABD +PK*C*BBD ) Z4P=G2P/POK +2*H2P*ABD +PK*CP*BBD 0.5D0*G2/(P2*POK) C +0.5D0*C*BBD/POK C +2*H2*ABDP +PK*G*BBDP Z4U=G2S/POK +2*H2S*ABD +PK*CS*BBD Z5= (G2/(P2*POK) U*G2S/(P2*POK) KJ2*BBD/P0K +2*G2P/POK C 2*G2*PPD/POK +2*H0*ABD/P2 4*H0*MPD 4*H2*MPD C C*BBD/POK ) Z5P=G2P/(P2*POK) U*G2SP/(P2*P0K) +G2P*BBD/P0K C +2*G2PP/POK 2*G2P*PPD/POK C +2*H0P*ABD/P2 4*H0P*MPD 4*H2P*MPD CP*BBD/POK C +1.5D0*G2/(P4*POK) +1.5D0*U*G2S/(P4*POK) C 0.5DO*G2*BBD/(P2*POK) G2P/(P2*POK) C +G2*PPD/(P2*P0K) 2*H0*ABD/P4 +0.5D0* C*BBD/(P2*P0K) C +G2*BBDP/P0K 2*G2*PPDP/POK +2*H0*ABDP/P2 4*H0*MPDP C 4*H2*MPDP C*BBDP/POK Z5U=G2S/(P2*POK) U*G2SS/(P2*P0K) +G2S*BBD/P0K +2*G2SP/P0K C 2*G2S*PPD/POK +2*H0S*ABD/P2 4*H0S*IffD 4*H2S*MPD C CS*BBD/POK C G2S/(P2*POK) Z6= (G2S/P2) Z6P=G2SP/P2 G2S/P4
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lf'9 Z6U=G2SS/P2 Z7= (F*ABD/PK 4Â€0*BBD +G1/P2 +G1*BBD +U*G2*BBD) Z7P=FP*ABD/PK +GOP*BBD +G1P/P2 +G1P*BBD +U*G2P*BBD C +0.5DO*F*ABD/(P2*PK) G1/P4 F*ABDP/PK +GO*BBDP C +G1*BBDP +U*G2*BBDP Z7U=FS*ABD/PK +GOS*BBD +G1S/P2 +G1S*BBD +U*G2S*BBD C +G2*BBD Z8=(2*G0P U*G0S/P2 2*G0*PPD +GO*BBD +4*P0K*H1*MPD C 2*H1*ABD/PK +4*H2*POK*U*MPD +U*C*BBD ) Z8P=2*G0PP U*G0SP/P2 2*G0P*PPD +GOP*BBD +4*P0K*H1P*MPD C 2*H1P*ABD/PK +4*H2P*POK*U*MPD +U*CP*BBD C +U*G0S/P4 +2*H1*MPD/PK +H1*ABD/(P2*PK) +2*H2*U*MPD/PK C 2*G0*PPDP +GO*BBDP +4*P0K*H1*MPDP 2*H1*ABDP/PK C +4*H2*POK*U*MPDP +U*C*BBDP Z8U=2*G0SP U*G0SS/P2 2*G0S*PPD +GOS*BBD +4*P0K*H1S*MPD C 2*H1S*ABD/PK +4*H2S*POK*U*MPD +U*CS*BBD C G0S/P2 +4*P0K*H2*MPD +C*BBD Z9= (GOS/POK 2*H2*ABD PK*C*BBD ) Z9P=G0SP/POK 2*H2P*ABD PK*CP*BBD C 0.5DO*GOS/(P2*POK) 0.5DO*C*BBD/POK C 2*H2*ABDP PK*C*BBDP Z9U=G0SS/POK 2*H2S*ABD PK*CS*BBD Z10=(G2*BBD/POK 2*HO*ABD/P2 +C*BBD/POK ) Z10P=G2P*BBD/POK 2*HOP*ABD/P2 + CP*BBD/POK C +0.5DO*G2*BBD/(P2*POK) +2*H0*ABD/P4 0.5DO*C*BBD/(P2*POK) C G2*BBDP/POK 2*H0*ABDP/P2 + C*BBDP/POK Z10U=G2S*BBD/POK 2*H0S* ABD/P2 + CS*BBD/POK Zll = ( 4*H2*MPD C/(P2*POK) +2*CP/POK U*CS/(P2*POK) C 2*C*PPD/POK ) Z11P=4*H2P*MPD CP/(P2*POK) +2*CPP/POK U*CSP/(P2*POK) C 2*CP*PPD/POK +1.5DO*C/(P4*POK) CP/(P2*POK) C +1.5DO*U*CS/(P4*POK) C + C*PPD/(P2*POK) 4*H2*MPDP 2*C*PPDP/POK Z11U=4*H2S*MPD CS/(P2*POK) +2*CSP/POK U*CSS/(P2*POK) C 2*CS*PPD/POK C CS/(P2*POK) Z12=( GO*BBD 2*H1*ABD/PK +U*C*BBD ) Z12P=GOP*BBD 2*H1P*ABD/PK +U*CP*BBD +H1*ABD/ (P2*PK) C +GO*BBDP 2*H1*ABDP/PK +U*C*BBDP Z12U=GOS*BBD 2*H1S*ABD/PK +U*CS*BBD +C*BBD Z14=(CS/P2 ) Z14P=CSP/P2 CS/P4 Z14U=CSS/P2 Z13=(2*H2*ABD +C/POK +PK*C*BBD ) Z13P=2*H2P*ABD +CP/POK +PK*CP*BBD 0.5DO* C/ (P2*POK) C +0.5DO*C*BBD/POK C +2*H2*ABDP +PK*C*BBDP Z13U=2*H2S*ABD +CS/POK +PK*CS*BBD V1 = AD*Z1 +BD*(ZE5 +P0K*U*ZE14 +ZE14) V1P=AD*Z1P +ADP*Z1 +BDP*(ZE5 +P0K*U*ZE14 +ZE14) C +BD*(ZE5P +POK*U*ZE14P + ZE14P +0.5DO*U* ZE14/PK) V1U=AD*Z1U+ADU*Z1 C +BD*(ZE5U +P0K*ZE14 +POK.*U* ZE14U +ZE14U) C +BDU*(ZE5 +P0K*U*ZE14 +ZE14)
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170 V2=AB*Z2 +BD*(ZE1 +POK*U*ZE6 + ZE8 + ZE1 1 +ZE6 ) V2P=ADP*Z2 + Z2P*AD +BD*(ZE1P +0. 5D0*U* ZE6/PK +POK*U*ZE6P C +ZE8P+ZE11P + ZE6P ) +BDP*(ZE1+P0K*U*ZE6+ZE8+ZE1 1 +ZE6) V2U=ADU*Z2 +AD*Z2U +BDU*(ZE1+P0K*U*ZE6+ZE8+ZE1 1 + ZE6) C +BD*(ZE1U+P0K*ZE6+P0K*U*ZE6U+ZE8U+ZE11U+ZE6U) V3=AD*Z3 +BD*(ZE2 +PK*U*ZE7 +ZE13 +K2*ZE7 +K2*ZE11 ) V3P=AD*Z3P+ADP*Z3 +BDP*(ZE2 +PK*U*ZE7 + ZE13 +K2*ZE7 +K2*ZE11 ) C +BD*(ZE2P+PK*U*ZE7P +0.5DO*U*ZE7/POK + ZE13P C +K2*ZE7P+K2*ZE11P) V3U=ADU*Z3+AD*Z3U +BDU*(ZE2 +PK*U*ZE7 + ZE13 +K2*ZE7 +K2*ZE11 ) C +BD*(ZE2U+PK*ZE7+PK*U*ZE7U+ZE13U +K2*ZE7U +K2*ZE11U) V4=AD*Z4 +BD*(P2*ZE14 K2*ZE5 PK*U*ZE14 ) V4P=ADP*Z4+Z4P*AD +BDP*(P2*ZE14 K2*ZE5 PK*U*ZE14 ) c +BD*(P2*ZE14P ZE14 K2*ZE5P 0.5D0*U*ZE14/POK PK*U*ZE14P) V4U=ADU*Z4+Z4U*AD +BDU*(P2*ZE14 K2*ZE5 PK*U*ZE14) C +BD*(P2*ZE14UK2*ZE5U PK*ZE14 PK*U*ZE14U) V5=AD*Z5 +BD*(P2*ZE6 ZE10 ZE12 K2*ZE1 PK*U*ZE6 ) V5P=ADP*Z5+Z5P*AD+BDP*(P2*ZE6 ZE10 ZE12 K2*ZE1 PK*U*ZE6) C +BD*(ZE6P2*ZE6PZE10PZE12PK2*ZE1PPK*U*ZE6P C U*0.5D0*ZE6/POK ) V5U=ADU*Z5+Z5U*ADfBDU*(P2*ZE6 ZE10 ZE12 K2*ZE1 PK*U*ZE6) C +BD*(P2*ZE6UZE10UZE12UK2*ZE1UPK*ZE6 PK*U*ZE6U) V6=AD*Z6 +BD*((P2+PK*U)*ZE7 ZE9 ZE2 ZE10 ) V6P=ADP*Z6+Z6P*ATHBDP*((P2+PK*U)*ZE7 ZE9 ZE2 ZE10 ) C +BD*( ZE7 U*0.5DO*ZE7/POK (P2+PK*U)*ZE7P ZE9P ZE2P C ZE10P ) V6U=AD*Z6U+ADU*Z6+BDU*((P2+PK*U)*ZE7 ZE9 ZE2 ZE10 ) C +BD*(PK*ZE7 (P2+PK*U)*ZE7U ZE9U ZE2UZE10U) V7 = AD*Z7 +BD*(ZE4 +POK*U*ZE10 (P2+PK*U)*ZE1 1 +ZE10 ) V7U=ADU*Z7+AD*Z7U +BDU*(ZE4 +POK*U*ZE10 C (P2+PK*U)*ZE11 +ZE10) C +BD*(ZE4U +POK*ZE10 +P0K*U*ZE10U C PK*ZE11 (P2+PK*U)*ZE11U C +ZE10U) V7P=AD*Z7P+ADP*Z7 +BDP*(ZE4 +P0K*U*ZE10 (P2+PK*U)*ZE1 1 C +ZE10 ) C +BD*(ZE4P+0.5D0*U*ZE10/PK +POK*U*ZE10P (P2+PK*U)*ZE1 IP C ZE11 0.5D0*U*ZE11/P0K +ZE10P ) V8=AD*Z8 +BD*( ZE4(P2+PK*U)*ZE8 +(1.D0 +P0K*U)*ZE12 ) V8P=ADP*Z8+AD*Z8P +BDP*( ZE4(P2+PK*U)*ZE8 C +(1.D0+P0K*U)*ZE12 ) C +BD*( ZE4P ZE8 0.5D0*U*ZE8/POK (P2+PK.*U)*ZE8P + C (l.DCHP0K*U)*ZE12P +U*0.5D0*ZE12/PK ) V8U=ADU*Z8+AD*Z8U +BDU*(ZE4 (P2+PK*U)*ZE8 C +(1.D0+P0K*U)*ZE12) C +BD*( ZE4U PK*ZE8 (P2+PK*U)*ZE8U +( 1 .D0+P0K*U)*ZE12U C +P0K*ZE12) V9=AD*Z9 +BD*((K2+PK*U)*ZE9 (P2+PK*U)*ZE13 +K2* ZE4 ) V9P=ADP*Z9 +AD*Z9P +BDP*( (K2+PK*U)*ZE9 C (P2+PK*U)*ZE13 +K2*ZE4) C +BD*(0.5D0*U*ZE9/POK +(K2+PK*U)*ZE9P ZE13 (P2+PK*U)*ZE13P C +K2*ZE4P U*0.5DO*ZE13/POK ) V9U=ADU*Z9+Z9U*AD +BDU*( (K2+PK*U)* ZE9 (P2+PK*U)* ZE13 +K2*ZE4) C +BD*( (K2+PK*U)*ZE9U +PK* ZE9 (P2+PK*U)* ZE1 3U
