can now "draw" the completed graph on a sphere with h handles so that lines intersect ..... By Furry theorem, this is also true if the Lagrangian contains fermion ...
I
SLAC-PUB-2376 August 1979 (T)
GROUP WEIGHT P. Butera;
G. M. Cicuta+
Stanford Stanford
AND VANISHING
Linear
University,
GRAPHS
*
and M. Enriotti'
Accelerator Stanford,
Center
California
94305
ABSTRACT
Various
properties
Feynman graphs
vanishing Lie
in non-Abelian
Infinitely
discussed.
weight
group.
expansion
is
channels
with
of the group weight
of
gauge theories
many skeleton
are exhibited
for
graphs every
are with
compact
The l/N2
dependence
of the topological
related
to an l/NZ
expansion
the exchange
of definite
in some
quantum
numbers.
(Submitted *
t
to Phys.
This work was supported by the Department number DE-AC03-76SF00515.
On leave from Istituto Nazionale Sezione di Milano, ITALY. § On leave from Istituto di Fisica,
di Fisica Universita
Rev. D)
of Energy
under
contract
Nucleare, di Milano,
ITALY.
-21. In the perturbative
analysis
represent
every
depending
on the gauge group,
graphic
Introduction
Feynman graph G as the product times
method to compute WG for
described.
'
the weight
is
convenient
of a weight
Lie
WG
An efficient
groups
method to discuss
factor
to
has been
further
properties
of
factor.2 II
we show that
theory
are polynomials
relate
this
amplitude
property
III
many skeleton
graphs
theory.
feature
be of help
perturbative
order
In Section to exchanged scattering
that
of the scattering
sight
at first
typical
weight
surprisingly,
in every
of non-Abelian
of perturbation
infinitely
non-Abelian
gauge
gauge theories
and may
theory
although
a rough
the number of non-vanishing
graphs
grows with
IV we derive with
expansion
N) and we
that,
much faster
states
from a factor
quantum numbers are exchanged.
in the analysis suggests
of a SU(N) gauge
where definite
have vanishing is
factors
possibly
to the topological
we observe
This
the weight
inN2(apart
in the channels
In Section
than
the number of the vanishing
the projection
definite
operators
the
ones.
corresponding
quantum numbers in gluon-gluon
elastic
in SU(N) gauge theory. Planarity
II. In this an arbitrary
section graph
At order vertices
it
a momentum integral.
most simple
We make use of that
In Section
estimate
of gauge theories
(in
of three-gluon
and Topological
we discuss
Expansion
the dependence
on N of the weight
WG of
in the SU(N) gauge theory.3-8
v in the coupling the usual vertices)l
constant
way the four-gluon some of which
g, the graph has v trilinear vertex being
is replaced
three-gluon
by couples
vertices
v
Et'
I
-3the others
being
quark-quark-gluon
v
p= pg+pq
propagators
(we do not
count
pg gluon
and pq quark
propagators.
If
n= 3v- 2p, the group described
by Cvitanovic
WG as a linear having
factor
without
internal
vertices
only
involves
=
ijk
(see Fig.
that
of rank n.
which
is lines,
The graphical the generic
set of independent
tensors,
method tensor
tensors
are associated
the evaluation
the two steps:
to graphs
2 Tr
( TiTjTk
internal
representation
- TkTjTi gluon
of the group
(a) to re-express
of the fundamental
all
(b) to replace
lines),
g1uons.l
in terms if
The Feynman graph has
way to express
of a complete
In the SU(N) gauge theory any graph
.
the graph has n external
is an efficient
the basis
4
the external
WG is a tensor
combination
the same rank,
v=vg+v
4'
factor
WG for
the three-gluon
(see Fig.
1):
)
lines
(2
with
gluon
projection
l
1)
operators
2):
(2.2)
WG is
as the sum of 2vpg
then expressed
As an example,
3(a)
Fig.
theory,
with
six
In Fig.
3(b)
there
shows a graph at order
three-gluon
vertices
is one of the 2
steps
(a) and (b).
In the "double
. i.e.,
fermion
unconnected
lines, are also lines.
loops
each contributing index
paths
"double
a factor called
and nine
15
"double
line"
to the rest
boundaries
There are no boundaries
line"
for
graphs
gluon
graphs
are called
index
after loops,
and to external
windows.
are attached
propagators.
obtained
may appear
of the graph
which
graphs.
10 in perturbation
internal
graphs
N, which
g
line"
There
to the external
where the external
sources
are
-4all
color
singlets.
