W.
J.
SPRY
the results
of this
experiment
A=use/ebs=
(1.
0&0.
2)
)&10"
sec
'
if
the
limits
placed
on. Pi'
r—
rl'
are
taken
seriously.
This assumes the
radius
of
the mesonic
Bohr
orbit
b=2.
2&(10
"
cm and that
tis
——
8X10'
cm/sec;
o'e
=
(8ir/9k')
(tie/ri) (pis
—
cr
s)'.
Panofsky's
experiment
shows that
this
capture
rate should
equal
the
capture
rate
for the
competing
process
p(w,
y)n.
Bernardini"
has
discussed
the cross sections for
p
(y,
ir+)e,
d
(y,
ir+)2n,
and d
(y,
ir
)
2p
for
E~
between 170
and
190 Mev
in the
laboratory system.
If
it is assumed
that the
ratio of
m
to
++
production
obtained
from
the
second
reaction is the
same
as
the
photoproduction
ratio between
the free
neutron and free
proton
then
the
principle
of detailed
balance
and this
ratio
can be
used to
predict
the
corresponding
cross sections for
p(w,
y)ts.
If
these
cross
sections are
extrapolated
to
the
energy
of
Panofsky's
experiment
Bernardini
obtains
a
capture
rate that
requires
the initial
slope
of
Pie
—
n
to
be
&(9.
2')r)'
in
contrast to the value
of
—
(16.
5')r)'
obtained from this experiment.
Bethe and
Noyes"
have
given
an
argument
to
explain
this
discrepancy
in terms
of
Marshak's'
suggestion.
In
brief this
argument
assumes
that the
slope
of
P
—
n
obtained
from this
experiment
cannot be
extrapolated
'6
H. Bethe
and
H. P.
Noyes,
Proceedings
of the Fourth Annual
Rochester
Conference,
1954,
University
of Rochester.
"R.
Marshak, Phys.
Rev.
88,
1208
(1952).
to
the
energy
of
Panofsky's
experiment,
but that
this
initial
slope
is
&
(9.
2')i)'.
They
then fit this initial
slope
and the
data of this
experiment
with a smooth
curve
for
pie
n
is
zersus
t)'.
When the values of
p
and
ni'
from
higher
energies
are
extrapolated
with
this
restric-
tion on
their
difference,
it is difIicult to fit
the data
without
assuming
that 0.
~'
varies less
rapidly
than
p'
and
that
p
varies more
rapidly
than
r)'
in
the
energy
region
between 20
and 42 Mev. For the most
probable
fit
under these assumptions
Pie
changes
sign
between
20
and
30 Mev.
This
energy
dependence
for pie
suggests
a
Jastrow"
potential
for this
phase
shift.
ACKNOWLEDGMENTS
A
number
of
people
aided the author in
this experi-
ment. Thanks
are due
especially
to Dr. A. Roberts
and
Dr.
J.
Tinlot
for their
help
and advice
throughout
the
experiment.
The
author
also
wishes to thank Dr.
J.
French and
Dr. H.
P.
Noyes
for
their
helpful
discussions
of the results.
Others who
assisted and who contributed
numerous
suggestions
were~Dr.
E.
Hafner,
Dr. F.
Tenney,
R.
Santirocco,
andj,
D.
Nelson.
W.
Coombs,
K.
Enslein,
L.
Braun,
H.
VonThenen,
and F. Palmer
were
responsible
for
construction of
parts
of the
equip-
ment and
for the reliable
operation
of
the
cyclotron.
's
R.
Jastrow,
Phys.
Rev.
81,
1165
(1951).
PH
YSI
GAL
REVIEW
VOLUME
95,
NUMBER 5
SEPTEM
8ER
1,
19/4
Quantum
Electrodynamics
at
Small
Distances~
M.
GELL-MANN't
AND
F. E.
Low
I'hysics
Department,
Un&Jersity
of
Illinois, Urbana,
Illinois
(Received
April
1,
1954)
The
renormalized
propagation
functions
D~q
and
Sgg
for
photons
and
electrons,
respectively,
are
in-
vestigated
for momenta much
greater
than
the mass
of the electron.
It
is found
that
in
this
region
the
indi-
vidual
terms of the perturbation
series
to
all orders in
the
coupling
constant
take on
very
simple
asymptotic
forms. An
attempt
to sum
the entire series
is
only partially
successful. It
is found that the
series
satisfy
certain
functional
equations
by
virtue of the
renormalizability of the
theory.