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171 C PK*ZE13 +K2*ZE4U) V10=AD*Z10 +BD*( ZE3 +ZE10 +K2*ZE11 ) V10P=AD*Z10P+ADP*Z10 +BDP*( ZE3 +ZE10 +K2*ZE11) C +BD*( ZE3P + ZE10P +K2*ZE11P ) V10U=AD*Z10U +ADU*Z10 +BDU*( ZE3 +ZE10 +K2*ZE11 ) C +BD*( ZE3U +ZE10U +K2*ZE11U ) V11 = AD*Z11 +BD*(ZE12 ZE3 +K2*ZE8) V11P=ADP*Z11 +AD*Z11P +BDP*(ZE12 ZE3 +K2*ZE8) C +BD*(ZE12P ZE3P +K2*ZE8P) V11U=ADU*Z11 +AD*Z11U +BDU*(ZE12 ZE3 +K2*ZE8) C +BD*(ZE12U ZE3U +K2*ZE8U) V12=AD*Z12 +BD*(+(1.D0+P0K*U)*ZE3 +ZE4 ) V12P=ADP*Z12 +AD*Z12P +BDP*( ( 1 . D0+P0K*U)*ZE3 +ZE4) C +BD*((1.D0+P0K*U)*ZE3P +0.5DO*U*ZE3/PK +ZE4P) V12U=ADU*Z12 +AD*Z12U +BDU*( ( 1 . D0+P0K*U)*ZE3 + ZE4) C +BD*((1.D0+P0K*U)*ZE3U +P0K*ZE3 +ZE4U) V13=AD*Z13 +BD*(K2*ZE4+(P2+PK*U)*ZE3) V13P=ADP*Z13 +Z13P*AD +BD*(K2* ZE4P +ZE3 +0.5D0*U* ZE3/P0K C +(P2+PK*U)*ZE3P) +BDP*(K2*ZE4+(P2+PK*U)*ZE3) V13U=ADU*Z13 +AD*Z13U +BDU*( K2*ZE4 +(P2+PK*U)*ZE3) C +BD*(K2*ZE4U +(P2+PK*U)*ZE3U +PK*ZE3) V14=AD*Z14 +BD*(ZE9 +ZE13 ZE3) V14P=ADP*Z14+AD*Z14P +BDP*( ZE9 +ZE13 ZE3) C +BD*(ZE9P +ZE13P ZE3P) V14U=ADU*Z14 +AD*Z14U +BDU*( ZE9+ZE13 ZE3) C +BD*(ZE9U +ZE13U ZE3U) W1 = AD*ZE1 +BD*((P2+PK*U)*Z2 ( 1 . DO+POK*U)* Z5 Z7 Z8 ) W1P=AD*ZE1P+ADP*ZE1 +BD*(Z2 U*0. 5D0* Z2/P0K (P2+PK*U)*Z2P C 0.5DO*U*Z5/PK (l.D0+P0K*U)*Z5P Z7P Z8P) C +BDP*((P2+PK*U)*Z2 C (l.D0+P0K*U)*Z5 Z7 Z8) W1U=ADU*ZE1 +AD*ZE1U +BDU*((P2+PK*U)*Z2 ( 1 . D0+P0K*U)*Z5 Z7 Z8) C +BD*(PK*Z2 (P2+PK*U)*Z2U P0K*Z5 ( 1 .D0+P0K*U)*Z5U Z7U Z8U) W2=AD*ZE2 +BD*((P2+PK*U)*Z3 (K2+PK*U)*Z6 K2* Z7 Z9 ) W2P=ADP*ZE2 +AD*ZE2P C +BDP*((P2+PK*U)*Z3 (K2+PK*U)*Z6 K2*Z7 Z9) C +BD*tZ3 0.5D0*U*Z3/POK (P2+PK*U)*Z3P 0.5DO*U*Z6/POK C (K2+PK*U)*Z6P K2*Z7P Z9P ) W2U=ADU*ZE2 +AD*ZE2U +BDU*((P2+PK*U)*Z3 (K2+PK*U)*Z6 C K2*Z7Z9) C +BD*(PK*Z3 (P2+PK*U)*Z3U (K2+PK*U)*Z6U PK*Z6 K2*Z7U Z9U) W3=AD*ZE3 +BD*( Z13 +K2*Z12) W3P=ADP*ZE3 +AD*ZE3P +BDP*(Z13 +K2*Z12 ) +BD*( Z13P+K2* Z12P) W3U=ADU*ZE3 +AD*ZE3U +BDU*( Z13+K2*Z12 ) +BD*( Z13U+K2*Z12U) W4=AD*ZE4+BD*(+(P2+PK*U)*Z12 ( 1 . D0+P0K*U)*Z13 ) W4P=ADP*ZE4 +AD*ZE4P +BDP*( (P2+PK*U)*Z12 ( 1 . DO+POK*U)* Z13) C +BD*( Z12 +0.5D0*U*Z12/P0K +(P2+PK*U)*Z12P ( 1 . D0+P0K*U)*Z13P C U*0.5D0*Z13/PK ) W4U=ADU*ZE4 +AD*ZE4U +BDU*( (P2+PK*U)* Zl 2 ( 1 ,DO+POK*U)* Z13) C +BD*(PK*Z12 +(P2+PK*U)*Z12U ( 1 . DO+POK*U)* Z13U P0K*Z13 ) W5=AD*ZE5 +BD*((P2+PK*U)*Z1 ( 1 . DO+POK*U)* Z4 ) W5P=ADP*ZE5 +AD*ZE5P +BDP*(( P2+PK*U)* Zl ( 1 . DO+POK*U)* Z4 )
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172 C +BD*(Z1 0.5DO*U*Z1/POK (P2+PK*U)*Z1P ( 1 . DO+POK*U)* Z4P C 0.5DO*U*Z4/PK ) W5U=ADU*ZE5+AD*ZE5U +BDU*((P2+PK*U)*Z1 ( 1 . DO+POK*U)* Z4) c +BD*(PK*Z1 (P2+PK*U)*Z1U ( 1 . DO+POK*U)* Z4U P0K*Z4) W6=AD*ZE6+BD*(K2*Z2 Z5 Z10 Zll ) W6P=ADP*ZE6 +AD*ZE6P +BDP*( K2* Z2 Z5Z10Z11) C +BD*(K2*Z2P Z5P ZIOP Z11P) W6U=ADU*ZE6 +AD*ZE6U +BDU*( K2*Z2 75 ZIO Zll) C +BD*( K2*Z2U Z5U ZIOU Z11U) W7 = AD*ZE7+BD*(Z3 Z6 ZIO Z14 ) W7P=ADP*ZE7 +AD*ZE7P +BDP*( Z3 Z6 ZIO Z14) C +BD*(Z3P Z6P ZIOP Z14P ) W7U=ADU*ZE7 +AD*ZE7U +BDU*(Z3 Z6 ZIO Z14 ) C +BD*(Z3U Z6U ZIOU Z14U) W8=AD*ZE8 +BD*(Z8 +(1 .DO+POK*U)*Zll + Z12 ) W8P=ADP*ZE8 +AD*ZE8P +BDP*(Z8 +( 1 .DO+POK*U)* Zll +Z12 ) C +BD*( Z8P +(1.D0+P0K*U)*Z11P K).5DO*U*Zl 1/PK +Z12P) W8U=ADU*ZE8 +AD*ZE8U +BDU*( Z8 +( 1 .DO+POK*U)* Zll +Z12) C +BD*( Z8U +P0K*Z11 +(1.D0+P0K*U)*Z11U +Z12U ) W9 = AB*ZE9 +BD*(Z9 +(P2+PK*U)*Z14 +Z13 ) W9P=ADP*ZE9 +AD*ZE9P +BDP*(Z9 +(P2+PK*U)*Z14 +Z13) C +BD*(Z9P +Z14 +U*0.5DO*Z14/POK +(P2+PK*U)*Z14P +Z13P) W9U=ADQ*ZE9 +AD*ZE9U +BDU*(Z9 +(P2+PK*U)*Z14 +Z13) C +BD*( Z9U +(P2+PK*U)*Z14U +PK*Z14 +Z13U) W10=AD*ZE10 +BD*(K2*Z7 +(P2+PK*U)*Z10 Z13 ) W10P=ADP*ZE10 +AD*ZE10P +BDP*(K2*Z7 +(P2+PK*U)* Z10Z13) C +BD*(K2*Z7P +Z10 +0.5DO*U* ZIO/POK +(P2+PK*U)*Z10P Z13P) W10U=ADU*ZE10 +AD*ZE10U +BDU*(K2*Z7 +(P2+PK*U)* ZIO Z13) C +BD*(K2*Z7U +PK*Z10 +(P2+PK*U)*Z10U Z13U) W11=AD*ZE11 +BD*(Z7 +( 1 . DO+POK*U)* ZIO Z12 ) W11P=ADP*ZE11 +AD*ZE11P +BDP*(Z7 +( 1 . DO+POK*U)* ZIO Z12) C +BD*(Z7P +(1.DO+POK*U)*Z10P +0.5DO*U* ZIO/PK Z12P) W11U=ADU*ZE11 +AD*ZE11U +BDU*( Z7 +( 1 . DO+POK*U)* ZIO Z12) c +BD *( _27U +(1.DO+POK*U)*Z10U +POK*Z10 Z12U ) W12=AD*ZE12 +BD*(K2*Z8 +(P2+PK*U)*Z11 +Z13 ) W12P=ADP*ZE12 +AD*ZE12P +BDP*(K2*Z3 +(P2+PK*U)* Zll +Z13) C +BD*( K2*Z8P +Z11 +0.5D0*U*Z11/P0K +(P2+PK*U)*Z1 IP +Z13P) W12U=ADU*ZE12 +AD*ZE12U +BDO*( K2*Z8 +(P2+PK*U)*Z11 +Z13) C +BD*( K2*Z8U +PK*Z11 +(P2+PK*U)*Z1 1U +Z13U ) W13=AD*ZE13 +BD*(Z9 +(K2+PK*U)*Z14 +K2*Z12 ) W13P=AD*ZE13P+ADP*ZE13 +BDP*(Z9 +(K2+PK*U)*Z14 +K2*Z12) C +BD*(Z9P +0.5DO*U*Z14/POK +(K2+PK*U)*Z14P +K2*Z12P ) W13U=ADU*ZE13 +AD*ZE13U +BDU*( Z9 +(K2+PK*U)*Z14 +K2*Z12) C +BD*(Z9U +PK*Z14 +(K2+PK*U)*Z14U +K2*Z12U) W14=AD*ZE14 +BD*( K2*Z1 ZA ) W14P=ADP*ZE14 +AD*ZE14P +BDP*(K2*Z1ZA) +BD*(K2* ZIPZAP) W14U=ADU*ZE14 +AD*ZE14U +BDU*( K2*Z1ZA) +BD*(K2* Z1UZAU) CALL TETO(P2,U,K2) RETURN END SUBROUTINE TETO(P2,U,K2) IMPLICIT REAL*8 (AH.OZ) REAL* 8 K2 COMMON/ SIGNPK/ SIGN
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173 C0MM0N/SUBTT/TE(H),TES(14),TEP(14),A0UT,AP0UT,APP,B0UT,BP0UT, C TO(14),TOS(14),TOP(14) COMMON/TWS/W1,W2,W3,W4,W5,W6,W7,W8,W9,W10,W11,W12,W13 > W14 C , Wl P , W2P , W3P , W4P , W5P , W6P , W7P , W8P , W9P , Wl OP , Wl IP , Wl 2P , Wl 3P , Wl 4P C ,W1U,W2U,W3U,W4U,W5U,W6U,W7U,W8U,W9U,W10U,W11U,W12U,W13U,W14U COMMON/TVS/V1,V2,V3,V4,V5,V6,V7,V8,V9,V10,V11,V12,V13,V14 C ,V1P,V2P,V3P,V4P,V5P,V6P,V7P,V8P,V9P,V10P,V11P,V12P,V13P,V14P C ,V1U,V2U,V3U,V4U,V5U,V6U,V7U,V8U,V9U,V10U,V11U,V12U,V13U,V14U P4=P2*P2 P6=P2*P4 PK=SIGN*DSQRT(P2*K2) POK=DSQRT(P2/K2) TE(1)=( 1.5DO*Wl 0.5D0*POK*U*W6 0.5DO*W7 0.5DO*W8 1.5DO*Wll) TEP(1)=(1.5D0*W1P 0.5DO*POK*U*W6P 0.25DO*U*W6/PK 0.5DO*W7P C 0.5DO*W8P 1.5D0*W11P ) TES(1)=1.5D0*W1U 0.5DO*POK*W6 0.5DO*POK*U*W6U 0.5DO*W7U C 0.5DO*W8U 1.5D0*W11U TE(2)=( 