Each boundary
the number of external the "double boundary, Basic It
is very
called
line"
notions
external
only
is possible
of the graph.
Fig.
has h, =l,
graph
embedded on a sphere
in terms
graph on a sphere
is
with
(embedding).
it
one more vertex,
(see Fig. h handles
iff
hm= 0. g
hm handles
may be regarded Then the Euler
P+F=
2-2h graph,
(p+n)+
One
4). so that
of the degree
is planar
of the graph.
of the original
of non-planarity.
The minimum hm for
V, of edges P and faces
(v+l)-
which
of non-
The graph in The completed as a polyhedron
formula
relating
F holds: (2.3)
m
is:
(f+n-l>=
2-2hm
(2.4)
where f Z F-n+l. The multitude
of "double
graph has a different
line"
number of faces
graphs
originating
and handles
but
from a Feynman the same number
of f+2h Indeed fM = 2-v+ (if
fM is
=
p-v+2
the number of faces
= ranges
t-f+2
(2.5)
between
p- 2hm and the minimum f m = 0 (if odd).
to
For example,
may be embedded on a torus.
with
Vwhich
and degree
of G are incident
The graph
that
the number of vertices
rank equal
has one window and one
is a characterization
whose edges are the lines
path.
the graph G by adding
lines
at vertices
planarity 3(a)
of planarity
the completed
intersect
3(b)
with
3(c).
to "complete"
PW,- where all
a tensor
to the index
in Fig.
are those
convenient
the embedding
attached
configuration
as shown in Fig.
can now "draw" lines
lines
represents
the maximum
fM is even)
or fm = 1
-5The quark of faces
loops
in the original
of the "double
line"
graph do not
graph.
to the number
Then =
f-q
where q, b, w represent
contribute
b+w
the number of quark
(2.6)
loops,
boundaries
and
windows. The group
theoretic
weight
V WG
=
g
WG can then be expressed:
NW c
c
where T(m) is one of the basis the "double
line"
graphs
of the configuration arising
from steps By use of
'G
=
tensor,
w windows
the summation
extends
cm counts
and the factors
over
all
the multiplicity
of 2 and (-l/N)
(a) and (b). and (2.6)
f-q-b 'rn N.,
c
Tcm)
and the coefficient
with
(2.5)
V
g
m
the power NW can be‘rewritten:
,(m)
(2.7) The coefficient line" step
cw may contain
configurations
which
a dependence
arise
on N only
from the singlet
for
those
subtraction
"double term in
(b): 2 (Ti);
For the first
(Ti);--
+
such replacement,
the "double
v-2 vertices
and p-3 propagators,
integer
different
with
parity
6; 6;
then its
line"
configuration
invariant
from the original
has
f+ 2h is a positive
graph or any configuration
-6-
where instead
the replacement
(2.8)
has been made. Therefore
(l/N)
where the singlet
times
the value
subtraction
term has been used,
the power -of N as the graph with both
can be written
on N.
in the form
The same argument
terms
and Eq.
independent It
is
(2.7)
then
clear
(b= 0) then Eq.
decreasing take
for
for
with
(2.8)
everywhere
coefficients
multiple
the general
configuration
has the same parity
the replacement (2.7)
line
and
cm now independent
use of the singlet graph,
of
with
subtraction
coefficient
cm
on N.
number of boundaries, only
holds
holds
of the double
that for
contribution
one is
instance
(2.7)‘
powers of N2.
the limit
if
interested
processes
arranges
in processes with
the contribution
Or one may sum over
N-tm with
g2N=y2
to amplitudes
with
fixed fixed
color
with
singlet
fixed sources
of each graph
the perturbation
and one would
obtain
number of boundaries
in
series that
and
the
and quark
loops
n
are arranged degree
of non-planarity.
A simple (Eq. i)
(2.2))
powers of NL and are associated
in decreasing
may be substituted
one three-gluon
connects vertex
increasing
4-0
remark may shorten
the gluon
with
the computation by the simpler
two three-gluon with
vertices
one quark-quark-gluon
of WC. replacement or ii)
Step (2.8)
the gluon
vertex.
