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272 November 7, 2013
10.6.2 Counting one-loop diagrams
The SDe approach to counting diagrams has a number of interesting or useful
applications, one of which we discuss here. We can extend the treatment of the
previous section as follows. For the case of purely gluonic QCD the number of
one-loop diagrams including their symmetry factors can be counted by iterating
the appropriate Schwinger-Dyson equation :
Φ(J) = J + 1 2 Φ2 + 1 6 Φ3 + ¯h 2 (1 + Φ) Φ′ (10.77)
and taking care to discard terms of order ¯h 2 or higher.
gluonic 20-point function is given by
As an example, the
N(19) = N 0 (19) + ¯h N 1 (19) + O (¯h 2) ,
N 0 (19) = 11081983532721088487500 ,
N 1 (19) = 2900013601350201168582750 . (10.78)
The number N 0 (19) is the actual number of diagrams since tree diagrams always
have unit symmetry factor ; but the number N 1 (19) underestimates the actual
number of diagrams since the symmetry factors are not trivial. We can see,
however, that the only possible nonntrivial symmetry factor at the one-loop
level is 1/2, as evidenced by the factor ¯h/2 in Eq.(10.77). Inspection tells us
that in this theory the only elementary Feynman diagrams that have symmetry
factor 1/2 are
E 1 = , E 2 = , E 3 = ,
E 4 = , E 5 = .
All diagrams that contain one of these elementaries as a subgraph will have a
symmetry factor 1/2, and it will suffice to determine their number and multiply
it by two 19 . Alternatively, we may get rid of all such diagrams, and work with
the difference. This is the more useful approach ; and it illustrates how we may
go about using counterterms to impose constraints on the structure of Feynman
diagrams. The procedure is best explained by going through it step by step.
In the first place, it will become necessary to again distinguish betwee threeand
four-point vertices. We therefore modify Eq.(10.77) be reinserting labels
for these couplings:
Φ(J) = J + g 3
2 Φ2 + g 4
6 Φ3 + ¯h 2 (g 3 + g 4 Φ) Φ ′ (10.79)
19 This relies, of course, on the fact that there can be no diagrams containing two (or more)
of the elementaries, since that would be a two-loop diagram (or even higher).