If
photon
self-energy
parts
are
omitted
from
the series,
so
that
Dec=ax,
then
Sec
has the
asymptotic
form ALp'/m'j"pep
pg
',
where
A =A
(eP)
and
a=n (eis).
When
all
diagrams
are
included,
less
specific
results are found. One
conclusion
is
that the
shape
of the
charge
distribution
surrounding
a test
charge
in the vacuum
does
not,
at
small
dis-
tances, depend
on
the
coupling
constant
except through
a scale factor.
The
behavior of the
propagation
functions
for
large
momenta
is related to the
magnitude
of the
renormalization constants
in the
theory.
Thus
it
is shown
that the
unrenormalized
coupling
constant
e0'/471.
Ac,
which
appears
in
perturbation
theory
as
a power
series in the
renormalized
coupling
constant
eP/4vkc
with
divergent
coetficients,
may
behave
in
either of two
ways:
(a)
It
may
really be
infinite as perturbation
theory
indicates;
(b)
It
may
be a
finite number
independent
of
e&'/47i-kc.
l. INTRODUCTION
'
'T
is a
well-known fact
that
according
to
quantum
~ ~
electrodynamics
the electrostatic
potential
between
two classical
test
charges
in
the vacuum is
not
given
exactly
by
Coulomb's
law. The
deviations
are
due to
*
This
work
was
supported
by
grants
from the U. S.
Once
of
Naval Research and the U.
S.Atomic
Energy
Commission.
)Now
at
Department
of
Physics
and
Institute
for Nuclear
Studies,
University
of
Chicago.
vacuum polarization.
They
were
calculated
to first
order
in
the
coupling
constant
o.
by
Serber'
and
Uehling'
shortly
after the erst
discussion
of
vacuum polarization
by
Dirac'
and Heisenberg.
4
We
may
express
their
re-
sults
by
writing
a formula
for the
potential
energy
be-
'
R. Serber,
Phys.
Rev.
48,
49
(1935).
'
A. E.
Uehling,
Phys.
Rev.
48,
55
(1935).
'
P.
A. M. Dirac, )Proc. Cambridge
Phil. Soc.
30,
150(1934).
'
W.
Heisenberg,
lZ.
Physik
90,
'209
(1934).
QUAN
"I
UM
ELECTRODYNAMICS
AT
SMALL DISTANCES
tween
two
heavy point
test
bodies,
with renormalized
charges
q
and q',
separated
by
a distance r:
qq
t
n
t"
(Msc')
&'
V(r)
=
~
1+
—
'
exp
—
r)
4zv
)
3g
~
is
i
(
Ji'
)
2ms)
(
415s)
I
de
X
1+
&Vs)
E
3fs)
&Vs
+O(ii')+
(1
1)'
Here
n=
eis/4s.
lie=1/137 is
the
renormalized
fine
struc-
ture constant
and nz is the renormalized
(observed)
rest mass
of the electron.
If r«
fi/mc,
then
(1.
1)
takes the
simple
asymptotic
form,
qq'
2rr
(
h
)
V(.
)
=
4mr
3n.
&
mcr)
+O(
')+
.
(12)'
where
y
—
1.
781.
We
shall discuss the behavior
of
the
entire series
(1.
2),
to
all orders in
the
coupling
constant,
making
use
of
certain
simple
properties
that it
possesses
in
virtue of the
approximation r«A/mc
Thes.e
proper-
ties are
intimately
connected
with
the
concept
of
charge
renormalization. The relation between
(1.
2)
and
charge
renormalization can
be made clear
by
the
following
physical
argument:
A
test
body
of
"bare
charge"
qo
polarizes
the
vacuum,
surrounding
itself
by
a
neutral cloud
of electrons and
positrons;
some
of
these,
with a net
charge
bq,
of
the
same
sign
as
qo,
escape
to
in6nity,
leaving
a net
charge
—
bq
in the
part
of the
cloud which
is
closely
bound to
the
test
body
(within
a
distance
5/mc).
If we
observe
the
body
from a
distance much
greater
than
5/mc,
we
see an effective
charge
q
equal
to
(qs
—
bq),
the
renormal-
ized
charge.