(b) when connects
I -7-
Proofs (i)2f
are straightforward:
jikfklm
-4Tr(TjTiTk-
= =
-2TrlT.T.T J
For the point
(ii)
TkTiTj)
ilm
Tr(TkTITm-
T - TjTiTmTl-
TmTITk)
TiTjTITm+TiTjTmTl)
(2.9)
we have:
* i(
fkij
2i
(Tk]i
=
1*
CT,,T~$
in all
Therefore, connect
=
quark
a simple
(2.10)
(without
vg to 2 .
quarks).
functions,
ly 6ab and fabc) ,
where there
the number of "double
lines,
and three-point
generic
- TjTiTk)
Feynman graphs
graph is reduced
gauge theory
Tr(TkTiTj
WG =
line"
Of course
where there
graphs
this
Then one finds is just
which
that
for
directly
originating
happens
from a
in a pure SLJ(N)
two-point
one basic
WG of the generic
the group weight
Feynman graph
are no gluons
tensor
functions Crespective-
Feynman graph is
is
6ab (Ng2)'
'2'
cp (N2),-'
at order
g
2s
(2.11)
P=O
[s/21 fabc
WG =
g(Ng2)'
c
cp(N2)-Pat
order
g2'+l
(2.12)
P=O
where the leading the graph As it basic
coefficient
is planar. is
tensors,
from zero
if
and only
if
lo
shown in Section three
c o is different
of which
IV,
for
(A,B,C)
the 4-point
function
have one boundary
one has six
and three
(D,E,F)
-8-
have two boundaries.
At order
g
2&2
[s/21
Cd21 wG = g2(g2N)'
Ac
a,(N')-'+D
Es/21
c
bp(N2)-P+G
dp(N2)-'+E
'r
0
ep(N2)-'+F
'2
n-point
functions
one has taken boundaries
care of the N factors
operators
do not mix basis the proper
elastic
amplitude
scattering from an overall
tion
theory,
while
with
the last
is
finitely
gluon
many skeleton
Lie
group.
theory. a subgraph
It
loses
is convenient
vanishing
of this
will
or
on the order
in N2,
in perturbaassociated
on N2.
Graphs gauge theories
vanishing
it
be a polynomial
dependence
weight.
the anitsymmetry
rules1
pro-
the gluon-gluon
in the channels
the simpler
to restrict
weight
will
four
boundaries
Therefore
N.
in non-Abelian
using
the first
different
channels
Vanishing
the results
From the graphical with
factor
with
amplitude
graphs with
way by only so that
the number 'of
IV,
independent
the scattering
easy to show that
vertex
pure
normalization
two projectors
in an obvious
tensors
in those
III. It
in the same form after
with
in Section
(4.3)-(4.8)
they mix them with
apart
associated
(2.13)
tensors.
As one can see from jection
WG expressed
have weights
of the basic
II
fp(N2)-' 0
0 higher
cp (N2)-'
c
0
0
t
one finds
section first
is obvious also
there
are in-
They are identified
property
of the three-
hold
any compact
for
to a pure gauge non-Abelian that
a graph containing
have vanishing
weight.
-9-
Furthermore,
since
there
single
one of rank
three,
weight.
In fact
trary
large
class Let
It
graph
us consider
may be obtained
produce
non-Abelian
graphs
While
for
with
vanishing
graphs it
if
arbi-
may be difficult
the vanishing
theory,
two and a
of rank
vanishing
are made.
conditions
in a general
of vanishing
will
insertions
and sufficient
tensor
to skeleton
we may restrict
or vertex
necessary
of a skeleton
independent
each such skeleton
self-energy
to give
is a single
of the weight
our remarks
select
a
of a graph G(see Fig.
5).
graphs.
the weight by partial
TG
al . ..Uk Tl...Tm
saturation
of the weights
of the subgraphs
Gl and G2
TG
5 . ..Uk
A simple
=T
Tl.e.Tm
sufficient
three-gluon
diagram
four-gluon nature
tensor
associated
with
of the external examples
gluons
with
any three-gluon
(see Figs.
6b-9b).
every through
lying
the bare
planar,
indices.
graphs
Green function
vertex I'
f
times
lines,
of indices
plane.
for
any a
four-leg is (say a,B)
The lowest
are shown in Fig.
three-gluon
and
Because of the
or non-planar,
the couple
G1
vertex
two of the external in
T
is obtained
of a three-gluon
in the symmetry
skeleton
that
two corresponding
weight
in the two saturated
WG symmetric not
in
a vanishing
the product
of symmetry
of such symmetric
with
and antisymmetric
vertex
a tensor
By convolution
graph
is
of the three-gluon a plane
lently
which
Tl...Tm
of TG is
the vanishing
In particular
symmetric
graph with
for
symmetric
a indices.