1.5DO*W2 +0.5DO*P2*W6 +0.5DO*PK*U*W7 +1.5DO*W10 C +0.5D0*W12 +0.5D0*W14 ) TEP(2)=0.5DO*( 3*W2P +W6 +P2*W6P +PK*U*W7P +0.5DO*U*W7/POK C +3*W10P +W12P +W14P ) TES(2)=0.5D0*( 3*W2U +P2*W6U +PK*U*W7U +PK*W7 C +3*W10U +W12U +W14U) TE(3)=( 1.5DO*W3 0.5D0*W12 0.5DO*W13 +0.5D0*W14 ) TEP(3)=0.5DO*( 3*W3P W12P W13P +W14P) TES(3)=0.5DO*( 3*W3U W12U W13U +W14U) TE(4)=( 2.5DO*W4 0.5DO*W5 +0.5DO*W9 +0.5DO*POK*U*W12 C 0.5DO*P2*W8 0.5DO*POK*U*W13 ) TEP(4)=0.5D0*(5*W4P W5P +W9P +POK*U*W12P +0.5DO*U*W12/PK C P2*W8P W8 POK*U*W13P 0.5DO*U*W13/PK ) TES(4)=0.5DO*(5*W4U W5U +W9U +POK*U*W12U +POK*W12 P2*W8U C POK*U*W13U POK*W13 ) TE(5)=( 1.5DO*W5 1.5DO*W4 0.5DO*POK*U*W12 +0.5DO*P2*W8 C +0.5DO*POK*U*W13 0.5DO*W9 ) TEP(5)=0.5DO*(3*W5P 3*W4P POK*U*W12P U*0.5DO*W12/PK +W8 C +P2*W8P +POK*U*W13P +0.5D0*U*W13/PK W9P ) TES(5)= 0.5DO*( 3*W5U 3*W4U POK*W12 POK*U*W12U +P2*W8U C +POK*W13 +POK*U*W13U W9U ) TE(6)=W6 TEP(6)=W6P TES(6)=W6U TE(7)=W7 TEP(7)=W7P TES(7)=W7U TE(8)=W8 TEP(8)=W8P TES(8)=W8U TE(9)=W9 TEP(9)=W9P TES(9)=W9U TE(10)=( 2.5DO*W10 +0.5DO*W2 +0. 5DO*P2*W6 +0. 5DO*PK*U*W7 C +0.5D0*W14 +0.5D0*W12 ) TEP(10)=0.5D0*( 5*W10P +W2P +W6 +P2*W6P +PK*U*W7P C +0.5DO*U*W7/POK +W14P +W12P )
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174 TES(10)=0.5D0*( 5*W10U +W2U +P2*W6U +PK*U*W7U +PK*W7 C +W14U +W12U ) TE(11)=(2.5D0*W11 0.5D0*W1 +0. 5D0*POK*U*W6 +0.5DO*W7 C +0.5D0*W8) TEP(11)=0.5D0*(5*W11P W1P +P0K*U*W6P +0.5D0*U*W6/PK +W7P +W8P) TES(11)=0.5D0*(5*W11UW1U +POK*W6 +POK*U*W6U +W7U +W8U ) TE(12)=( 1.5D0*W12 0.5D0*W3 0.5D0*W14 +0.5D0*W13 ) TEP(12)=0.5D0*( 3*W12P W3P W14P +W13P ) TES(12)=0.5D0*(3*W12U W3U W14U +W13U ) TE(13)=( 1.5D0*W13 0.5DO*W3 0.5DO*W14 +0.5D0*W12 ) TEP(13)=0.5DO*(3*W13P W3P W14P +W12P) TES(13)=0.5D0*(3*W13U W3U W14U +W12U) TE(14)Â«( 1.5D0*W14 +0.5D0*W3 0.5DO*W12 0.5D0*W13 ) TEP(14)=0.5D0*(3*W14P +W3P W12P W13P) TES(14)=0.5D0*(3*W14U +W3U W12U W13U) T0(1)=(1.5D0*V1 0.5DO*V8 +0.5D0*POK*U*Vll +V12 +0.5D0*V14 ) TOP(1)=0.5DO*(3*V1P V8P +P0K*U*V11P +0.5D0*U*V11/PK C +2*V12P+V14P) TOS(1)=0.5DO*(3*V1U V8U +P0K*V11 +P0K*U*V11U +2*V12U+V14U) TO(2)=V2 T0P(2)=V2P TOS(2)=V2U TO(3)=(1.5D0*V3 0.5D0*V5 V10 0.5D0*V11 ) TOP(3)=0.5D0*( 3*V3P V5P 2*V10P VHP ) TOS(3)=0.5D0*(3*V3U V5U 2*V10U V11U) TO(4)=(1.5D0*V4 0.5D0*V9 0.5D0*P2*V11 V13 0.5D0*PK*U*V14) TOP(4)=0.5D0*(3*V4P V9P Vll P2*V11P2*V13P0.5DO*U*V14/POK C PK*U*V14P ) TOS(4)=0.5DO*(3*V4U V9U P2*V11U 2*V13U PK*V14 PK*U*V14U) TO(5)=(1.5DO*V5 0.5DO*V3 +V10 +0.5D0*V11 ) TOP(5)=0.5D0*(3*V5P V3P +2*V10P +V11P) TOS(5)=0.5D0*(3*V5U V3U +2*V10U +V11U) TO(6)=V6 TOP(6)=V6P TOS(6)=V6U TO(7)=( 3.DO*V7 +0.5D0*V1 +0.5D0*P2*V2 +0.5D0*POK*U*V3 C +0.5DO*POK*U*V5 +0.5DO*V6 +0.5D0*V8 ) TOP(7)=0.5D0*( 6.DO*V7P +V1P +P2*V2P +V2 +0.5D0*U*V3/PK C +POK*U*V3P +0.5D0*U*V5/PK +POK*U*V5P +V6P +V8P) TOS(7)=0.5D0*( 6.DO*V7U +V1U +P2*V2U C +POK*V3 +POK*U*V3U +POK*V5 C +POK*U*V5U +V6U +V8U ) TO(8)=(1.5DO*V8 0.5D0*V1 0.5DO*POK*U*V11 V12 0.5D0*V14 ) TOP(8)=0.5DO*(3*V8PV1P P0K*U*V11P C 0.5D0*U*Vll/PK 2*V12P V14P) TOS(8)=0.5D0*(3*V8U V1U P0K*U*V11U P0K*V11 2*V12U V14U) TO(9)=(1.5D0*V9 0.5D0*V4+0.5D0*P2*V11 +V13 +0.5D0*PK*U*V14 ) TOP(9)=0.5DO*(3*V9P V4P +V11 +P2*V11P +2*V13P +PK*U*V14P C +0.5D0*U*V14/POK ) TOS(9)=0.5DO*(3*V9U V4U +P2*V11U +2*V13U +PK*U*V14U +PK*V14) TO(10)=(1.5DO*V10 0.5DO*V3 +0.5D0*V5 +0.5D0*V10 +0.5D0*V11 ) TOP(10)=0.5D0*( 3*V10P V3P +V5P +V10F +V11P) TOS(10)=0.5DO*( 3*V10U V3U +V5U +V10U +V11U) T0(11)=V11
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175 T0P(11)=V11P T0S(11)=V11U TO(12)=(1.5D0*V12 +0.5D0*V1 0.5D0*V8 +0.5D0*V12 C +0.5DO*POK*U*V11 +0.5D0*V14 ) TOP(12)=0.5DO*( 3*V12P +V1P V8P +V12P +P0K*U*V11P +0.5D0*U* C Vll/PK +V14P) TOS(12)=0.5D0*( 3*V12U +V1U V8U +V12U C +P0K*V11 +P0K*U*V11U +V14U) T0(13)=( 1.5D0*V13 0.5D0*V4 +0.5D0*V13 +0. 5D0*P2*V11 C +0.5D0*PK*U*V14 +0.5D0*V9 ) TOP(13)=0.5DO*( 3*V13P V4P +V13P +V11 +P2*V11P +PK*U*V14P C +0.5D0*U*V14/POK +V9P ) TOS(13)=0.5DO*( 3*V13U V4U +V13U +P2*V11U +PK*V14 +PK*U*V14U C +V9U ) T0(14)=V14 TOP(14)=V14P T0S(14)=V14U 50 CONTINUE 99 CONTINUE IK 1=0 IF(IKI.EQ.l) WRITE (6,666) (TOP( IS) , IS=1 ,14) ,(TOS( IR) , IR=1 , 14) 666 FORMAT (' TOP.TOS' .6D15.6) IF(IKI.EQ.1)WRITE (6,556) (TEP( IS) , IS=1 , 14) ,(TES( IR) , IR=1 , 14) 556 FORMAT (' TEP,TES' .6D15.6) RETURN END C THIS IS THE ELECTRON PROPAGATOR FUNCTION A(P2) FUNCTION XA(R) IMPLICIT REAL*8 (AH.OZ) COMMON/ SCALE/ SKL IF (DABS(R).LT.0.1D6) GO TO 1 SE=1.74517D3 SM=1.D0*SKL SM2=SM*SM XR=DABS(1.D0R/SM2) XP=SE*(SM2R)/R XA=XR**XP C XA=XA*(SKL)**(2.D0*XP) XA=XA*SKL RETURN 1 XA=1.D0 C XA=0.0D0 RETURN END C THESE TWO FUNCTIONS DEFINE THE DERIVATIVES OF THE ELECTRON PROPAGATOR C FUNCTION A FUNCTION XAP(R) IMPLICIT REAL*8 (AH.OZ) COMMON/ SCALE/ SKL SE=1.74517D3 SM=1.D0*SKL SM2=SM*SM XR=DABS(1.D0R/SM2) XAP=(SE*SM2*DLOG(XR)/R**2 SE/R)*XA(R)
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176 RETURN END FUNCTION XAPP(R) IMPLICIT REAL*8 (AH.OZ) COMMON/ SCALE/ SKL SE=1.74517D3 SM=l.DO*SKL SM2=SM*SM XR=DABS(l.DOR/SM2) XAPP=(2*SE*SM2*DLOG(XR)/R**3 C +SE*SM2/((SM2R)*R**2) +SE/R**2 )*XA(R) C +(SE*SM2*DLOG(XR)/R**2 SE/R )*XAP(R) RETURN END C THIS IS THE ELECTRON PROPAGATOR FUNCTION B(P2) FUNCTION XB(R) IMPLICIT REAL*8 (AH.OZ) XB=1.0DO RETURN END FUNCTION XBP(R) IMPLICIT REAL*8 (AH.OZ) XBP=O.ODO RETURN END