However,
as
we
inspect
more
closely
and
penetrate through
the cloud to the core of the test
body,
the
charge
that we see inside
approaches
the bare
charge
qo,
concentrated in a
point
at
the center. It is
clear, then,
that
the
potential
V(r),
in
Eqs. (1.
1)
and
(1.
2),
must
approach
qsqs'/4rr
as r
approaches
zero.
Thus,
using
(1.
1),
we
may
write
2rr
(h/mc
)
qoqo'=qq'
1+
—
inl
I
s
—
in'
+O(ot
),
(1.
3)
&0)
2.
REPRESENTATIONS
OF THE PROPAGATION
FUNCTIONS
The
modi6ed Coulomb
potential
discussed in Sec. I
can
be
expressed
in terms of the
hnite modi6ed
photon
propagation
function
Dr
c(P',
eis) that
includes vacuum
polarization sects
to
all orders in the
coupling
constant.
(Here
P'
is
the
square
of
a
four-vector
momentum
Pli.
)
The function
Dp&
is calculated
by
summing
all Feyn-
man
diagrams
that
begin
and end with
a
single photon
line,
renormalizing
to
all orders. The
potential
is
given
by'
qq'
r
V
(r)
=
~
O'Pe'&'Di
o(P'
eis)
(2')s~
(2.
1)
of the
bare
charge
es
and the bare
(or
mechanical)
mass
mo
of the
electron. The renormalizability
of the
theory
consists in
the fact
that,
when the
observable
quantities
are re-expressed in
terms of the renormalized
param-
eters
e~
and
ns,
no
divergences
appear,
at least when
a
power
series
expansion
in
eis/4n.
kc
is used. The
proof
of
renormalizability has
been
given
by
Dyson,
Salam,
'
and
Ward.
'
We
shall
make
particular
use
in
Secs. III
and
IV
of the
elegant techniques
of Ward.
We
shall show that the
fact of
renormalizability
gives
considerable
information about the behavior of the
complete
series
(1.
2).
It
may
be
objected
to an in.
-
vestigation.
of this sort that
while
(1.
2)
is valid for
r«k/mc,
the first
few
terms should
suffice
for calcula-
tion unless r is as
small as
e
"'5/mc,
a
ridiculously
small
distance.
We
have
no
reason,
in
fact,
to believe
that
at such distances
quantum
electrodynamics has
any
validity
whatever,
particularly
since interactions
of
the
electromagnetic
field with
particles
other
than
the
electron
are
ignored. However,
a
study
of
the
mathematical character
of
the
theory
at small
distances
may
prove
useful in
constructing
future
theories.
Moreover,
in other 6eld
theories now
being
considered,
such as
the relativistic
pseudoscalar
meson
theory,
conclusions similar to ours
may
be
reached,
and
the
characteristic
distance
at
which
they
become useful
is
much
greater,
on
account of the
largeness
of
the
coup-
ling
constant.
In this
paper
we shall be
mainly
concerned with
quantum
electrodynamics,
simply
because
gauge
in-
variance and
charge
conservation
simplify
the
calcula-
tions to
a considerable extent.
Actually,
our
considera-
tions
apply
to
any
renormalizable 6eld
theory,
and we
shall
from
time to
time indicate the
form
they
would
take in meson
theory.
where the individual
terms in
the
series
diverge
loga-
rithmically
in a
familiar
way.
The
occurrence of
these
logarithmic
divergences
will
play
an
important
role
in
our work.
Such
divergences
occur in
quantum
electrodynamics
whenever
observable
quantities
are
expressed
in
terms
s
J.
Schwinger, Phys.
Rev.
75,
651
(1949).
where
p
is a
three-dimensional
vector.
If
we
were to sum all the
Feynman
diagrams
that
make
up
Dpg
without
renormalizing
the
charge
we
s
F.
J.
Dyson,
Phys.
Rev.
75,
1756
(1949).
'
A.
Salarn, Phys.
Rev.
84,
426
(1951).
s
J.
C.
Ward,
Proc.
Phys.
Soc.
(London)
A64,
54
(1951);
see
also
Phys.
Rev.
84,
897
(1951}.
'
From this
point
on,
@re
take
A=c=1.
i302
M.
GELL
—
MANN
AN D F.
E.
LOW
would
obtain the
divergent function D r'(P',
e
ss),
which
is
related to the 6nite
propagation
function
by
Dyson's
equations
(2.