TG2 alme.an
O1o--Ok alaeean
condition
TG2 are respectively saturated
G1
-caB
) one obtains Tc-6
orderll
6a-9a.
(or equivaa vanishing
-lO-
This
procedure
symmetric
four-leg
suggests
in the symmetry
number of skeletons generic
be,
plane
is
neglects
with
vertices
on the symmetry
lead
to three-point
on the specific
the estimated
none of these
symmetry expects path
transforming
for
the
are as many as the R of the vanishing
the three-point
a) there b) not
function,
are symmetric
all
symmetric
(see for
having ratio
points
skeletons
four-point
example Fig.
a symmetric
7b or 9b),
there
tensor.13
R, (b) would
are nonWhile
decrease
can substantially
10a.
that
for
four-point
When convoluted
graphs, plane
a non-Abelian
(a) and It
it.
change the very
with
as in Fig.
lob.
must preserve
also
only
gauge theory
as the fundamental
one may look
vanishing
vertices
seems rough
estimate.
We can now consider
in Fig.
constant,
where x(m) N m! is
gauge group of the theory,
increase
Again
the two external
the ratio
skeletons
(c) would
fermions
that:
plane,
skeletons
previous
n in the coupling
of estimate
ones,
the facts
four-point
that
level
Therefore
symmetric
however
x[F),
of the number of such
n
estimate
c) depending
roughly
m.
This
skeletons
only
the non-vanishing
at order
order
with
in this
at order
skeletons-versus would
skeletons
(which
graphs)12
rough estimate
At large
skeletons.
the number of symmetric lying
a very
special
in the vanishing
gluon
representation graphs
graphs
with
the three-gluon Since
the direction
will
still
a multiplet
vertex,
give
plane,
they around
in the fermion
a gluon
of
of the group. a symmetry
the reflection
ways of replacing
gluon
with
as
originate the path,
one
path with
a fermion
a vanishing
graph.
-llOne may note around
the, symmetry
one may still of weights it
plane
produce depending
in Fig.
no symmetry llb
lla plane
does not preserve
a vanishing
remark
and vanishing
weight
instance, related
to that
We finally properties spectral
II)
(as it
mention
that
graph in
of graphs, trilinear
external
“vacuum"
inequivalent
fabc
vertex
way15 It
matrix.
graph
with
or
(ha]:.
in the definition
three-point
of graphs
lines
graph by
of the type
was mentioned
in an algebraic
vanishing
three
the vanishing
the study
of the adjacency
cages with
weight
of the
the graph has
the three-point
a new vanishing
by one more trigluon
the five
in the SU(N) case,
although
a vanishing
from another
llb.
class
properties
the weight
(a,B),
one more coupling
projection
in Fig.
order
in
is
graph with
way one shows that
properties
of a peculiar
vanishing
any given
with
can be pursued
the two lowest completed
for
one may obtain
in this
which
ayB6
Therefore
one can obtain
in Section
graph,
of additional
For instance
tensor
(~,a).
that
Next by stereographic
"vacuum"
T
of the loop,
weight.14
the former
of planarity
the tensor
through
the direction
graph because
on the gauge group.
has vanishing
"completing"
even in some cases where the reflection
is a symmetric
We also
all
that
is easy to check that
graph
Fig.
however
of the graph.
in Fig.
definite
through
symmetry of the
to notice
already
vertex,
are just
the first
representatives
calles
cages.16
We checked
vertices,
exhibited
due to the mechanism previously
in Ref.
that
graphs,
sometimes
exhibited
12 is
the study
is amusing
For
in Ref.
described.
1, when
that
16, have
-12Basis
IV. In order
to discuss
l/N2
expansion
here
the SU(N) tensor
external
are associated all
for
processes
r,
of products
contains
of Ti matrices,
fermion
basis
fields.18
tensors,
A+ B
r= 4 external of the basis
gluons tensors
linearly
by the rules
(2.1)
which
Therefore,
for
basis,
but
independent.17 complete
antisymmetrizing)
are graphically is also
which
and (2.2)
(or
by symmetrizing
(no
The set of
tensor
are not
a
we write
reasons,
form a natural
this
fermion
true
if
loops
of
the Lagrangian
r= 4 .one has six
instead
of
i.e.:
[Tr(TaTdTcTb)
+ [T~(T,T~T~T~]
=
is actually
quantum numbers.
theorem,
By Furry
length.