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177 C THIS IS A MAIN TO FORM THE PERTURBATION SOLUTIONS C IT DOES ALL THE DERIVATIVES OF THE INTEGRALS ANALYTICALLY. C THIS PROGRAM WILL WORK FOR NEGATIVE MOMENTA. C YOU MUST SUPPLY P SQUARED, U AND K SQUARED. IF P2 IS NEGATIVE C SO MUST BE K2 AND YOU MUST GIVE THE SIGN OF THE MOMENTA C THIS PROGRAM SENDS ITS DATA TO JC. FUND WHERE IT IS STORED C FOR J C. MAIN TO USE. IMPLICIT REAL*8 (AH.OZ) REAL* 8 K2 COMMON/ SIGNPK/ SIGN COMMON/ FUNS/XF(8 ,10) COMMON/ SCALE/ SKL WRITE(6,701) 701 FORMATC P2 ,U,K2 , SKL, SIGN' ) READ(9,*) P2,U,K2, SKL, SIGN PK=SIGN*DSQRT(P2*K2) RL=P2+2.D0*PK*U +K2 SM1=XA(RL) SM2=XA(P2) WRITE(6,783) SMI ,SM2 ,RL,P2 783 FORMATC Al AND A2*,2D15.7/' PI SQ. AND P2 SQ.',2D15.7) THETA=0.0 WRITE(4,22)P2,U,K2,SKL,SIGN 22 FORMAT(D24.16,4D11.3) CALL PERTF(P2,U,K2,SM1,SM2,SKL) CALL PERTG(P2,U,K2,SM1,SM2,SKL) CALL PERTH(P2,U,K2,SM1,SM2,SKL) DO 10 1=1,10 XF(7,I)=XF(7,I) XF(8,I)=XF(1,I)/3.D0 10 XF(1,I) = (SM1+SM2)*XF(1,I)/2.D0 WRITE(6,784) XF(7,1) 784 FORMATC H2=',D15.7) DO 20 J=l,8 C WRITE(6,33)(XF(J,I),I=1,10) 20 WRITE(4,33)(XF(J,I),I=1,10) 33 FORMATC 3 D24. 16) STOP END C THIS PROGRAM DEFINES THE F AND I FUNCTIONS AND THEIR DERIVATIVES SUBROUTINE PERTF(P2 ,U,K2 ,SM1 ,SM2 ,SKL) IMPLICIT REAL*8 (AH,0Z) REAL*8 K2 COMMON/ S IGNPK/ S IGN COMMON/ FUNS/ X F( 8 , 1 ) COMMON/ INT/ DI( 40) C THIS IS PERTF EPS=2.322819D03 IOUT=l T1=1.D0 T2=2.D0 T3=3.D0 T4=4.D0 T5=5.D0
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178 SMA=0.5D0*(SM1+SM2) PK=SIGN*DSQRT(P2*K2) POK=DSQRT(P2/K2) CALL INTGRT(1,1,10,SM1,SM2,P2,U,K2) XF(l,l) = 0.75DO*EPS*PK*DI(l) XF(1 ,2)=0.75DO*EPS*(PK*DI(2) +0.5D0*DI( 1 )/POK) XF(l,3)=0.75DO*EPS*( PK*DI(3) +DI(2)/P0K C 0.25DO*DI(l)/(P2*POK)) XF(1 ,4)=0.75D0*EPS*(PK*DI(4) +1 .5DO*DI(3)/POK C 0.75D0*DI(2)/(P2*POK) +T3*DI( 1)/(8.DO*(P2*P2*POK) ) ) XF(1,5)=0.75D0*EPS*PK*DI(5) XF(l,6)=0.75DO*EPS*PK*DI(6) XF(l,7)=0.75DO*EPS*PK*DI(7) XF(1,8)=0.75D0*EPS*(PK*DI(8) +0.5D0*DI(5)/POK) XF(1,9)=0.75DO*EPS*(PK*DI(9) +0.5DO*DI(6)/POK) XF(l,10)=0.75DO*EPS*(PK*DI(10) C +DI(8)/P0K 0.25DO*DI(5)/(P2*POK)) IOUT=0 IF( IOUT.EQ. 1 )WRITE(6 ,77 ) (XF( 1 , IK) , IK=1 , 10) 77 FORMATC THE F' .D24.16/3D24. 16/3D24. 16/3D24. 16) RETURN END C THIS PROGRAM DEFINES THE G0.G1.G2 FUNCTIONS AND DERIVATIVES SUBROUTINE PERTG(P2 ,U,K2 ,SM1 ,SM2 ,SKL) IMPLICIT REAL*8 (AH,0Z) REAL*8 K2 COMMON/ S IGNPK/ S IGN COMMON/ INT/ DI( 40) COMMON/FUNS/ XF(8,10) DATA EPS/2. 322819D03/ AM2=SM1*SM2 PK=SIGN*DSQRT(P2*K2) P4=P2*P2 POK=DSQRT(P2/K2) CALL INTGRT(2, 1,30, SMI, SM2,P2,U,K2) T1=1.0D0 T2=2.0D0 T3=3.0D0 T4=4.0D0 T5=5.0D0 IOUT=0 IF(IOUT.EQ.l)WRITE(6,116)P2,U 116 FORMATC P2,U IN PERTG* ,2D18.8, 13 ,D14.5) WA=P2+T2*PK*U+K2 WAP=T1+U/P0K WAS=T2*PK WAPP=0 . 5 D0*U/ ( P2*POK) WAPPP=0.75DO*U/(P4*POK) WASP=1.D0/P0K WASPP=0.5D0/(P2*POK) W1=AM2/WAT1 W1P=AM2*WAP/WA**2 W1PP=T2*AM2*WAP*WAP/WA**3 AM2*WAPP/WA**2 761 FORMATC 3 Dl 5. 7)
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179 W1PPP=6.D0*AM2*WAP**3/WA**4 +6. DO* AM2*WAPP*WAP/WA**3 C AM2*WAPPP/WA**2 W1S=AM2*WAS/WA**2 W1SS=T2*AM2*WAS*WAS/WA**3 W1SSS=6.D0*AM2*WAS**3/WA**4 W1SP=T2*AM2*WAP*WAS/WA**3 AM2*WASP/WA**2 W1SSP=T2*AM2*WASP*WAS/WA**3 6. DO*AM2*WAP*WAS**2/WA**4 C +T2*AM2*WASP*WAS/WA**3 W1SPP=T2*AM2*WAPP*WAS/WA**3 +T2*AM2*WAP*WASP/WA**3 C 6.DO*AM2*WAP*WAP*WAS/WA**4 AM2*WASPP/WA**2 C +T2*AM2*WASP*WAP/WA**3 W2=DLOG( DABS( AM2WA) )DLOG( AM2 ) W2=DLOG( DABS( 1 .OWA) ) W2P=WAP/(AM2WA) W2PP=(WAP)**2/( AM2WA)**2 C WAPP/(AM2WA) W2PPP=T2*WAPP*WAP/(AM2WA)**2 C T2*(WAP)**3/(AM2WA)**3 WAPPP/( AM2WA) C (WAPP)*(WAP)/(AM2WA)**2 W2S=(WAS)/(AM2WA) W2SS=(WAS)**2/(AM2WA)**2 W2SSS=T2*(WAS)**3/(AM2WA)**3 W2SP=(WASP)/(AM2WA) (WAS)*(WAP)/( AM2WA)**2 W2SSP=T2*(WASP)*(WAS)/(AM2WA)**2 C T2*(WAP)*(WAS)**2/(AM2WA)**3 W2SPP=T2*(WASP)*(WAP)/(AM2WA)**2 C T2*(WAS)*(WAP)**2/(AM2WA)**3 C (WASPP)/(AM2WA) (WAS)*(WAPP)/( AM2WA)**2 W3=AM2/P2T1 W3P=AM2/P2**2 W3PP=T2*AM2/P2**3 W3PPP=6.DO*AM2/P2**4 W4=DLOG( DABS( AM2P2 ) )DLOG( AM2 ) W4=DLOG( DABS( 1 . D0P2) ) W4P=(T1)/(AM2P2) W4PP=T1/(AM2P2)**2 W4PPP=T2/(AM2P2)**3 W=W1*W2+W3*W4 WP=W1P*W2+W1*W2P +W3P*W4+W4P*W3 WPP= W1PP*W2 +T2*W1P*W2P +W1*W2PP +W3PP*W4 +T2*W3P*W4P C +W4PP*W3 WPPP=W1PPP*W2 +T3*W1PP*W2P +T3*W1P*W2PP +W1*W2PPP C +W3PPP*W4 +T3*W3PP*W4P +T3*W3P*W4PP +W3*W4PPP WS=W1S*W2 +W1*W2S WSS=W1SS*W2 +T2*W1S*W2S +W1*W2SS WSSS=W1SSS*W2 +T3*W1SS*W2S +T3*W1S*W2SS +W1*W2SSS WSP=W1SP*W2 +W1S*W2P +W1P*W2S +W1*W2SP +W1SP*W2 +W1SS*W2P WSSP=W1SSP*W2 +W1SS*W2P +W1SP*W2S +W1S*W2SP +W1SSP*W2 C +W1SSS*W2P C +W1SP*W2S +W1S*W2SP +W1P*W2SS +W1*W2SSP +W1SP*W2S +W1SS*W2SP WSPP=W1SPP*W2 +W1SP*W2P +W1PP*W2S +W1P*W2SP +W1SPP*W2 C +W1SSP*W2P C +W1SP*W2P +W1S*W2PP +W1P*W2SP +W1*W2SPP +W1SP*W2P +W1SS*W2PP CK=T2*P2*U+PK
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OCP=T2*U +1.DO/(T2*POK) CKPP=0.25D0/(P2*POK) CKPPP=0.37 5DO/(P4*POK) CKS=T2*P2 CKSP=T2 XF(2,1)=1.D0 +EPS*0.25D0*W +EPS*DI( 1 )/4. DO XF(2,2)=EPS*0.25DO*WP +EPS*DI(2)/4.DO XF(2,3)=EPS*0.25DO*WPP +EPS*DI(3)/4.DO XF(2,4)=EPS*0.25DO*WPPP +EPS*DI(4)/4.DO XF(2,5)=EPS*0.25DO*WS +EPS*DI(5)/4.DO XF(2,6)=EPS*0.25DO*WSS +EPS*DI(6)/4.DO XF(2,7)=EPS*0.25DO*WSSS +EPS*DI(7)/4.DO XF(2,8)=EPS*0.25DO*WSP +EPS*DI(8)/4.DO XF(2,9)=EPS*0.25DO*WSSP +EPS*DI(9)/4.DO XF(2,10)=EPS*0.25DO*WSPP +EPS*DI( 10)/4.D0 XF(3,1)=EPS*0.5DO*P2*DI(11) XF(3,2)=EPS*0.5DO*( DI( 11)+P2*DI(12) ) XF(3,3)=EPS*0.5DO*( T2*DI( 12)+P2*DI(13) ) XF(3,4)=EPS*0.5DO*( T3*DI(13)+P2*DI(14)) XF(3,5)=EPS*0.5D0*P2*DI(15) XF(3,6)=EPS*0.5DO*P2*DI(16) XF(3,7)=EPS*0.5DO*P2*DI(17) XF(3,8)=EPS*0.5DO*( DI(15) +P2*DI(18)) XF(3,9)=EPS*0.5DO*( DI(16) +DI(18) +P2*DI(19)) XF(3,10)=EPS*0.5DO*( DI(18) +DI(18) +P2*DI(20)) XF(4,1)=0.25D0*EPS*CX*DI(21) C +0.25DO*EPS*PK*DI(ll) XF(4,2)=0.25DO*EPS*( CKP*DI(21) +CK*DI(22) ) C +0.25DO*EPS*( 0.5DO*DI(11)/POK +PK*DI(12) ) XF(4,3)=0.25DO*EPS*( CKPP*DI(21) +T2*OCP*DI(22) C +CX*DI(23) ) +0.25DO*EPS*( 0.25D0*DI(ll)/(P2*POK) C +DI(12)/POK +PK*DI(13) ) XF(4,4)=0.25DO*EPS*( CKPPP*DI(21) +T3*OCPP*DI(22) C +T3*CKP*DI(23) +CK*DI(24)) +0.25DO*EPS*( C 0.375DO*DI(11)/(P4*POK) 0.75DO*DI( 12)/(P2*POK) C +1.5D0*DI(13)/POK C +PK*DI(14) ) XF(4,5)=0.25DO*EPS*( CKS*DI(21) +CK*DI(25) ) C +0.25D0*EPS*PK*DI(15) XF(4,6)=0.25DO*EPS*( CKS*DI(25)*T2 +CK*DI(26)) C +0.25D0*EPS*PK*DI(16) XF(4,7)=0.25DO*EPS*( T3*CKS*DI(26) +CK*DI(27)) C +0.25DO*EPS*PK*DI(17) XF(4,8)=0.25DO*EPS*( CKSP*DI(21) +CKS*DI(22) C +CKP*DI(25) +CK*DI(28) ) C +0.25DO*EPS*( PK*DI(18) +0.5DO*DI( 15)/POK) XF(4,9)=0.25DO*EPS*( CKSP*DI(25)*T2 +CKS*DI(28)*T2 C +CKP*DI(26) +QC*DI(29) ) C +0.25DO*EPS*( PK*DI(19) +0.5DO*DI( 16)/POK ) XF(4,10)=0.25DO*EPS*( CKSP*DI(22) +CKSP*DI(22) C +CKS*DI(23) +CKPP*DI(25) +(XP*DI(28) +(XP*DI(28) C +CK*DI(30) ) C +0.25DO*EPS*( 0.5DO*DI(18)/POK +PK*DI(20) C +0.5DO*DI(18)/POK 0.25DO*DI( 15)/(P2*POK) )
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181 RETURN END C THIS PROGRAM DEFINES THE H0,H1,H2 FUNCTIONS SUBROUTINE PERTH(P2 ,U ,K2 ,SM1 ,SM2 ,SKL) IMPLICIT REAL*8 (AH.OZ) REAL* 8 K2 COMMON/ S IGNPK/ S IGN COMMON/ FUNS/XF(8 , 10) COMMON/ INT/ DI( 40) EPS=2.322819D03 CALL INTGRT(3, 1,40, SMI, SM2,P2,U,K2) T1=1.D0 T2=2.D0 T3=3.D0 T4=4.D0 T5=5.D0 PK=S IGN*DSQRT( P2*K2 ) P4=P2*P2 POK=DSQRT(P2/K2) SMA=(SM1+SM2)/2.D0 CH0=EPS*SMA*P2*K2/8 . DO CH0P=EPS*SMA*K2/8 . DO CH1=EPS*SMA*K2*(0.5D0*PK+U*P2)/8.D0 CHlP=EPS*SMA*K2*(0.25D0/POK+U)/8.D0 CH1PP=EPS*SMA*K2*(0.125DO/(P2*POK))/8.DO CH1PPP=EPS*SMA*K2*(T3/ ( 16 . D0*P4*P0K) )/8 . DO CH1S=EPS*SMA*K2*P2/8.D0 CH1SP=EPS*SMA*K2/8.D0 CH1SPP=0.0D0 BH1=EPS*SMA*K2*PK/16.D0 BH1P=EPS*SMA*K2/ ( 32 . D0*POK) BH1PP=EPS*SMA*K2/ ( 64 . D0*P2*P0K) BH1PPP=1 .5D0*EPS*SMA*K2/ ( 64. D0*P4*POK) BH1S=0.0D0 BH1SP=0.0D0 BH1SPP=0.0D0 AH2=EPS*SMA*K2/8 . DO DH2=EPS*SMA*K2*K2/32.D0 BH2=EPS*SMA*K2* (K2+PK*U) / 1 6 . DO BH2P=EPS*SMA*K2*( . 5D0*U/POK) / 1 6 . DO BH2PP=EPS*SMA*K2*(0 .25D0*U/ (P2*P0K) ) / 1 6. DO BH2PPP=EPS*SMA*K2* ( T3*U/ ( 8 . D0*P4*POK) ) / 1 6 . DO BH2S=EPS*SMA*K2*PK/ 1 6. DO BH2SP=EPS*SMA*K2/(32.D0*P0K) BH2SPP=EPS*SMA*K2*(0.5D0/(P2*POK))/32.D0 CH2=EPS*SMA*K2*(T2*PK*U+K2)/32.D0 CH2P=EPS*SMA*K2*U/ (32 . DO*POK) CH2PP=EPS*SMA*K2* (U/ ( T2*P2*POK) ) / 32 . DO CH2PPP=EPS*SMA*K2*(0.7 5DO*U/(P4*POK))/32.D0 CH2S=EPS*SMA*K2*(T2*PK)/32.D0 CH2SP=EPS*SMA*K2/ ( 32 . D0*POK) CH2SPP=EPS*SMA*K2*(0.5D0/(P2*POK))/32.D0 CH2SPP=EPS*SMA*K2*(0.5DO/(P2*POK))/32.DO XF(5,1)=CH0*DI(1)
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182 XF(5,2) = CH0*DI(2) +CH0P*DI(1) XF(5,3) = CH0*DI(3) +T2*CH0P*DI(2) XF(5,4)=CH0*DI(4) +T3*CH0P*DI(3) XF(5,5) = CH0*DI(5) XF(5,6) = CHO*DI(6) XF(5,7) = CH0*DI(7) XF(5,8)=CH0*DI(8) + CH0P*DI(5) XF(5,9) = CH0*DI(9) +CH0P*DI(6) XF(5,10) = CH0*DI(10) +T2*CH0P*DI(8) XF(6,1) = CH1*DI(1)+BH1*DI(21) XF(6,2) = CH1*DI(2) +CH1P*DI(1) C +BH1*DI(22) +BH1P*DI(21) XF(6,3) = CH1*DI(3) +T2*CH1P*DI(2) +CH1PP*DI(1) C +BH1*DI(23) +T2*BH1P*DI(22) +BH1PP*DI(21) XF(6,4) = CH1*DI(4) +T3*CH1P*DI(3)+T3*CH1PP*DI(2)+CH1PPP*DI(1) C +BH1*DI(24) +T3*BH1P*DI(23)+T3*BH1PP*DI(22)+BH1PPP*DI(21) XF(6,5)=CH1*DI(5)+CH1S*DI(1)+BH1*DI(25)+BH1S*DI(21) XF(6,6)=T2*CH1S*DI(5)+CH1*DI(6) +T2*BH1S*DI(25)+BH1*DI(26) XF(6,7)=T3*CH1S*DI(6)+CH1*DI(7) C +T3*BH1S*DI(26) +BH1*DI(27) XF(6,8) = CH1*DI(8) +CH1P*DI(5)+CH1SP*DI(1) +CH1S*DI(2) C +BH1*DI(28) +BH1P*DI(25)+BH1SP*DI(21) +BH1S*DI(22) XF(6,9)=T2*CH1SP*DI(5) +CH1P*DI(6) +T2*CH1S*DI(8)+CH1*DI(9) C +T2*BH1SP*DI(25) +BH1P*DI(26) +T2*BH1S*DI(28) +BH1*DI(29) XF(6,10)=CH1PP*DI(5) +T2*CH1P*DI(8) +CH1*DI(10) +CH1SPP C *DI(1) +CH1SP*DI(2) +CH1SP*DI(2) +CH1S*DI(3) C +BH1PP*DI(25) +T2*BH1P*DI(28) +BH1*DI(30) +BH1SPP C *DI(21) +BH1SP*DI(22) +BH1SP*DI(22) +BH1S*DI(23) XF(7,1) = CH2*DI(1)+BH2*DI(21) XF(7,2) = CH2*DI(2) +CH2P*DI(1) C +BH2*DI(22) +BH2P*DI(21) XF(7,3) = CH2*DI(3) +T2*CH2P*DI(2) +CH2PP*DI(1) C +BH2*DI(23) +T2*BH2P*DI(22) +BH2PP*DI(21) XF(7,4) = CH2*DI(4) +T3*CH2P*DI(3)+T3*CH2PP*DI(2)+CH2PPP*DI( 1) C +BH2*DI(24) +T3*BH2P*DI(23)+T3*BH2PP*DI(22)+BH2PPP*DI(21) XF(7,5) = CH2*DI(5)+CH2S*DI(1)+BH2*DI(25)+BH2S*DI(21) XF(7 ,6)=T2*CH2S*DI(5)+CH2*DI(6) +T2*BH2S*DI(25)+BH2*DI(26) XF(7,7)=T3*CH2S*DI(6) + CH2*DI(7) C +T3*BH2S*DI(26) +BH2*DI(27) XF(7,8) = CH2*DI(8) +CH2P*DI(5)+CH2SP*DI(1) +CH2S*DI(2) C +BH2*DI(28) +BH2P*DI(25)+BH2SP*DI(21) +BH2S*DI(22) XF(7,9)=T2*CH2SP*DI(5) +CH2P*DI(6) +T2*CH2S*DI(8) + CH2*DI(9) C +T2*BH2SP*DI(25) +BH2P*DI(26) +T2*BH2S*DI(28) +BH2*DI(29) XF(7,10) = CH2PP*DI(5) +T2*CH2P*DI(8) +CH2*DI(10) +CH2SPP C *DI(1) +CH2SP*DI(2) +CH2SP*DI(2) +CH2S*DI(3) C +BH2PP*DI(25) +T2*BH2P*DI(28) +BH2*DI(30) +BH2SPP C *DI(21) +BH2SP*DI(22) +BH2SP*DI(22) +BH2S*DI(23) DO 151 IE=1,10 151 XF(7,IE)=XF(7,IE)+AH2*DI(10+IE) +DH2*DI(30+IE) IOUT=0 IF( IOUT.NE.DGO TO 103 DO 101 IF=5,7 101 WRITE(6,102)(XF(IF,ID) ,ID=1,10) 102 FORMATC THE H0,H1,H2' , Dl 5.7/3D1 5.7/3D15.7/3D1 5.7// )