2)
Dp'(p',
eos)
=
ZsDr c
(p',
eis),
ez
=Zaeo
&
(2.
3)
gl
=
ZsgO
7
(2.
4)
where
Z3
is a
power
series in
ez2
with
divergent
coefFi-
cients. The
bare and
renormalized
charges
of
a test
body
satisfy
a relation
similar
to
(2.3):
or
smaller,
for
large
momenta,
than the
corresponding
free-particle
propagation
functions.
The
propagation
functions for
the
electron
behave
quite
similarly
to
the
photon
functions
we
have been
discussing. Analogous
to
Dp'
is the
divergent
electron
propagation
function
SF'(p,
es').
It
is
obtained
by
sum-
ming
all
Feynma,
n
diagrams
beginning
and
ending
in a
single
electron
line,
renormalizing the
mass
of
the
electron,
but not its
charge,
to all
orders.
Corresponding
to
Drc
there is the
finite
function
Src(p,
eis),
related
to
Sr'
by
an
equation
similar
to
(2.
2):
so
that
Zs
'
is
just
the
bracketed
quantity
in
(1.3).
The
function
Dpq
can be
represented
in
the
form
Sr'(P,
ep')
=
Z2S
pc(P,
eis).
(2.
7)
1
Drc(p',
ei')
=-
16
t'M2
)
de
+
fl
e
(
ms
j
M2
Ps+
M2
where
f
is real
and
positive;
the
quantity
Zs
may
be
expressed
in
terms
of
f
through
the
relation
The
quantity
Z2,
like
Z3,
appears
as
a
power
series
in
e~'
with divergent
coefficients.
It
does
not,
however,
contribute to
charge
renormalization.
A
parametric
representation of
S~g,
resembling
Eq.
(2.
5)
for
Drc,
is derived
in
Appendix
A and
re-
produced
here:
1 r
g(M/m,
eis)
&II
S~c(p,
ei')=
+
l
ipP+m
ie
—
"
ipP+M
ie
M—
(M'
p
dM'
Z;=1+
~
j"j,
,
s
)
(m'
j
M'
(2.
6)
l"
h(M/m,
eis)
dM
(2 8)
imp
M+ie
M—
These
equations
have been
presented
and
derived,
in
a
slightly
diGerent form
by
Kallen.
"
Their derivation
is
completely analogous
to
the derivation
given
in
Ap-
pendix
A
of
Eqs. (2.
8)
and
(2.
9)
for
the
propagation
function of
the
electron,
which is
discussed below.
We
see
from
(2.
5)
and
(2.
6)
that a virtual
photon
propagates
like
a particle
with a
probability
distribu-
tion of
virtual masses. In
Dp~,
the distribution
is
not normalized, but in the unrenormalized
propagation
function
DI; =Z3DI:~
the
probabilities
are normalized
to
1.
The normalization
integral
is
just
the
formally
divergent
quantity Z3
.
In
Dpz,
it is the coefficient
of
1/(p'
—
ie)
that
is
1,
corresponding
to the fact
that the
potential
V(r)
in
(2.
1)
at
large
distances
is
simply
qg'/4irr.
It has been
remarked""
that
Z3
'
must be
greater
than
unity,
a result
that
follows
immediately
from
Eq.
(2.6).
To this
property
of
the renormalization
con-
stant
there
corresponds
a
simple
property
of the
finite function
Dro,
to
wit,
that as p'~~,
the quan-
tity
p'Drc
approaches
Zs
'.
If
Zs
'
is in
fact
infinite,
as
it
appears
to be when
expanded
in
a
power
series,
then
DI:
~
is more
singu1ar
than the free
photon
propaga-
tion
function
Dr=1/(p'
—
ie).
In
any
case,
Dro
can
never
be less
singular
than
Dp,
nor even smaller
asymp-
totically.
This is
a
general
property
of
existing
field
theories;
it is of
particular
interest in connection with
the
hope
often
expressed
that in meson
theory
the
exact modified
propagation
functions are less
singular,
'0
G.
Ksllen,
Helv.
Phys.
Acta
25,
417
(1952).
"J.
Schwinger
(private
communication from
R. Glauber).
Both
g
and
h
are
real;
in
meson
theory
they
are
posi-
tive,
but
in
quantum
electrodynamics
they
may
assume
negative
values.