with
so obtained
bases are obtained
expansion
completeness
of definite
r Ti matrices
r 2 N
l/N
combinations
in a pure SU(N) gauge theory
even (odd)
nine
over
of rank
and independent traces
and for
and the linear
traces
the tensors Actually
basis
II)
to the exchange
distinct
and Projectors
the cases where the
(see Section
quarks)
Tensors
f Tr(TaTbTcTd) 1
+ T~(T,T~T~T~)I
(4.1) C
=
i
CTr(TaTdTbTc)
+ Tr(TaTcTbTd)l
D = bab‘&j’ E = 6ac6bd’ F = 6ad%c They are shown in Fig. One may define "vertical" fermion
channel loop
is
13.
a product (that
of basis
tensors
as a convolution
is KL= H means Hacbd=KactuL
symmetrized,
this
product
is
tubd 1.
commutative,
in the Since
D acts
the
as the
-13identity
and one easily
A2 =
&
=
AB
finds: - &
(D+F)
AC
=
AF
=
A,
BE
=
CE =
(B+C)
$(B+C)+
BF
=
-I- 1 E 16N2
LE 16N2
C,
CF
$ (N-i)
=
B
E (4.2)
AE
FE
=GE,
B2 = c2 =
BC =
E2
=
The linear orthogonal
;
- + (B+C)
-I-
+
D
(3
+ i)
I31
($++)El
(N2 - 1) E combinations
of the basis
operators
and that
tensors
definite
with
the dimension
of the irreducible
that
represent
quantum numbers in the "vertical"
with
property
=
E,
[NC - $ (B+C)
projection
of the product
F2
=
the exchange channel
representations
(N2- 1) 8 (N2- 1) and by the symmetry
in the exchange
of the indices E
'N2-l , A =
4 (B -C) N
are mutually of a state
are here labelled in the decomposLtion
(or antisymmetry)
a and c (or b and d).lg (pomeron channel)
(antisymm.
adjoint
(4.3)
channel)
(4.4)
-14-
= w-%-N2-4 CB + c
p&1,3
(symm. adjoint
+ '(N2-4)(N2-1)+ 4
(N2-4)(N2-1) 4
'N2(N-l)(N+3) -4
, s
,A
= 2A-&
(B-C)
channel)
++
(4.5)
(D-F)
(4 -6)
=
(B+C)
i-i
(D+F)
1 + 2(N+ l)(N+
2) E (4.7)
'N2(N-3)(N+l) 4
, s
=-2A-&(B+C)+$
1 l)(N-
(D+F)+2(N-
2)
E (4.8)
for
the last only
projection
operator
vanishes
8(A+B+C)
=
is not because
present
and indeed ( valid
of the relation20
in SU(3))
In N- 2, more relations with
N2(N-3)(N+l) 4
N= 3 the representation
only
three
channels
exist
D+E+F
(see for
associated
instance
Ref.
1) and one is left
P1 , s, P3 , A, P5 , s.
with
Acknowledgment We thank was supported
the SLAC Theory by the Department
DE-AC03-76SF00515.
Group for
their
of Energy
hospitality.
under
contract
This number
work
1.
P. Cvitanovic,
2.
Part
Phys.
of the matter
Rev. G, in Section
one of us, G. Cicuta, 3.
The subject
which
would
II
section
usually
starting
correspond
in Cvitanovic
to the color
singlet
it
Furthermore,
up to equations Section
external
with
and/or
of the type
of
line"
Feynman graph,
Therefore, it
the study althqugh
this
may be useful,
previous
(2.13)
to ,various
by restricting
sources.
seem that
by
in the
and our approach
known literature,
does not
described
been discussed
from a '!double
to the same weight,
overlaps
was briefly
(1979).
has already
contribution
section
and III
SLAC-PUB-2328
of this
literature4-a
1536 (3976).
analyses
were carried
or the conclusion
of
II.
4.
G. 't
Hooft,
5.
G. Veneziano,
Nucl.
Phys.
Phys.
z,
Lett.
461 (1974).
g,
220 (1974)
and Nucl.
Phys,
fi,
365 (1974). 6.
M. Ciafaloni,
G. Marchesini,
G. Veneziano,
Nucl.
Phys.
B98, 472
and 493 (1975). 7.
G. P. Canning,
8.
Many papers
of the topological
few years.
Many of them are quoted
Physics
Reports
HUTP 79/A007 9.