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183 103 CONTINUE RETURN END C THE FOLLOWING PROGRAMS PERFORM INTEGRATIONS OVER BETA SUBROUTINE INTGRT( IZ, I.M.SM1 ,SM2 ,P2 ,U,K2) IMPLICIT REAL*8 (AH.OZ) REAL*8 K2 COMMON/ SIGNPK./ SIGN COMMON/ INT/ DI( 40) PK=SIGN*DSQRT(P2*K2) POK=DSQRT(P2/K2) AM2=SM1*SM2 C LOCATE SINGULARITIES S2=2.0DO S3=2.0D0 Sl=l.DO+(AM2P2)/(0.5DO*K2 +PK*U) CA=K2/4.0D0 CB=0.5D0*K2 +PK*U CC=P2+PK*U +K2/4.0D0 CE=AM2K2/4.D0 CF=K2/4.D0 DE= CB* CB4 . DO* CA* CC IF(DE.GT.O)DESQRT=DSQRT(DE) IF(DE.GT.0.0D0)S2=(CB+DSQRT(DE))/(2.0D0*CA) IF(DE.GT.0.0D0)S3=(CBDSQRT(DE))/(2.0D0*CA) IF(Sl.LT.l.ODO.AND.Sl.GT.l.ODO) WRITE(6,43) SI 43 FORMATC LAMBDA HAS A VALUE BETWEEN 1 AND 1 AT WHICH C (Zl) GOES TO ZERO. LAMDA =' ,D15.7) IF(S2.LT.1.0D0.AND.S2.GT.1.0D0)WRITE(6,44) S2 IF(S3.LT.1.0D0.AND.S3.GT.1.0D0) WRITE(6,44) S3 44 FORMAT( ' LAMBDA HAS A VALUE IN THE RANGE 1 TO 1 WHERE C Z=ZERO ; LAMBDA=* .D15.7) BGN=1.0D0 WDT=2.D0 CALL INTGRL(IZ,I,M,SM1,SM2,P2,U,K2,BGN,1.0D0) IOUT=0 62 FORMATC THE DI INTEGRALS' ,3 Dl 5. 7) IF(IOUT.EQ.2) WRITE(6,62) (DI( IR) , IR=I,M) RETURN END SUBROUTINE INTGRL( IZ, I,M,SM1 ,SM2 ,P2,U,K2 , BGN.EN) IMPLICIT REAL*8 (AH.OZ) REAL*8 K2 COMMON/GAUSST/XX(96) ,WX(96) COMMON/ INT/ D 1(40) DIMENSION A(98) DO 300 N=I,M CALL GAUSS(BGN, EN, SLOPE) SGAUSS=0.0D0 DO 100 IL=1,96 SL=XX(IL) A(IL) = FN(IZ,N,SL,SM1,SM2,P2,U,K2) 100 SGAUSS=SGAUSS +A( IL)*WX( IL)
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300 DI(N) = SGAUSS*SLOPE RETURN END SUBROUTINE GAUSS(XMIN,XMAX, SLOPE) IMPLICIT REAL*8 (AH.OZ) COMMON /GAUSST/ XX(96), DIMENSION XI(48) ,WI(48) DATA XI(1),XI(2),XI(3),XI(4),XI(5),XI(6) * /.1627674484960297D1..4881298513604973D1, * .8129749546442556D1..1136958501106659D0, * .14597371465489694D0..17809688236761860D0/ DATA XI(7) ,XI(8) ,XI(9) ,XI(10) ,XI(11) ,XI( 12) * /.21003131046056720DO, .24174315616384001DO, * .27319881259104914D0, .30436494435449635D0, * .33520852289262542D0, .36569686147231364D0/ DATA XI(13) ,XI(14) ,XI(15) ,XI(16) ,XI(17) ,XI(18) * /.39579764982890860D0..42547898840730055D0, * .45470942216774301D0, .48345797392059636D0, * .511 694177154667 67D0, .53938810832435744DO/ DATA XI(19) ,XI(20) ,XI(21) ,XI(22) ,XI(23) ,XI(24) * /.56651041856139717D0,.59303236477757208D0, * .61892584012546857D0, .64416340378496711D0, * . 66871831004391 615D0, .69256453664217156D0/ DATA XI(25) ,XI(26) ,XI(27) ,XI(28) ,XI(29) ,XI(30) * /.71567681234896763D0..73803064374440013D0, * .75960234117664750D0..78036904386743322D0, * .80030874413914082D0,. 819400310737931 68D0/ DATA XI(31) ,XI(32) ,XI(33) ,XI(34) ,XI(35) ,XI(36) * /.83762351122818712D0..85495903343460146D0, * .87138850590929650D0, .88689451740242042D0, * .90146063531585234D0, .91507142312089807D0/ DATA XI(37) ,XI(38) ,XI(39) ,XI(40) ,XI(41) ,XI(42) * /.92771245672230869DO, .93937033975275522DO, * .95003271778443764DO, .95968829144874254D0, * .96832682846326421D0..97593917458513647DO/ DATA XI(43) ,XI(44) ,XI(45) ,XI(46) ,XI(47) ,XI(48) * /.98251726356301468D0..98805412632962380D0, * .99254390032376262D0, .99598184298720929D0, * .99836437586318168D0, .99968950388323077D0/ DATA WI(1),WI(2),WI(3),WI(4),WI(5),WI(6) * /.03255061449236317D0,.03251611871386884D0, * .03244716371406427D0..03234382256857593D0, * .03220620479403025D0, .03203445623199266D0/ DATA WI(7) ,WI(8) ,WI(9) ,WI(10) ,WI(11) ,WI(12) * /. 03182875889441 101D0,. 031589330770727 17D0, * .03131642559686136D0, .03101033258631384D0, * .03067137612366915D0, .03029991542082759D0/ DATA WI(13) ,WI(14) ,WI(15) ,WI(16) ,WI(17) ,WI(18) * /.02989634413632839D0,. 0294610899581 6791 DO, * .02899461415055524D0..02849741106508539D0, * . 02797000761 684833D0, .02741296272602924DO/ DATA WI(19) ,WI(20),WI(21) ,WI(22) , WI(23) ,WI(24) * /. 026826866725591 7 6D0,. 02621234073567241 DO,
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185 * .02557003600534936D0, .02490063322248361D0 , * .02420484179236469D0, .02348339908592622DO/ DATA WI(25) ,WI(26) ,WI(27) ,WI(28) ,WI(29) ,WI(30) * /.02273706965832937D0..02196664443874435DO, * . 02 117293989219130DO,. 020356797 15433332D0, * .01951908114014502D0..01866067962741147D0/ DATA WI(31) ,WI(32) ,WI(33) ,WI(34) ,WI(35) ,WI(36) * /.01778250231604526D0,.01688547986424517D0, * .01597056290256229D0, .01503872102699494DO, * .01409094177231486D0, .01312822956696157D0/ DATA WI(37) ,WI(38) ,WI(39) ,WI(40) ,WI(41) ,WI(42) * /.01215160467108832D0..01116210209983850DO, * .01016077053500842DO, .00914867123078339D0, * .00812687692569876D0, .00709647079115386DO/ DATA WI(43) ,WI(44) ,WI(45) ,WI(46) ,WI(47) ,WI(48) * /.00605854550423596D0,.00501420274292752D0, * .00396455433844469DO..O0291073181793495D0, * .00185396078894692DO, .00079679206555201DO/ SL0PE=.5D0*(XMAXXMIN) Y INTER= . 5D0* ( XMAX+XMIN) DO 40 1=1,96 IF(I48)10,10,20 10 LL=49I SGN=1.D0 GO TO 30 20 LL=I48 SGN=1.D0 30 X=SGN*XI(LL) XX( I)=SLOPE*X+YINTER 40 WX(I)=WI(LL) RETURN END C THIS PROGRAM DEFINES THE VARIOUS BETA INTEGRALS FUNCTION FN(IZ,N,S,SM1,SM2,P2,U,K2) IMPLICIT REAL*8 (AH,0Z) REAL*8 K2.LOZ COMMON/ S IGNPK/ S IGN Tl=l.DO T2=2.D0 T3=3.D0 T4=4.D0 T5=5.D0 AM2=SM1*SM2 PK=S IGN*DSQRT( P2*K2 ) P4=P2*P2 POK=DSQRT(P2/K2) YU2=AM2 (T1S*S)*K2/T4 X=P2 +PK*U*(T1+S) +K2*(T1+S)*(T1+S)/T4 Z=X/YU2 BZ=(T1Z) LOZ=DLOG(DABS(BZ)) DP=(T1+0.5DO*U*(T1+S)/POK)/YU2 DPP=0.25DO*U*(T1+S)/(YU2*P2*POK) DPPP=T3*U*(T1+S)/(8.DO*YU2*P4*POK)