Z2
can
be
expressed
in terms
of
g
and
h
through
the relation
fM
)dM
t"
pM
)dM
Zs
'=1+,
'
gl
—
&i'
I
+
t
h]
—,
ei'
I
(2.
9)
Em'
')
M
Again
we
have
a sort of
probability
distribution of
virtual
masses
with a
formally
divergent
normalization
integral.
As before,
the modified
propagation
function
is more
singular,
or
at least
asymptotically
greater,
than
the
free-particle
propagation
function,
since
Z~
'~&
1,
except
possibly
in
quantum
electrodynamics.
Equations
(2.
8)
and
(2.
9),
like
(2.
5)
and
(2.
6),
are
similar
to ones
derived
by
Kallen.
"
However,
our
nota-
tion
and
approach
are
perhaps
suffi.
ciently
diGerent
from
his
to warrant separate
treatment.
It should be
noted
that
Kallen's
paper
contains
a
further
equation,
(70),
which,
in
our notation
expresses
the mechanical
mass
mo
of
the electron in terms of
g
and
h:
dM
t.
"
dM
m.
=
my
Mg
—
+
(
M)a-
3f
3f .
dM
-
dM
&&
1+
g
+
7i .
(2.
10)
%e
see that
mo
is
simply
the
mean
virtual
mass
of
the
electron.
QUANTUM
ELECTRODYNAMICS AT
SMALL DISTANCES i303
It
may
be
remarked that
a
quantity
analogous
to
iso
can be constructed for the
photon
6eld,
that
is,
the
mean
squared
virtual mass of the
photon:"
dM'
p"
dM'
fM'
1+
f
.
(2.
11)
M'
p
M'
While
gauge
invariance forbids the
occurrence of a
mechanical
mass of
the
photon
in the
theory,
it is
well
known that a
quadratically
divergent
quantity
that
looks
like
the
square
of a
mechanical mass
frequently
turns
up
in calculations
and
must
be
discarded. That
quantity
is
just
pp,
as
given
by
(2.
11).
An
equation
similar
to
(2.
11)
holds
in pseudoscalar
meson
theory,
where
p,
o'
is
really
the
square
of a
mechanical mass:
d3P
o"
dM'
pp'=
p'+
'
fM'
1+
i
f
.
(2.
12)
~
g„s
M'
g„s
M'
Here
p'
is the observed meson
mass.
Evidently
(2.
12)
implies
that
p,
o'&p'.
3.
EXAMPLE:
QUANTUM
ELECTRODYNAMICS
VfITHOUT PHOTON
SELF-ENERGY PARTS
Before
examining
the
asymptotic
forms of the singu-
lar
functions in the
full
theory
of
quantum
electro-
dynamics
let us
consider
a simpli6ed
but
still
re-
normalizable form of the
theory
in which all
photon
self-energy
parts
are omitted. A
photon
self-energy
part
is
a
portion
of a
Feynman diagram
which is connected
to
the
remainder
of the
diagram
by
two and
only
two
photon
lines.
By
omitting
such
parts,
we
electively
set
function
D&z
de6ned
by
Dpi,
(p')
=
Dp
(p'))P/('A'+
p'
i
—
g)
In
a
given
calculation,
if
X
is
large
enough,
quantities
that would be finite in
the absence of
a
cutoff remain
unchanged
while
logarithmically
divergent quantities
become
finite
logarithmic
functions of
X'.
Thus,
if we
calculate
Sg'(p) using
a
Feynman
cutoff
with X'»lp'l
and
X'))m'
and
drop
terms
that
ap-
proach
zero as
X'
approaches infinity,
we must
And
a
relation
similar
to
(2.7):
&I
~(p)=en&»c(p),
(3.
3)
where
the
finite function
Sgg
has remained
unchanged
by
the
cut-off
process,
while
the
infinite constant
Z&
has been converted to
the finite
quantity
2'»,
which
is
a
function
of
X'/re'
(The
. reader who
is
not
impressed
with the
rigor
of these
arguments
should
refer
to the
next
section,
where a
more
satisfactory
cut-oG
proce-
dure is introduced.
