We
will
define
a
class
A
of
such
functions,
whose
asymptotic
behavior
for
high
p
may
be
specified
in
a
certain
manner,
by
means
of
cer
tain
"asymptotic
coefficients.
"
(The
integrands
of
covariant
perturbation
theory,
constructed
according
to
the
Feynman
rules,
with
the
I
real
variables
taken
as
all
Furthermore,
it
is
obvious
that
for
any
Li,
L2,
the
vector
Lizi+L2
will
not
be
orthogonal
to
V
or
V'
for
sufficiently
large
g~,
so
that
for
qj
large
enough,
n(Ligi+L,
)
=—
3.
(21)
What
is
not
obvious,
and
indeed
is
not
contained
in
(7),
is
that
we
can
find
numbers
bi(Li,
L2,
W),
b~(L,
,
L2,
W),
and
3E(Li,
L2,
W),
such
that
(12)
holds,
or
alternatively,
by
comparison
with
(19)
that
M(Liyi+L2,
W)
~&
M(Li)Lg,
W)gi
iL»,
b(Liin+L2,
W)
~&
b2(Li)L2,
W),
(22)
n(Ligi+L,
)
=
—
3.
for
all
qi&
bi(Li,
L2,
W).
The
statement
(7)
alone
would
of
course
allow
us
to
determine
the
convergence
of
Z(p');
however
we
need
(11)
for
alternatively
(22))
to
determine
its
asymptotic
behavior.
The
proof
that
the
asymptotic
behavior
(4)
of
Z(p')
can be
directly
obtained
from
(8)
and
(11)
alone,
with
out
knowing
any
other
properties
of
the
function
f(p',
p"),
will
be
reserved
until
the
next
section.
We
shall
show
there
that
(4)
follows
immediately
in
a
simple
application
of
the
general
asymptotic
theorem.
LSee
(III15).
]
HIGH
—
ENERGY
BEHAVIOR
IN
QUANTUM
FIELD
THEORY
components
of
all
internal
and
external
momenta,
are
shown
in
Sec.
U
to
belong
to
A„,
providing
that
all
energy
integration
contours
may
be
rotated
up
to
the
imaginary
axis.
)
The
exact
definition
of
the
classes
A„
and.
of
the
"asymptotic
coe%cients"
is
chosen
in
just
such
a
way
that
we
will
be
able
to
prove
the
asymptotic
theorem,
which
says
that
if
a
function
belongs
to
A
„,
any
sufficiently
convergent
integral
over
k
of
its
argu
ments
belongs
to
A
„~,
and
which
provides
a
rule
for
calculating
the
convergence
properties
and
asymptotic
coeKcients
of
the
integral
in
terms
of
the
asymptotic
coeKcients
of
the
integrand.
Definition
The
function
f(P)
is
said
to
belong
to
the
class
A
if
to
every
subspace
Sf
R„
there
corresponds
a
pair
of
coefTicients,
a
"power"
u(S)
and
a
"logarithmic
power"
p(S),
and
for
any
choice
of
222~&22
independent
vectors
Ll
L„and
finite
region
W
in
R
we
have
f(L~~"
~+L~
~+"
+~
L.
+C)
—
0{sf
a(L&)(lnrf1)S(L1)rf
aif
Ll,
L2f)(inrf2)S(f
Ll,
Lsl)
X
'
'
'g
0'(j
LI,
Lg
~
~
L2rb))
(ln
lP((
LI
L2
'
Lm))4
'''nm
(
if
sf
1
..
rf
tend
independently
to
infinity
with
C
confined
to
W.
[Here
n({L1
L,
})
and
p({L,
L„})
are
the
asymptotic
coeKcients
associated
with
the
subspace
{Ll
L„}
spanned
by
the
vectors
Ll
.
L,
.
]
More
pre
cisely,
there
exists
a
set
of
numbers
b&

b
)1
and
M)
0
(depending
on
L,
L
and
W
but
not
of
course
on
the
211
rf„),
such
that
coe%cients
of
f(P)
for
P
~
~
along
typical
directions
in
5.
In
the
special
example
given
in
Sec.
II,
we
saw
that
n(R2)=
—
3,
where
R2
was
the
whole
p',
p"
plane.
Furthermore,
this
was
also
the
value
of
n(L)
for
almost
all
vectors
L
in
R2,
the
only
exceptions
being
L
(1,
0)
(with
n=
—
2)
and
L
(1,
1)
(with
n=
—
1),
as
shown
in
Fig.
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 Fall '05
 Energy, Radiation, asymptotic behavior, J Li