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DU=PK*(T1+S)/YU2 DUP=0.5D0*(T1+S)/(YU2*P0K) DUPP=0.25D0*(Tl+S)/(YU2*P2*POK) W1=T1/Z W11=T1/Z**2 W12=T2/Z**3 W13=6.D0/Z**4 W14=24.D0/Z**5 W15=120.D0/Z**6 W2=(T1/Z**2)*L0Z W21=W12*LOZ +W11/BZ W22=W13*L0Z +T2*W12/BZ +W11/BZ**2 W23=W14*LOZ +T3*W13/BZ +T3*W12/BZ**2 +T2*W11/BZ**3 W24=W15*LOZ +T4*W14/BZ +6. D0*W13/BZ**2 +8. D0*W12/BZ**3 C +6.D0*W11/BZ**4 W3=T1/(Z*BZ) W31=W1/BZ**2 +W11/BZ W32=T2*W1/BZ**3 +T2*W11/BZ**2 +W12/BZ W33=6.D0*W1/BZ**4+6.D0*W11/BZ**3 +T3*W12/BZ**2 C +W13/BZ W4=W1*L0Z/YU2 W41=W11*L0Z/YU2 W1/BZ/YU2 W42=W12*LOZ/YU2 T2*W11/BZ/YU2 W1/(BZ*BZ*YU2) W43=W13*LOZ/YU2 T3*W12/BZ/YU2 T3*W11/ C (BZ*BZ*YU2) T2*W1/(YU2*BZ**3) W44=W14*LOZ/YU2 T4*W13/BZ/YU2 6.D0*W12/(YU2*BZ**2) C 8.D0*W11/(YU2*BZ**3) 6.D0*W1/(YU2*BZ**4) W5=L0Z W51=T1/BZ W52=T1/BZ**2 W53=T2/BZ**3 W54=6.D0/BZ**4 IF(IZ.NE.2) GO TO 73 B1=(AM2YU2)*(W2+YU2*W4W1)/YU2 +W5 W4*YU2 W4*K2/2.DO B2=(AM2YU2)*(W21+YU2*W41W11)/YU2 +W51 W41*YU2 W41* C K2/2.D0 B3=(AM2YU2)*(W22+YU2*W42W12)/YU2 +W52 W42*YU2 W42* C K2/2.D0 B4=(AM2YU2)*(W23+YU2*W43W13)/YU2 +W53 W43*YU2 W43* C K2/2.DO B5=(AM2YU2)*(W21YU2*W41+W11)/YU2**2 B6=(AM2YU2)*(W22YU2*W42+W12)/YU2**2 B7=(AM2YU2)*(W23YU2*W43+W13)/YU2**2 B8=(AM2YU2)*(W24YU2*W44+W14)/YU2**2 B9=(AM2YU2)*(W21 YU2*W41 +W11 W2 W3)/YU2**2 B10=(AM2YU2)*(W22 YU2*W42 +W12 W21 W31 )/YU2**2 B11=(AM2YU2)*(W23 YU2*W43 +W13 W22 W32 )/YU2**2 B12=(AM2YU2)*(W24 YU2*W44 +W14 W23 W33 )/YU2**2 IF(N.EQ.1)FN=B1 TF(N.EQ.2)FN=B2*DP IF(N.EQ.3)FN=B2*DPP +B3*DP*DP IF(N.EQ.4)FN=B2*DPPP +T3*B3*DPP*DP +B4*DP**3 IF(N.EQ.5)FN=B2*DU IF(N.EQ.6)FN=B3*DU*DU
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187 IF(N.EQ.7)FN=B4*DU**3 IF(N.EQ.8)FN=B2*DUP +B3*DU*DP IF(N.EQ.9)FN=B3*DUP*DU +B3*DU*DUP +B4*DU*DU*DP IF(N.EQ.10)FN=B2*DUPP +T2*B3*DUP*DP +B3*DU*DPP +B4*DU*DP**2 IF(N.EQ.11)FN=B5 IF(N.EQ.12)FN=B6*DP IF(N.EQ.13)FN=B6*DPP +B7*DP**2 IF(N.EQ.14)FN=B6*DPPP +T3*B7*DPP*DP +B8*DP**3 IF(N.EQ.15)FN=B6*DU IF(N.EQ.16)FN=B7*DU**2 IF( N.EQ. 17 ) FN=B8*DU**3 IF(N.EQ.18)FN=B6*DUP +B7*DU*DP IF(N.EQ.19)FN=T2*B7*DUP*DU +B8*DU*DU*DP IF(N.EQ.20)FN=B6*DUPP +T2*B7*DUP*DP +B8*DU*DP*DP C +B7*DU*DPP IF(N.EQ.21)FN=B9 IF(N.EQ.22)FN=B10*DP IF(N.EQ.23)FN=B10*DPP +B11*DP**2 IF(N.EQ.24)FN=B10*DPPP +T3*B1 1*DPP*DP +B12*DP**3 IF(N.EQ.25)FN=B10*DU IF(N.EQ.26)FN=B11*DU**2 IF(N.EQ.27) FN=B12*DU**3 IF(N.EQ.28)FN=B10*DUP +B11*DU*DP IF(N.EQ.29) FN=T2*B11*DUP*DU +B12*DU*DU*DP IF(N.EQ.30)FN=B10*DUPP +T2*B1 1*DUP*DP +B12*DU*DP*DP C +B11*DU*DPP RETURN 73 CONTINUE IF(IZ.NE.l) GO TO 72 IF(N.EQ.1)FN=(W1+W2)/YU2 IF(N.EQ.2)FN=(W11+W21)*DP/YU2 IF(N.EQ.3)FN=(W11+W21)*DPP/YU2 +(W12+W22)*DP*DP/YU2 IF(N.EQ.4)FN=(W11+W21)*DPPP/YU2 +T3*(W12+W22)*DPP*DP/YU2 C +(W13+W23)*DP*DP*DP/YU2 IF(N.EQ.5)FN=(W11+W21)*DU/YU2 IF(N.EQ.6) FN=(W12+W22)*DU*DU/YU2 IF(N.EQ.7)FN=(W13+W23)*DU*DU*DU/YU2 IF(N.EQ.8)FN=(W11+W21)*DUP/YU2 +( W12+W22)*DP*DU/YU2 IF(N.EQ.9)FN=(W12+W22)*DU*DUP/YU2 C +(W13+W23)*DP*DU*DU/YU2 +(W12+W22)*DUP*DU/YU2 IF(N.EQ.10)FN=(W11+W21)*DUPP/YU2+(W12+W22)*DPP*DU/YU2 C +(W12+W22)*T2*DUP*DP/YU2 +(W13+W23)*DU*DP*DP/YU2 RETURN 72 CONTINUE IF(IZ.NE.3)GO TO 12 WN1=( W2 1YU2* W4 1+W1 1 ) /YU2**2 WN2=(W22YU2*W42+W12)/YU2**2 WN3= ( W23YU2* W43+W13 ) /YU2**2 WN4=( W24YU2*W44+W14 ) /YU2**2 WN5=(W2+W1)/YU2 W4/2.DO WN6=(W21+W11)/YU2 W41/2.D0 WN7=(W22+W12)/YU2 W42/2.D0 WN8=(W23+W13)/YU2 W43/2.DO IF(N.EQ.1)FN=WN1
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IF(N.EQ.2)FN=WN2*DP IF(N.EQ.3)FN=WN2*DPP +WN3*DP**2 IF(N.EQ.4)FN=WN2*DPPP +T3*WN3*DPP*DP +WN4*DP**3 IF(N.EQ.5)FN=WN2*DU IF( N.EQ. 6) FN=WN3*DU*DU IF(N.EQ.7)FN=WN4*DU**3 IF(N.EQ.8)FN=WN2*DUP +WN3*DU*DP IF(N.EQ.9)FN=T2*WN3*DUP*DU +WN4*DU*DU*DP IF(N.EQ.10)FN=WN2*DUPP +T2*WN3*DUP*DP +WN3*DU*DPP C +WN4*DP*DP*DU IF(N.EQ.21)FN=S*(WN1) IF(N.EQ.22)FN=S*(WN2*DP) IF(N.EQ.23)FN=S*(WN2*DPP +WN3*DP**2) IF(N.EQ.24)FN=S*(WN2*DPPP +T3*WN3*DPP*DP +WN4*DP**3) IF(N.EQ.25)FN=S*(WN2*DU) IF(N.EQ.26)FN=S*(WN3*DU*DU) IF(N.EQ.27)FN=S*(WN4*DU**3) IF(N.EQ.28)FN=S*(WN2*DUP +WN3*DU*DP) IF(N.EQ.29)FN=S*(T2*WN3*DUP*DU +WN4*DU*DU*DP) IF(N.EQ.30)FN=S*(WN2*DUPP +T2*WN3*DUP*DP +WN3*DU*DPP C +WN4*DP*DP*DU) IF(N.EQ.31)FN=S*S*(WN1) IF(N.EQ.32)FN=S*S*(WN2*DP) IF(N.EQ.33)FN=S*S*(WN2*DPP +WN3*DP**2) IF(N.EQ.34)FN=S*S*(WN2*DPPP +T3*WN3*DPP*DP +WN4*DP**3) IF(N.EQ.35)FN=S*S*(WN2*DU) IF(N.EQ.36)FN=S*S*(WN3*DU*DU) IF(N.EQ.37)FN=S*S*(WN4*DU**3) IF(N.EQ.38)FN=S*S*(WN2*DUP +WN3*DU*DP) IF(N.EQ.39)FN=S*S*(T2*WN3*DUP*DU +WN4*DU*DU*DP) IF(N.EQ.40)FN=S*S*(WN2*DUPP +T2*WN3*DUP*DP +WN3*DU*DPP C +WN4*DU*DP*DP) IF(N.EQ.11)FN=WN5 IF(N.EQ.12)FN=WN6*DP IF(N.EQ.13)FN=WN6*DPP +WN7*DP**2 IF(N.EQ.14)FN=WN6*DPPP +T3*WN7*DPP*DP +WN8*DP**3 IF(N.EQ.15)FN=WN6*DU I F( N. EQ . 1 6 ) FN=WN7 * DU* DU IF(N.EQ.17)FN=WN8*DU**3 IF(N.EQ.18)FN=WN6*DUP +WN7*DU*DP IF(N.EQ.19)FN=T2*WN7*DUP*DU +WN8*DU*DU*DP IF(N.EQ.20)FN=WN6*DUPP +T2*WN7*DUP*DP +WN7*DU*DPP C +WN8*DU*DP*DP RETURN 12 CONTINUE WRITE(6,87) 87 FORMAT(* CORRECT VALUE IF IZ IN FNNS NOT FOUND') RETURN END : THIS DEFINES THE ELECTRON FUNCTION A FUNCTION XA(R) IMPLICIT REAL*8 (AH.OZ) COMMON/ SCALE/ SKL IF (DABS(R).LT.0.1D6) CO TO 1
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189 SE=1.74517D3 SM=1.D0*SKL SM2=SM*SM XR=DABS(1.D0R/SM2) XP=SE*(SM2R)/R XA=XR**XP XA=XA*(SKL)**(2.D0*XP) XA=XA*SKL RETURN 1 XA=1.D0 XA=O.ODO RETURN END
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APPENDIX D CALCULATION OF FOUR DIMENSIONAL INTEGRALS In the electron self energy integral, 2(p), and the vacuum polarization integral, n yv; (p ), various types of four dimensional integrals occur. These integrals may be evaluated if a Wick rotation is made to cause the real axis of the fourth component of the momentum to lie along the imaginary axis. This can be done by substituting o.4 p > ip . The integral can be performed than over four dimensional Euclidean space. Thus, d 4 p = dp dp ] dp 2 dp 3 * idp 4 dp ] dp 2 dp 3 . The momenta variables can be expressed as four dimensional hyperspherical coordinates so that, 1 x p = r 2 sin cos sin x 2 A p =r 2 sin6sin$sinx 3 x p = r 2 cos 6 sin x p 4 = r* cos X (DD 2 where r = p . (Hence r is a positive quantity.) The Jacobian of this transformation from Cartesian to hyperspherical coordinates is given by 190