)
Calculation
to the erst
few orders in
the
coupling
constant indicates
that
s2q
has
the
form
X'y
s,
~=1+e,
'I
a,
+b,
ln-
mg)
(
l'&'
+ei'
ag+fg»
—
+egl
»
—,
I
+"
(34)
ngg
&
m')
The
propagation
function
Spz may
also
be
calculated
to
fourth order in
ei,
for
l
p'l))te'
it has the
form
DF
c(P')
=D~(p')
=
1/(P'
gp)—
(3.
1)
I
p'1))~':
&gc(p)
1
Moreover,
there
is
no
charge
renormalization
left
in
the
theory,
so
that
and
(1+
ei'[a,
'+
f,
'
in(p'/m')
)
ivp
+
e,
'[ag'+
b,
'
ln
(p'/m')
02=
ei2
(3.
2)
+eg'(1
(P'/~'))'j+
)
(3
~)
Although
some finite effects of vacuum
polarization
(such
as its
contribution
to
the second-order
Lamb
shift)
have
been left
out,
others
(such
as the
scattering
of
light
by
light)
are
still
included.
If
the mass of the electron is now renormalized
the
only
divergence remaining
in the
theory
is
Z2.
It
has
been shown
by
Ward'
that in the calculation of
any
observable
quantity,
such as a cross
section,
Z2
cancels
out.
We
shall
nevertheless be concerned with
Z2
since
it
does
appear
in
a
calculation
of
the electron
propaga-
tion function
Sp'.
In order
to
deal with
Z2,
a
divergent
quantity,
we
shall make
use
of the relativistic high-momentum
cut-
off
procedure
introduced
by
Feynman,
which consists
of
replacing
the
photon
function
DI:
by
a modified
'2
We are indebted to
Dr.
Kallen for
a
discussion of this
point.
In order
to obtain some
understanding
of the
properties
of
Eq.
(3.
3),
let
us
substitute these
approximate
ex-
pressions
into that
equation.
Let
us
then
examine
what
happens
in the limit
m~0.
We
see that in
neither of
the
expressions
(3.
4)
and
(3.
5)
can nz
be
set
equal
to
zero
with
impunity;
that is
to
say,
both factors
on the
right-hand side of
(3.
3)
contain
logarithmic divergences
as m
—
&0.
Thus we should
naively expect
that
the
left-
hand side
have no
limit
as
m
—
+0.
Rather,
we should
expect
to
find
logarithmic divergences
to each order
in
e~~,
unless fantastic
cancellations,
involving
the
con-
stants
u~,
a~',
etc.
,
should
happen
to occur.
But such cancellations must indeed
occur,
since
a
direct
calculation of
5»'
with
m set
equal
to
zero
ex-
hibits no
divergences
whatever.
Instead,
each
Feynman
diagram
yields
a
term
equal
to
1/imp
times a finite
function
of
X'/p'.
lt
is clear
that
this
must
be
so,
since
X
1304
M.
GELL
—
MANN
AND
F.
E.
LOW
provides
an
ultraviolet cutoff
for
every integral,
while
p
provides
an
infrared
cutoG.
Let us
now
make
use of
the remarkable cancellations
that
we have
discussed,
but
in
such a
way
that
we
do
not
rely
on
the
specific
forms of
(3.
4)
and
(3.
5),
for which
we have so
far claimed
no
validity
beyond
fourth order
in
ei.
We
shall
consider the
asymptotic
region
X'))
I
p'I
))m2.
We
may
write
S-=
(I/svp)"
(p'im')
s,
),
—
—
ss
(X'/m')
.
(3.
6)
(3.
7)
Equation
(3.
3)
then
implies
the
following
functional
equation:
(3.
9)
s()
'/p')
=
ss(),
'/m')
'sc(p'/m')
The functional
equation
has
the
general
solution"
ss(X'/m')
=A(X'/m')
"=A
expL
—
e
ln(V/m')]
(3.10)
s,
(p'/m')
=B(p'/m')"=B
expLn
ln(p'/m')],
(3.11)
s
p.
'/p')
=
AB ()is/p')
—"
=AB
expL
—
e
1nots/P')].
(3.12)
Here
A, 8,
and e are functions of
e»2
alone.
If
all three
constants are
expanded
in
power
series in
e»2,
then
formulas
like
(3.
4)
and
(3.
5)
can
be
seen
to be valid
to
all orders in
e»2.