This
Phys.
definition
Graph Theory the benefit
K,
Rev. D12, 2505 (1975). expansion
263 (1978)
have appeared
in the past
in G. Chew and C. Rosenzweig,
and E. Witten,
Harvard
preprint
(1979). of planarity
appears
and Feynman Integrals, of relating
the notion
in
54 of Nakanishi's
Gordon and Breach, of planarity
book, 1971.
It
has
of Feynman graphs
to
I -16-
planarity
in graph
Kuratowski
theory,
theorems.
This
use in the literature
10.
11.
it
is not written.
All
sums in Eq.
of
5.
We recall to express
that
Then the occurrence results
P. Cvitanovic,
13.
For instance, to see that symmetric
This
vanishing
was found
B. Lautrup,
the entire
part
it
is
convenient
couplings.
graphs
four-gluon
couplings with
trilinear
Figs.
Algebraic
E. Poggio,
The projection
Graph Theory,
all
Elsevier simple
operators
in P. Yeung,
also
13, it
is easy
is a tensor be symmetric
in the
order
in a study H. Quinn,
graph which
of the quark
Phys.
Rev. g,
form 2286
12b and 13.
American for
Fig.
of a sixth
weight
J. A. Bondy and U. S. R. Murty,
basis
will
is a subgraph
16.
III,
tensor,
1939 (1978).
the graph may not have such symmetry.
to have a vanishing
their
Rev. g,
graph whose weight
of indices,
even though graph
Phys.
at the basis
N. Biggs,
appear
usually
some four
among different
15.
Tensor
graphs- with
R. Pearson,
by looking
by J. Carazzone,
(1975),
17,
method1
the common
although
in terms of trigluon
couplings
in one couple
couple
Appendix
with
[I4,
up to
the graphic
of vanishing
every
other
factor
agrees
only.
12.
14.
within
the use of
expansion
extend
from the cancellation
couplings
also
topological
(2.13)
four-gluon
possible
definition
about
(2.11)-
again
then making
Phys.
Lie
Cambridge
University
Graph Theory Publishing groups
in SU(3) for Rev. E,
with
Press,
1974.
Applications,
Co.
are described gluon-gluon
2306 (1976)
in Ref.
1.
scattering with
some misprints.
I -17-
18.
We thank
Michael
Dine
19.
We thank
Franc0
Buccella
20.
A. J. MacFarlane, 77 (1968).
for
a discussion for
A. Sudbery,
on this
a discussion P. H. Weisz,
point,
on this Corn.
point. Math.
Phys.
11,
-18-
Figure
Captions
1.
Graphical
representation
of Eq. (2.1).
2.
Graphical
representation
of Eq.
3.
(a)
A graph at order
(b)
One of the 215,'1double weight
(c)
of the graph
The graph vertex
5.
lines 6(b),
7(b),
8(a), that
11.
A graph with
10(a)
indices (b)
vertex four
(y,6)
with
one more
of non-planarity. as a convolution
order
graphs
through
with
four
the external
vanishing
in Figs.
6(a),
external lines
graphs
7(a),
8(a)
(y,6).
obtained and 9(a)
Tab'
external plane
gluons
and one fermion
through
the lines
by convolution
the three-gluon
(y,6).
of the graph in
vertex.
does not have a symmetry
but whose weight
loop,
is
a tensor
plane
through
symmetric
the in the
(cc,@).
A vanishing Fig.
f
by adding
G1 and G2.
graph obtained
A graph which lines
now exhibiting
considered
The corresponding
has a symmetry
A vanishing
degree
is
plane
of the graphs
(a)
(a)
its
The lowest
9(b).
a three-gluon
Fig.
3(b),
completed
of the subgraphs
with
(b)
is here
have a symmetry
which
to the group
3(a).
as in Fig.
to exhibit
9(a).
8(b),
contributions
in Fig.
graph G whose weight
by convolution
10.
3(a)
of Fig.
Pm in order
A generic
7(a),
line"
theory.
and windows.
of the weights
6(a),
in perturbation
The same contribution boundaries
4.
gl'
(2.2).
11(a)
graph obtained with
by convolution
the three-gluon
vertex.
of the graph
in
-19-
12.
This
three-leg
the vanishing projecting" 13.
Basis
tensors
graph may be proved graph of Fig. from a three-gluon for
gluon-gluon
11(b)
to vanish and next
by first
by "stereographic
vertex. scattering
completing
in SU(N).
I
Fig. 1
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Fig. 10
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Fig. 12
cl
C
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a
C
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d
+
b
d
a
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b
d
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a
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