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191 3/2 Â• 2 J = r sin e sin x The ranges of integration are, < < 2rr < 6 < tt < x < t 1 < r < Â°Â° . In summary, we can evaluate the following integral by an integration over the hyperspherical coordinates, f(p 2 )d 4 p = i /f(r)r 3/2 sin 6 sin 2 x dr de d* dp. (D2) A further complication occurs when a second independent vector, k, troduced. Le1 relating p do k is is introduced. Let R lie along the p axis, then the angle variable u = p 9 * k 1 = cos X Â• (D^ (P 2 k 2 ) 5 We would like to consider the evaluation of integrals which involve functions of this angle variable, as in /f(p 2 , u, k 2 )d 4 p. Considerable simplification of these integrals is obtained by expanding the integrands in terms of Gegenbauer polynomials. These have the following orthogonality relation, c](u) cj,(u) (1u 2 ) 1 du =Â§ 5 ab . (D4; 1
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192 The first three Gegenbauer polynomials are, c 1
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93 Similar steps taken to evaluate the other components lead to 11 = 22 = .33 = j44 and also I a3 = when a f 3It follows that x yv s g yv y J f(4) 2 dr . It is no more difficult to evaluate a more general situation where f is a function of u as well as r. In this case the expression of the integrans in terms of the Gegenbauer polynomials would be altered but the integration over u could be performed with the same ease due to the orthogonality relations. There is one last useful maneuver that should be mentioned. In both 9 ? the IT (k ) and Z(p) integrals the expression (pR) occurs in the denominators. This expression has an expansion in terms of Gegenbauer polynomials. Let (pkT 2 = i(1 2uz + z 2 ) where r< = the lesser of p , k 2 2 r > = the greater of p , k , Since (1 2 u z + z 2 )" 1 1 C ] (u)z n , we find, n=0 n
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94 (pk)" 2 =77 E C n (u)A r> n=0 n This enables one to evaluate such integrals as, [f(p 2 ,u,k 2 )d 4 p = iff(r,u,k 2 ) Z dÂ£rdr(lu 2 )"dusin( J (pR) l) n=0 n r> For the purposes of illustration, consider the case where f = f (p ) , KeX d % = iff(r) 2 c' n fr dr(lu 2 )"du sin d( (pfc) n=0 n r> 2, ni/f(r) c] I fJ^ r dr(lu 2 )Mu These methods described here are sufficient to calculate any of the four dimensional integrals which appear in Â£(p) and r yv (k ), provided that the form of the function f(p 2 ,u,k 2 ) is known. A tabulation of the integrals needed to calculate the self energy of the electron, Z(p), appears in the work of Green et al. 28 A tabulation of the integrals needed to calculate the vacuum polarization, r yv (k 2 ), is provided in the work of Yock.
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96 15. S. F. Edwards, Phys. Rec. 90 284 (1953); P. Yock, Nuovo Cim. 55 A 217 (1968); F. A. Kaempffer, Phys. Rev. D 25 439 (1982); Y. Frishman Phys. Rev. J38 1450 (1965) 16. J. Schwinger, Proc. Natl. Acad. Sci USA _3J 455 (1951) 17. J. D. Bjorken, and S. D. Drell , "Relativistic Quantum Mechanics," McGrawHill Book Company, New York, 1964; "Relativistic Fields," McGrawHill Book Company, New York, 1965. 18. H. S. Green, Proc. Phys. Soc 66 873 (1952), Phys. Rec. 95 5 , 48 (1954) 19. H. S. Green, Phys. Rev. 97 540 (1955) 20. S. K. Bose, and S. N. Biswas, J. Math. Phys. 6 1227 (1965) 21. J. Schwinger, Phys. Rev. Let. 3 296 (1959) 22. H. I. Akhiezer, and V. B. Berestetskii , Quantum Electrodynamics Interscience Publishers, New York, 1965. 23. G. Kalian, Helv. Phys. Acta, 26 755 (1953) 24. N. Nakanishi, Prog. Theoretical Phys. 35 1111 (1966), 38 881 (1967), and 50 1388 (1973) 25. L. Fox and D. F. Mayers, "Computing Methods for Scientists and Engineers," Clarendon Press, Oxford, 1968, pp. 176180. 26. Private communication from Ruben Mendez Placito, University of Puerto Rico, Recinto Universitario , Mayaguez. 27. University of Florida, Physica Research Report 6, "Solving the SchwingerDyson Equations." 28. H. S. Green, J. F. Cartier, and A. A. Broyles, Phys. Rec. D j8 1102 (1978) 29. J. Schwinger, Phys. Rev. 74 1429 (1948); Z. Koba, T. Tati , and S. Tomonaga, Progr. Theor. Phys. 2 198 (1957). 30. H. A. Lorentz, "The Theory of Electrons, Leipzig, 1909.
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BIOGRAPHICAL SKETCH Joan F. Cartier was born in Toronto, Canada, in 1950 but was for the most part raised in the outislands of the Bahamas. She obtained a Bachelor of Science degree with High Honors from the University of Florida in 1975 while studying chemistry. 197
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosopny. Charles E. Reid, Chairman Associate Professor of Chemistry I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosopny. Yngve Ohrr! Professor of Chemistry I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosopny. Arthur Broyles Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, a^a di^s elation for the degree of Doctor of Philosopny. Hendri/kjj. Monkhorst Associate Professor of Physics
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Â— 'Kermi n't Sigmon Associate Professor of Mathematics This dissertation was submitted to the Graduate Faculty of the Department of Chemistry in the College of Liberal Arts and Sciences and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. April 1983 Dean for Graduate Studies and Research
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