The constants
are
given,
to second
order in
e»,
by
the
equations
Moreover,
in
the
asymptotic
region,
we
may
drop
nz
entirely
in
S&
z,
since a limit exists as
ms/)is
and
m'/P'
approach
zero,
with
'As&)IP'I.
Thus
we
have
S
'=
(1/'~p)
Oi'/p')
(3.
8)
we
may
derive an
asymptotic
form for
the vertex oper-
ator'
I'„e(p,
p') for
equal
arguments.
We
use
Ward's'
relation
and obtain
1 8
I'„c(p,
p)
=
—
I
Spy(p)]
—
',
1C)
p
(3.
17)
I'.
c(P,
P)
=B
'(P'/m')
"(~.
2~P—
.
7P/P') (3
18)
A result
similar
to
this was found
by
Edwards,
'4
who
summed a small subset
of
the
diagrams
we consider
here.
We
may
note that
corresponding
to
the
increase
in
the
singularity
of
Sp
there
is a
decrease in
that of
F„.
The two are
obviously
tied
together
by
(3.
17).
It
is
therefore
highly
inadvisable to take
seriously
any
cal-
culation
using
a
modified
I'„and
unmodified
Sp',
or
vice versa.
It is
unfortunate
that
the inclusion of
photon
self-
energy
parts
(omitted
in this
section)
invalidates the
simple
results we have obtained
here. In
Sec.
V
we
shall derive
and
solve
the functional
equations
that
replace
(3.
9)
in
the
general
case,
but
the
solutions
give
much less detailed
information than
(3.
16).
In order to
treat
the
general
case,
we must
first
develop
(in
Sec.
IV)
a more
powerful
cut-off
technique
than
the
one we
have
used so
far.
4
WARD
S
METHODs
USED AS A CUTOFF
The
starting
point
of
Ward's
method of
renormaliza-
tion is a
set of four
integral
equations
derived
by
sum-
ming
Feynman
diagrams.
The
equations
involve
four
functions:
Sp'(p),
D~'(k),
the vertex
operator
I'„(pr,
ps),
and
a
function
W„(k)
defined
by
(4.
1)
8»
P
de
I:S~'(p)]
'=s ~
&~(p„p„')X—
(5m'M'
—
m4)+ .
(3.15)
16''
~
s
M4(M'
—
m')
2
(3.
13)
W„=
(el/r)k„)I
Dp'(k)]
'.
16~2
e'
I"
dM'
Two of the
equations
are
trivial, following
from
(4.
1)
A
=
1+
—
s+
(Sm'M'
—
m'),
(3.
14)
and
(3.
17),
respectively:
167r'
"
M4(M'
—
m')
B
(p'i
S~o(p)
=.
'&p
t.
ms)
"J.
C.
Maxwell,
Phil.
Mag.
(Series
4)
19,
19
(1860).
(3.16)
It
is now
apparent
that we
have
glossed
over
a
difhculty,
although
it turns
out
to be a minor one.
While AB is
perfectly
finite,
A and 8
separately
contain
infrared
divergences
that
must be cut
oG
by
the
intro-
duction
of a small fictitious
photon
mass
p.
These
di-
vergences
are
well-known
and
arise
from
the require-
ment
that
(i&p+m)S&&(p)
approach
unity
as
ipp+m
tends
to
zero,
while
the
point iyp+m=0
is in
fact
a
singularity
of
the
function
(iyp+m)Ss'(p).
From
the
asymptotic
form for
Sp~,
and
I'„(px+
p'(1
—
x), px+
p'(1
—
x)
)
(4.
2)
p»
I
D,
'(x)]-
=
dye„w„(xy),
(4 3)
where
p'
is
a
free electron
momentum, i.
e.
,
after
inte-
gration
p"
is
to be
replaced
by
—
m'
and
imp'
by
—
m,
where
m
is the experimental
electron mass.
No
further
mass
renormalization is
necessary.
"S.
F.
Edwards,
Phys.
Rev.
90,
284
(1953).
"The
purpose
of this section is to
justify
the
use of
a
cutoG
when
photon
self-energy
parts
are included.
The
reader who
is
willing
to take this
point
for
granted
need
devote
only
the briefest
attention to the material between
Eqs.
(4.
3)
and
(4.6).
The
re-
mainder of the section contains some
simple
but
important
alge-
braic
manipulation
of the
cut-off
propagation
functions.